Pseudo Symmetric Multifuntors: Coherence and Examples by Diego Fernando Manco Berrío A dissertation accepted and approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Dissertation Committee: Daniel Dugger, Chair Angélica Osorno, Core Member Nicolas Addington, Core Member Boris Botvinnik, Core Member Robert Lipshitz, Core Member Thanh Nguyen, Institutional Representative University of Oregon Spring 2024 © 2024 Diego Fernando Manco Berrío This work is openly licensed via a Creative Commons Attribution-Noncommercial-Noderivs 4.0 International License 2 DISSERTATION ABSTRACT Diego Fernando Manco Berrío Doctor of Philosophy in Mathematics Title: Pseudo Symmetric Multifunctors: Coherence and Examples Donald Yau introduced pseudo symmetric Cat-multifunctors and proved that Mandell’s inverse K-theory multifunctor is stably equivalent to a pseudo symmet- ric one. We prove a coherence result for pseudo symmetric Cat-multifunctors in the form of a 2-adjunction. As a consequence, we obtain that pseudo symmetric Cat-multifunctors preserve En-algebras parameterized by Σ-free Cat-operads at the cost of changing the parameterizing Cat-operad O by O × EΣ∗, where EΣ∗ is the categorical Barrat-Eccles operad. Since Mandell’s inverse K-theory is pseudo symmetric we derive that En-algebras parameterized by free En Cat-operads in the symmetric monoidal category of Γ-categories can be realized, up to stable equiv- alence, as the K-theory of some En-algebra in the multicategory of permutative categories. This result can be regarded as a multiplicative version of a theorem by Thomason that says that any connective spectrum can be realized as the K-theory of a suitable symmetric monoidal category up to stable equivalence. Our coherence theorem also allows for a simple description of a 2-category defined by Yau which has Cat-multicategories as 0-cells and pseudo symmetric Cat-multifunctors as 1- cells. We also provide new examples of pseudo symmetric Cat-multifunctors by proving that the free algebra functor of a symmetric, pseudo commutative, strong 2-monad, as defined by Hyland and Power, can be seen as a pseudo symmetric Cat-multifunctor. This result can be interpreted as a coherence result for sym- metric, pseudo commutative, strong 2-monads and it implies a coherence result for pseudo commutative, strong 2-monads conjectured by Hyland and Power. 3 CURRICULUM VITAE NAME OF AUTHOR: Diego Fernando Manco Berrío GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED: University of Oregon, Eugene, OR, USA Universidad Nacional de Colombia, Bogotá, Colombia Universidad de Antioquia, Medellín, Colombia DEGREES AWARDED: Doctor of Philosophy, Mathematics, 2024, University of Oregon Master of Science, Mathemtaics, 2021, University of Oregon Maestría en Ciencias, Matemáticas, 2016, Universidad Nacional de Colombia Pregrado, Matemáticas, 2012, Universidad de Antioquia AREAS OF SPECIAL INTEREST: Algebraic Topology Algberaic K-theory Category theory PROFESSIONAL EXPERIENCE: Graduate Employee, University of Oregon, Eugene, OR, USA 2018-2024 Lecturer, Universidad de Antioquia, Medellín, ANT, Colombia, 2015-2018 GRANTS, AWARDS, AND HONORS: Anderson Mathematics PhD Student Research Award, 2023 Fulbright-COLCIENCIAS scholarship 2018-2022 Honorable Mention for a Master’s Thesis at Universidad Nacional de Colombia, 2016 4 ACKNOWLEDGMENTS First and foremost I would like to thank my advisor Angélica Osorno who gen- erously took me under her wing for the last two years without being required to or having the time for advising PhD students. Thanks for believing in me and un- knowingly and gracefully guiding me out of a dark place. Second, I would like to thank my advisor Dan Dugger. He is the single person from which I have learned the most math from and taught me all the things that I wanted to learn about Al- gebraic Topology when I decided to come to the United States, and a lot of other things I didn’t know I wanted to learn. I would also like to thank Donald Yau for helping me, encouraging me and for asking me a question that I have yet to answer. During these years I also learned a lot from the classes I took with Robert Lipshitz, Jon Brundan, and Alexander Polishchuck. During the first four years of the dura- tion of my PhD I was supported by a Fulbright-COLCIENCIAS scholarship funded by the Colombian government. I would like to thank my friends from the PhD program, specially Nico, Juan Sebastián, Duca, Holt, Gary, Sean, and Nate. I wouldn’t have made it without their help support. Thanks to Alonso who was in a lot of senses my mentor while I was in Eugene, I wouldn’t have gotten a job without his help. I hope I don’t forget any of my friends: Anto, Cami, Ale, Liz, Tami, Tanys, Theresa, Danielito, Gaby Brown, Mau, Adri, Santi, Naty, Luis, Nayeli, Wally, Jorge. Thanks for your support and for making me feel at home and taking care of me. Thanks also to Tomi and Valen who helped me in all kinds of ways despite the dis- tance. I want to thank my parents and my sister, for raising me and growing up with me in Giraldo, a beautiful town torn, like so many others, by a horrific war. Thank you for protecting me, for believing in me and for showing me that one should live, think, feel and fight in spite of... and that, as Lispector wrote, "it is sometimes the in spite of which pushes us forward". Thanks also to Gabi for keeping me company in spite of the distance, for tak- ing care of me, for her support and her love and for watching El Marginal with me during a depressive episode that almost managed to finish my career. 5 Para Mamá, Papá y Naty 6 TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. COHERENCE FOR PSEUDO SYMMETRIC MULTIFUNCTORS . . . . 14 2.1. Symmetric and pseudo symmetric Multifunctors . . . . . . . . . . . . 14 2.2. Equivalent definition of Pseudo Symmetry . . . . . . . . . . . . . . . 28 2.3. Applications to inverse K-heory . . . . . . . . . . . . . . . . . . . . . 44 3. COHERENCE FOR SYMMETRIC PSEUDO COMMUTATIVE MONADS 47 3.1. Symmetric pseudo commutative 2-monads . . . . . . . . . . . . . . . 47 3.2. The free T -algebra functor as a multifunctor . . . . . . . . . . . . . . 64 3.3. Pseudo symmetry of the free T -algebra multifunctor . . . . . . . . . . 78 REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7 CHAPTER 1 INTRODUCTION Algebraic K-theory can be seen as a technique to build spectra from algebraic data. Segal’s infinite loop space machine [Seg74] constructs spectra from symmet- ric monoidal categories. Permutative categories, i.e., symmetric monoidal categories that are strictly associative and unital, provide another input for K-theory, which a construction of May [May74] turns into spectra. These two K-theory constructions are equivalent [MT78]. We are interested in multiplicative structures in spectra, so it is natural to ask what kind of conditions are necessary to impose on a per- mutative category so that its K-theory is, for example, an E∞-ring spectrum. Al- though this question was first tackled by May [May77] who defined bipermutative categories, it wasn’t until later that an answer was provided independently by May [May09], and Elmendorf and Mandell [EM06]. They proved that the K-theory of a bipermutative category is an E∞-ring. Elmendorf and Mandell also describe the categorical input that gives rise to a range of rings, modules and algebras in spectra after applying May’s K-theory construction. In their apporach, Elmendorf and Mandell use two main tools. On the one hand, the homotopy theory of spectra was made more transparent with the in- troduction of the modern symmetric monoidal model categories of spectra. These model categories allow for the treatment of various multiplicative structures in spectra and thus, are a natural target for Elmendorf and Mandell’s K-theory con- struction. On the other hand, the domain of their construction, Perm, is not a symmetric monoidal category, although it is in a 2-categorical sense [GJO22]. The usual definitions of ring, algebra and module that use monoidal structures are thus not available. To overcome this obstruction, Elmendorf and Mandell introduced multicategories in homotopy theory. Multicategories are a generalization of cate- gories that allow for maps with multiple inputs, even in the absence of a symmet- ric monoidal structure. In a sense, they allow the handling of multilinear maps in the absence of tensor products. They are a generalization of symmetric monoidal categories and operads at the same time, with operads being multicategories with one object. One can define multiplicative structures in a given multicategory, with multifunctors between multicategories preserving these structures. Elmendorf and Mandell define K-theory as a multifunctor from the multicategory Perm to sym- metric spectra. Multifunctoriality implies that K-theory preserves multiplicative 8 structures. Thomason proved that every connective spectrum can be realized as the K- theory of some permutative category [Tho95]. Mandell [Man10] made Thomason’s construction functorial by providing a stable homotopy inverse functor P to K- theory. The functor P takes as input Γ-categories (modelling connective spectra [Tho80; Cis99; BF78]) and its target is Perm. Elmendorf [Elm21], and indepen- dently Johnson and Yau [JY22] extended P to a Cat-enriched multifunctor, but one that is not symmetric: it is not compatible with the permutation of elements in the domains of multicategory mapping spaces. To account for this Yau [Yau24b] introduced pseudo symmetric multifunctors, where there is compatibility only up to coherent natural isomorphisms, and he proved that Mandell’s inverse K-theory multifunctor P is pseudo symmetric in this sense. A natural question to ask about pseudo symmetric multifunctors is whether they preserve multiplicative structure and if so, in what sense. We answer this question by proving a coherence result for pseudo symmetric multifunctors. If F : M → N is a pseudo symmetric multifunctor between multicategories enriched in Cat, we prove that the natural isomorphisms attesting the pseudo symmetry of F assemble together to give a symmetric multifunctor ϕ(F ) : M×EΣ∗ → N satisfying a universal property, where EΣ∗ is the categorical Barratt-Eccles operad defined in Example 2.1.5. We can also think about our result as a rigidification result. We can rigidify F and turn it into a symmetric multifunctor ϕ(F ), at the cost of changing its domain. This is the main result of Chapter 2. Theorem 1.0.1. (Theorem 2.2.3) Let M be a Cat-enriched multicategory. There is a pseudo symmetric multifunctor ηM : M → M× EΣ∗ such that for every Cat- enriched multicategory N and every pseudo symmetric multifunctor F : M → N , there exists a unique symmetric Cat-enriched multifunctor ϕ(F ) : M × EΣ∗ → N such that the following diagram commutes: M× EΣ∗ ηM ϕ(F ) M N . F That is, F = ϕ(F ) ◦ ηM as pseudo symmetric multifunctors. 9 Thus, if O is an operad in Cat, pseudo symmetric algebras in a Cat-enriched multicategory M over O (i.e., pseudo symmetric multifunctors O → M) can be rigidified to get symmetric algebras in M over O×EΣ∗. The following result, which appears as Corollary 2.3.7, holds since multiplying by EΣ∗ sends the commutative operad {∗} to the E∞ operad EΣ∗ and sends Σ-free En-operads in Cat, like the ones defined in [Ber96] and [BFSV03], to En-operads. Corollary 1.0.2. (Corollary 2.3.7) Let F : M → N be a Cat-enriched pseudo symmetric multifunctor. Then, 1. F sends commutative monoids to E∞-algebras. 2. F sends En-algebras over Σ-free En Cat-operads to En-algebras for n = 1, 2, . . . , ∞. In this sense, F preserves symmetric En-algebras parameterized by Σ-free En- operads at the cost of changing the parameterizing operad. This corollary extends our understanding of the behavior of inverse K-theory since it implies that the in- verse K-theory pseudo symmetric multifunctor P from [Yau24b] sends commutative monoids to E∞-algebras and sends En-algebras (n = 1, 2, . . . ) parameterized by Σ- free operads to En-algebras. Since P provides a stable inverse to K-theory, and K- theory is a symmetric multifunctor, this implies that every symmetric En-algebra parameterized by a Σ-free Cat-operad in Γ-categories is stably equivalent to the K- theory of a symmetric En-algebra in permutative categories for n = 1, 2, . . . ,∞. Our result can thus be seen as a multiplicative version of Thomason’s theorem [Tho95]. This also shows how Theorem 1.0.1 can be used to grasp the behavior of pseudo symmetric multifunctors on structures parameterized by symmetric operads in general. This rigidification structure can be extended in a 2-categorical sense. In [Yau24b] Yau defines the 2-category Cat-Multicat having Cat-enriched multicategories as 0-cells, symmetric multifunctors as 1-cells and multinatural transformations as 2- cells. He also defines the 2-category Cat-Multicatps with 0-cells Cat-enriched multicategories, 1-cells pseudo symmetric multifunctors, and 2-cells pseudo sym- metric Cat-multinatural transformations. Every symmetric Cat-enriched multi- functor (respectively, multinatural transformation) is canonically a pseudo sym- metric multifunctor (respectively multinatural transformation), so there is a 2- functorial inclusion j : Cat-Multicat → Cat-Multicatps. Taking into account 10 these 2-categorical structures, we can extend our previous result by providing a left 2-adjoint ψ to j, which, at the 0-cell level, sends a multicategory M to ψ(M) = M× EΣ∗. Theorem 1.0.3. (Corollary 2.2.5 and Theorem 2.2.7) The 2-categorical inclusion j : Cat -Multicat → Cat-Multicatps admits a left 2-adjoint ψ : Cat−Multicatps → Cat−Multicat with ψ(M) = M × EΣ∗ for M a Cat-multicategory. In particular, for Cat- multicategories M and N we have an isomorphism of categories Cat-Multicatps(M,N ) ∼= Cat-Multicat(M× EΣ∗,N ). An important consequence of this theorem is that we can give a very simple and compact description of the 2-category Cat-Multicatps solely in terms of sym- metric Cat-multifunctors and Cat-mutinatural transformations, which we do in Definition 2.2.8. The question about the existence of a multiplicative equivariant K-theory ma- chine taking some algebraic input to G-spectra and preserving multiplicative struc- tures has received some attention in recent years. On the one hand Barwick, Glas- man and Shah [BGS20] and Kong, May, and Zou [KMZ24] among others provide examples of multiplicative structures in G-spectra built from various kinds of in- puts, but don’t provide a systematic approach. On the other hand the works of Guillou, May, Merling and Osorno [GMMO23], and Yau [Yau24a] introduce equiv- ariant multiplicative K-theory machines using multicategories. The K-theory ma- chine of [GMMO23] is a non-symmetric multifunctor from the multicategory of al- gebras and pseudo morphisms over a pseudo commutative operad enriched in G- Cat to orthogonal G-spectra. It is conjectured to be a pseudo symmetric functor. Yau’s G-equivariant K-theory machine is a symmetric multifunctor with domain the multicategory of pseudo algebras over a pseudo commutative operad in G-Cat and produces orthogonal spectra. In both constructions pseudo commutative oper- ads play an important role. Pseudo commutative operads were defined by Corner and Gurski [CG23]. These are operads whose associated monads are pseudo commutative. Now, commutative monads were introduced by Anders Kock in [Koc70] and are designed to capture the concept of a monoidal 2-monad. Let’s make this a little more precise. Suppose 11 that K is a 2-category with finite products, and consider K endowed with the sym- metric monoidal structure induced by products. Monoidal 2-monads T : K → K are strong in the sense that there is a 2-natural transfomation with components t2 : A × TB → T (A × B) for A,B objects of K. Of course, since K is symmetric monoidal, there is also a 2-natural transformation with components t1 : TA × B → T (A × B). Strong 2-monads can be regarded as monoidal 2-functors in two differ- ent ways, with the binary components being given by the two 1-cells that form the boundary of the following diagram: TA× tTB 1 T (A× TtTB) 2 T 2(A×B) t2 µ T (TA×B) T 2(A×B) µ T (A×B).Tt1 A commutative 2-monad is one where the previous diagram commutes for any A,B objects of K. It is a theorem of Kock [Koc70] that a strong 2-monad T is commutative if and only if T is a monoidal 2-monad. There are a lot of examples of 2-monads T : Cat → Cat, e.g., the 2-monad for symmetric strict monoidal cat- egories, that fail to be commutative but that are so up to coherent isomorphisms. Such monads are called pseudo commutative and they were introduce by Hyland and Power [HP02]. From the point of view of multiplicative equivariant algebraic K-theory, the most important feature of pseudo commutative 2-monads (and hence pseudo com- mutative operads) is that they allow for the definition of a multicategory of alge- bras. Blackwell, Kelly and Power define and study a 2-category of algebras T -Alg for T : K → K a 2-monad [BKP02]. Hyland and Power [HP02] extend T -Alg to a non symmetric Cat-multicategory when T is pseudo commutative. If T satisfies an- other technical condition of being symmetric, then the Cat-multicategory T -Alg is symmetric. When K = Cat and T is accesible, the Cat-multicategory structure in T -Alg arises from a monoidal bicategorical structure on T -Alg [Bou02]. The main theorem in Chapter 3 is the following. Theorem 1.0.4. (Theorem 3.3.18) Let T : K → K be a strong, pseudo commu- tative, symmetric 2-monad. Then, the free algebra multifunctor T : K → T -Alg is pseudo symmetric. This implies that the free algebra functor for pseudo commutative operads, like those used in [GMMO23] and [Yau24a] is pseudo symmetric. In Remark 3.3.14, 12 we explain how this result implies a coherence result for non symmetric, strong pseudo commutative 2-monads as was conjectured, but not stated clearly or proved in [HP02]. The results in this thesis, together with the author’s current work on the def- inition and coherence of pseudo symmetric Cat-multicategories could be used to prove that the multiplicative equivariant K-theory machine of [GMMO23] preserves multiplicative structures. This can also be useful in proving that this machine is equivalent to the one constructed by Yau [Yau24a]. In Chapter 2, we prove the coherence theorem for pseudo symmetric Cat-multifunctors and extract some 2-categorical consequences as well as some applications to K- theory. In Chapter 3 we prove a coherence for symmetric pseudo commutative 2- monads, and we show how this coherence can be interpreted as the pseudo symme- try of the free algebra multifunctor associated with the 2-monad. 13 CHAPTER 2 COHERENCE FOR PSEUDO SYMMETRIC MULTIFUNCTORS In this chapter we prove a coherence theorem for pseudo symmetric multi- functors as defined by Yau [Yau24a]. In Section 2.1 we introduce multicategories, multifunctors and their enriched versions in Cat as well as pseudo symmetric Cat- enriched multifunctors [Yau24b]. In Section 2.2 we prove the coherence theorem and extract some 2-categorical consequences. By work of Donald Yau [Yau24b] Mandell’s inverse K-theory is stably equivalent to a pseudo symmetric Cat-enriched multifunctor. We use Section 2.3 to develop the K-theoretical consequences of ap- plying our coherence theorem to Mandell’s inverse K-theory multifunctor. 2.1 Symmetric and pseudo symmetric Multifunctors We begin by reviewing the definition of multicategory enriched in a symmet- ric monoidal category. In the following definition (C, 1,⊕, λ, ρ, ξ) is a symmetric monoidal category with ⊕ : C × C → C the monoidal product, 1 the monoidal unit, λ the left unit isomorphism, ρ the right unit isomorphism and ξ the sym- metry. In this paper we will consider only categories enriched over Cat with the monoidal structure given by products, but we use a general monoidal category in the definition to make explicit the fact that this definition doesn’t make use of the 2-categorical structure of Cat. Remark 2.1.1. We will also use the following notation: if σ ∈ Σn and τi ∈ Σk fori 1 ≤ i ≤ n, σ⟨τ1, . . . , τn⟩ ∈ Σk1+···+kn is the permutation that permutes n blocks of lengths k1, . . . , kn according to σ and each block of length ki according to τi. Definition 2.1.2. If C is a symmetric monoidal category, a C-multicategory (M, γ, 1) consists of the following data. • A class of objects Ob(M). • For every n ≥ 0, ⟨a⟩ = ⟨ai⟩n ni=1 ∈ Ob(M) and b ∈ Ob(M), an object in C denoted by M(⟨a⟩; b) = M(a1, . . . , an; b). We will write ⟨a⟩ instead of ⟨a ni⟩i=1 when n is clear from the context or irrele- vant. [In the case C = Cat, an object f of M(⟨a⟩; b) will be called an n-ary 14 1-cell with input ⟨a⟩ and output b and will be denoted as f : ⟨a⟩ → b. Simi- larly, we will call α : f → g in M(⟨a⟩; b)(f, g) an n-ary 2-cell.] • For each n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), and σ ∈ Σn, a C-isomorphism M(⟨a⟩; b) σ∼ M(⟨a⟩σ; b)= called the right σ action or the symmetric group action. Here ⟨a⟩σ = ⟨a1, . . . , an⟩σ = ⟨aσ(1), . . . , aσ(n)⟩. [In the case C = Cat we write fσ for the image of an n-ary 1-cell f : ⟨a⟩ → b in M and similarly for 2-cells.] • For each object a ∈ Ob(M), a morphism 1 1 a M(a; a) called the a-unit. In the case C = Cat we notice that if a ∈ Ob(M), 1a : a → a is a 1-ary 1-cell while if f : ⟨a⟩ → b is an n-ary 1-cell, then 1f : f → f is an n-ary 2-cell in M(⟨a⟩; b)(f, f) so our notation is unambiguous. • For every c ∈ Ob(M), n ≥ 0, ⟨b⟩ = ⟨b ⟩nj j=1 ∈ Ob(M)n, kj ≥ 0 for 1 ≤ j ≤ n, and ⟨ ka j kjj⟩ = ⟨aj,i⟩i=1 ∈ Ob(M) for 1 ≤ j ≤ n, a morphism in C, ⊗n M(⟨b⟩ ⊗ M ⟨ γ; c) ( aj⟩; bj) M(⟨a⟩; c), j=1 ∑ where we adopt the convention that ⟨a⟩ ∈ Ob(M)k, where k = ni=1 kj, denotes the concatenation of the varying aj’s for j = 1, . . . , n. We write this as ⟨a⟩ = ⟨a1, . . . , an⟩ = ⟨⟨a nj⟩⟩j=1 = ⟨a1,1, . . . , a1,k1 , a2,1, . . . , an−1,kn−1an,1, . . . , an,kn⟩. The previous data are required to satisfy the following axioms. • Symmetric group action: For every n ≥ 0, ⟨a⟩ ∈ Ob(M), b ∈ Ob(M), and σ, τ in Σn the following diagram commutes in C : 15 M(⟨a⟩; b) σ M(⟨a⟩σ; b) τ στ M(⟨a⟩στ ; b). We also require the identity permutation idn ∈ Σn to act as the identity mor- phism on M(⟨a⟩; b). • Associativity: For every d ∈ Ob(M), n ≥ 1, ⟨c⟩ = ⟨cj⟩n ∈ Ob(M)nj=1 , kj ≥ 0 for 1 ≤ j ≤ n with kj ≥ 1 for at least one k j, ⟨bj⟩ = ⟨b ⟩ jj,i i=1 ∈ Ob(M)kj for 1 ≤ j ≤ n, li,j ≥ 0 for 1 ≤ j ≤ n and 1 ≤ i ≤ kj, and ⟨aj,i⟩ = ⟨ l a i,jj,i,p⟩p=1 ∈ Ob(M)li,j for 1 ≤ j ≤ n and 1 ≤ i ≤ kj, the following associativity diagram commutes in C: (⊗ ) ⊗ (⊗ )n n kj M(⟨c⟩; d)⊗ M(⟨bj⟩; cj) ⊗ M(⟨aj,i⟩; bj,i) j=1 j=1 i=1 γ⊗1 ⊗ (n ⊗ )kj ∼= M(⟨b⟩; c)⊗ M(⟨aj,i⟩); bj,i j=1 i=1 ⊗ ( )n ⊗kj M(⟨c⟩; d)⊗ M(⟨bj⟩; cj)⊗ M(⟨aj,i⟩; bj,i) γ j=1 ⊗ i=1 1⊗ nj=1 γ⊗n M(⟨c⟩; d)⊗ M(⟨aj⟩; cj) γ M(⟨a⟩; b). j=1 (2.1.1) • Unity: Suppose b ∈ Ob(M) and ⟨a⟩ = ⟨a ⟩nj j=1 ∈ Ob(M), then the following right unity diagram commutes in C : ⊗n M(⟨a⟩; b)⊗ 1 ⊗ j=1n id⊗ 1 ∼a ⊗ =jj=1 n M(⟨a⟩; b)⊗ M(aj; aj) γ M(⟨a⟩; b). j=1 16 With b, ⟨a⟩ as before, we also demand that the following left unity diagram commutes in C. 1⊗M(⟨a⟩; b) 1b⊗id λ M(b; b)⊗M(⟨a⟩; b) γ M(⟨a⟩; b). • Top equivariance: For every c ∈ Ob(M), n ≥ 1, ⟨b⟩ = ⟨b n nj⟩j=1 ∈ Ob(M) , kj ≥ 0 for k 1 ≤ j ≤ n, ⟨a j kjj⟩ = ⟨aj,i⟩i=1 ∈ Ob(M) for 1 ≤ j ≤ n, and σ ∈ Σn, the following diagram commutes: ⊗n −1 ⊗n M(⟨b⟩; c)⊗ M ⟨ ⟩ σ⊗σ( aj ; bj) M(⟨b⟩σ; c)⊗ M(⟨aσ(j)⟩; bσ(j)) j=1 j=1 γ γ M(⟨a1⟩, . . . , ⟨an⟩; c) 〈 〉 M(⟨aσ(1)⟩, . . . , ⟨aσ(n)⟩; c). σ idk ,...,idσ(1) kσ(n) (2.1.2) Here σ−1 is the unique isomorphism in C, given by the coherence theorem for symmetric monoidal categories, that permutes the factors M(⟨aj⟩, bj) accord- ing to σ−1. • Bottom equivariance: For ⟨aj⟩, ⟨b⟩ and c as in Top equivariance (2.1.2), the following diagram commutes: ⊗ ⊗nn id⊗ τj ⊗n M(⟨b⟩; c)⊗ M(⟨aj⟩ j=1 ; bj) M(⟨b⟩, c)⊗ M(⟨aj⟩τj; bj) j=1 j=1 (2.1.3) γ γ M(⟨a1⟩, . . . , ⟨an⟩; c) 〈 〉 M(⟨a1⟩τ1, . . . , ⟨an⟩τn; c). idn τ1,...,τn This concludes the definition of a C-multicategory. Remark 2.1.3. A C-operad is a C-multicategory with one object. If O is a C- operad, its n-ary operations will be denoted by On ∈ Ob(C). A non symmetric C-multicategory (C-operad) is defined in the same way as a C-multicategory (C- operad) excluding the data of the symmetric group action as well as the symmetric group, top and bottom equivariance coherence axioms. We will only be concerned 17 with symmetric multicategories and operads. C-multicategories are often referred to as colored operads, with the objects of the C-multicategory being referred to as colors and C-operads having just one color. Example 2.1.4. As examples of Set-operads, where Set has the monoidal struc- ture induced by products in Set, we have the commutative operad Comm = {∗} with Commn = {∗}. Another example is the associative operad Ass = Σ∗ with Assn = Σn, with the right action of the symmetric product given by right multi- plication and γ defined in the following wa∏y. If n ≥ 1 and k1, . . . , kn natural num- bers with k =∏Σni=1ki, we define γ : Σn × ( ni=1 Σk ) → Σk given for σ ∈ Σn andi ⟨τ1, . . . , τ ⟩ ∈ nn i=1Σk byi γ(σ, ⟨ρ ⟩n ni i=1) = σ⟨ρi⟩i=1 = σ⟨ρ1, . . . , ρn⟩, as in Remark 2.1.1. When n is clear from the context we will write σ⟨ρi⟩ = σ⟨ρ ni⟩i=1. Example 2.1.5. We will consider Cat-multicategories where the monoidal struc- ture in Cat is given by products. One source of examples is the forgetful functor Ob : Cat → Set which forgets the morphism structure and remembers only the object set. Its right adjoint E : Set → Cat is the functor that takes a set A to EA, the category with objects Ob(EA) = A, and with a unique isomorphism be- tween each pair of objects. E sends a morphism f : A → B of sets to the func- tor Ef : EA → EB, the only functor such that f = Ob(Ef). E preserves prod- ucts, and thus, if O is a Set-operad, EO is a Cat-operad. Similarly, if M is a Set- multicategory, EM is a Cat-multicategory with the same collection of objects as M. We will call EComm = {∗} the commutative Cat-operad. The Barratt-Eccles op- erad is the Cat-operad EΣ∗ = EAss. Example 2.1.6. Another source of examples for multicategories are symmetric monoidal categories, and thus also permutative categories. Each symmetric monoidal category C has an associated Set-multicategory End(C), whose objects agree with the objects of C and such that for ⟨a⟩ ∈ Ob(C)n and b ∈ Ob(C), End(C)(⟨a⟩; b) = C(a1 ⊗ · · · ⊗ an, b). Here we take a1 ⊗ · · · ⊗ an with the leftmost parenthesization. Any fixed parenthe- sization would work. An empty string of objects is interpreted as the monoidal unit 1 ∈ Ob(C). 18 Next, we define 1-cells between C-multicategories that preserve the action of the symmetric group. These are called symmetric C-multifunctors. Definition 2.1.7. A symmetric C-multifunctor F : M → N between C-multicategories M and N consists of the following data. • An object assignment F : Ob(M) → Ob(N ). • For each n ≥ 0, ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M) a C morphism M(⟨a⟩; b) F N (⟨Fa⟩;Fb). These data are required to preserve units, composition, and the action of the sym- metric group. • Units: For each object a ∈ Ob(M), F (1a) = 1Fa, i.e., the following diagram commutes in C : M(a, a) 1a F 1 N (Fa, Fa). 1Fa • Composition: For every c ∈ Ob(M), n ≥ 0, ⟨b⟩ = ⟨b nj⟩j=1 ∈ Ob(M)n, kj ≥ 0 for k1 ≤ j ≤ n, and ⟨aj⟩ = ⟨a ⟩ jj,i i=1 ∈ Ob(M)kj for 1 ≤ j ≤ n and 1 ≤ i ≤ n, the following diagram commutes in C : ⊗ ⊗nn F⊗ Fj=1 ⊗nM(⟨b⟩; c)⊗ M(⟨aj⟩; bj) N (⟨Fb⟩;Fc)⊗ N (⟨Faj⟩;Fbj) j=1 j=1 γ γ M(⟨a⟩; c) N (⟨Fa⟩;Fc). F (2.1.4) • Symmetric Group Action: For each ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M) the following diagram commutes in C : 19 M(⟨a⟩; b) F N (⟨Fa⟩;Fb) ∼= σ ∼= σ M(⟨a⟩σ; b) N (⟨Fa⟩σ;Fb). F Definition 2.1.8. Let O be a C-operad and a M be a C-multicategory. A sym- metric algebra in M over O is a symmetric multifunctor O → M. Symmetric algebras are usually called algebras, but we add the adjective sym- metric to distinguish them from pseudo symmetric algebras, which will be defined later. Example 2.1.9. Since their introduction by May [May72], operads have been used to characterize certain categories as the categories of symmetric algebras over a cer- tain operad. For example, symmetric algebras over Comm in Set are commutative monoids. Symmetric algebras over Σ∗ in Set are associative monoids. Symmetric algebras over the Barrat-Eccles operad EΣ∗ in Cat are precisely permutative cate- gories [May74]. Next we define composition of C-multifunctors. Definition 2.1.10. We define the horizontal composition of C-multifunctors in the following way. • Let F : M → N , and G : N → Q be C-multifunctors, we define the C- multifunctor GF : M → Q on objects as the composition Ob(M) F Ob(N ) G Ob(Q), and its component functors for ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M) as the composite M(⟨a⟩; b) F N (⟨Fa⟩;Fb) G Q(⟨GFa⟩;GFb). • The identity C-multifunctor 1M : M → M is defined as the identity assign- ment on objects with the identity functors as component functors. Next we define 2-cells between C-multifunctors. These will be the 2-cells of a 2-category with 0-cells C-multicategories and 1-cells C-multifunctors. 20 Definition 2.1.11. [Yau24b, Def. 3.2.5] For (symmetric) C-multifunctors F,G : M → N , we define a C-multinatural transformation θ : F ⇒ G as the data of a compo- nent morphism θa : 1 → N (Fa,Ga) in C for each a ∈ Ob(M) subject to the com- mutativity of the following diagram in C for each ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M), 1⊗M θ ⊗F(⟨a⟩; b) b N (Fb;Gb)⊗N (⟨Fa⟩;Fb) ∼= γ M(⟨a⟩, b) N (⟨Fa⟩;Gb). =∼ ⊗ ⊗ ⊗ γG⊗ θa M j(⟨a⟩; b)⊗ nj=1 1 N (⟨Ga⟩;Gb)⊗ n j=1 N (Faj;Gaj) We define the identity multinatural transformation 1F : F → F as having com- ponent (1F )a = 1Fa for a an object of M. Remark 2.1.12. When C = Cat, and given F,G : M → N Cat-multifunctors and the data of a 1-ary 1-cell θa : Fa → Ga for each a ∈ Ob(M), the commutativity of the diagram in the previous definition means that for every n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M) and each 1-cell f : ⟨a⟩ → b, γ(Gf ; ⟨θa ⟩) = γ(θb;Ff) (2.1.5)j holds in N (⟨Fa⟩;Gb) and that, for every 2-cell α : f → g in M(⟨a⟩; b)(f, g), γ(Gα; ⟨1θa ⟩) = γ(1θ ;Fα) (2.1.6)j b in N (⟨Fa⟩;Gb). We can express (2.1.5) diagrammatically as the commutativity of the square ⟨θa ⟩ ⟨ jFa⟩ ⟨Ga⟩ Ff Gf θ Fb b Gb, where the composition of adjacent 1-cells is done through γ and a square represents an equality between composite 1-cells. In the same fashion, and using (2.1.5), we can express (2.1.6) as the equality of multicategorical pasting diagrams ⟨θaj ⟩ ⟨θa ⟩⟨Fa⟩ ⟨Ga⟩ ⟨Fa⟩ j ⟨Ga⟩ Fα Ff Fg Gg = GαFf Gf Gg θ Fb Gb Fb b Gb. θb 21 Here the concatenation of adjacent 2-cells is done through γ, and an arrow labeled with the 1-cell h is interpreted as the 2-cell 1h : h → h. For example, the left hand side diagram represents γ(1θ , Fα) while the right hand side represents γ(Gα, ⟨θα ⟩).b j The empty squares represent equalities between composite 1-cells. Next, we define horizontal and vertical compositions of C-multinatural trans- formations. Definition 2.1.13. [Yau24b, Def. 3.2.7] Suppose given θ : F ⇒ G, ζ : G ⇒ H C-multinatural transformations with F,G,H : M → N C-multifunctors. The vertical composition ζθ : F ⇒ H is defined as having as component at each a ∈ Ob(M) (ζθ)a, the composite 1 = ∼ 1⊗ ζa⊗θa1 N (Ga;Ha)⊗N γ(Fa;Ga) N (Fa;Ha). Suppose that θ : F ⇒ G and ζ : F ′ ⇒ G′ are C-multinatural transformations with F,G : M → N and F ′, G′ : N → Q C-multifunctors. The horizontal composition ζ ∗ θ : F ′F ⇒ G′G is defined as the C-multinatural transformation with component at each a ∈ Ob(M), given by the composite (ζ∗θ)a 1 Q(F ′Fa;G′Ga) =∼ γ ζ ⊗θ 1⊗F ′Ga a 1⊗ 1 Q(F ′Ga;G′Ga)⊗N (Fa;Ga) Q(F ′Ga;G′Ga)⊗Q(F ′Fa;F ′Ga). Remark 2.1.14. When C = Cat and given θ : F ⇒ G, ζ : G ⇒ H Cat-multinatural transformations with F,G,H : M → N C-multifunctors and a ∈ Ob(M), (ζθ)a = γ(ζa, θa.) (2.1.7) On the other hand, if θ : F ⇒ G and ζ : F ′ ⇒ G′ are Cat-multinatural transforma- tions with F,G : M → N and F ′, G′ : N → Q Cat-multifunctors, (ζ ∗ θ)a = γ(ζGa;F ′θa). (2.1.8) Yau proves in [Yau24b] that Definitions 2.1.2, 2.1.7, 2.1.10 and 2.1.13 assemble together to give the 2-category C-Multicat, with 0-cells consisting of C-multicategories, 1-cells symmetric C-multifunctors, and 2-cells C-multinatural transformations. 22 There is a non symmetric variant where we drop the requirement that the C-multifunctors preserve the symmetric group action, as well as dropping the coherence axioms re- lated to the symmetric group action, but we won’t refer to this 2-category again. For the rest of the article we fix our symmetric monoidal category C to be Cat, with the symmetric monoidal structure induced by products. In this context we can define a pseudo symmetric variant of this 2-category, namely Cat-Multicatps using the 2-categorical structure of Cat. The objects of Cat-Multicatps are still Cat-multicategories, but the 1-cells are pseudo symmetric Cat-multifunctors: Cat- multifunctors where we only require that they preserve the symmetric group action up to coherent isomorphisms. Definition 2.1.15. [Yau24b, Def. 4.1.1] Suppose that M,N are Cat-multicategories. A pseudo symmetric Cat-multifunctor F : M → N consists of the following data: • A function on object sets F : Ob(M) → Ob(N ). • For each ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M), a component functor M(⟨a⟩; b) F // N (⟨Fa⟩;Fb). • For each σ ∈ Σn, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), a natural isomorphism Fσ,⟨a⟩,b M(⟨a⟩; b) F N (⟨Fa⟩;Fb) =∼ σ σ Fσ,⟨a⟩,b M(⟨a⟩σ; b) N (⟨Fa⟩σ;Fb). F When ⟨a⟩ and b are clear from the context we write simply Fσ, and if f ∈ Ob(M(⟨a⟩, b)) we will denote by Fσ,⟨a⟩,b;f = Fσ;f : F (fσ) → F (f)σ the 2- cell in N (⟨Fa⟩σ;Fb) corresponding to the component of Fσ at f. Naturality for Fσ means that given α : f → g a 2-cell in M(⟨a⟩; b)(f, g), the following diagram commutes in N (⟨Fa⟩σ; b) : Fσ;f F (fσ) F (f)σ F (ασ) (Fα)σ (2.1.9) F (gσ) F (g)σ. Fσ;g 23 These data are subject to the same axioms of unit and composition preservation (2.1.4) as a symmetric Cat-multifunctor, but we replace the symmetric group ac- tion preservation axiom by the following four axioms. • Unit permutation: Let n ≥ 0, ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M), then Fidn,⟨a⟩,b = 1F . (2.1.10) • Product permutation: This axiom expresses the coherence of the natural isomorphisms Fσ, for varying σ, with respect to the symmetric group action. Let n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M) and σ, τ ∈ Σn. Then, the following equality of pasting diagrams holds. M(⟨a⟩; b) F N (⟨Fa⟩;Fb) M(⟨a⟩; b) F N (⟨Fa⟩;Fb) σ σ Fσ M(⟨a⟩σ; b) N (⟨Fa⟩σ;Fb) = στ στ F Fστ τ τ Fτ M(⟨a⟩στ ; b) N (⟨Fa⟩στ ;Fb) M(⟨a⟩στ ; b) N (⟨Fa⟩στ ;Fb). F F Thus, for every 1-cell f ∈ Ob(M(⟨a⟩; b)), the following diagram of 2-cells commutes in N (⟨Fa⟩;Fb): F (fσ)τ Fτ ;fσ (Fσ;f )τ (2.1.11) F (fστ) F (f)στ. Fστ ;f • Top equivariance: For every c ∈ Ob(M), n ≥ 0, ⟨b⟩ = ⟨bj⟩n nj=1 ∈ Ob(M) , k kj ≥ 0 for 1 ≤ j ≤ n, and ⟨a j kjj⟩ = ⟨aj,i⟩i=1 ∈ Ob(M) for 1 ≤ j ≤ n, and σ ∈ Σn, the following two pasting diagrams are equal. 24 ∏ ∏F× j F M(⟨b⟩; c)× n ∏ j=1 M(⟨aj⟩; bj) N (⟨Fb⟩;Fc)× n j=1 N (⟨Faj⟩;Fbj) γ γ M(⟨⟨a ⟩⟩n F j j=1; c) N (⟨⟨Faj⟩⟩nj=1;Fc) σ⟨idk ⟩ σ⟨id ⟩σ(j) kF σ(j)σ⟨idk ⟩σ(j) M(⟨⟨a nσ(j)⟩⟩j=1; c) N (⟨⟨Faσ(j)⟩⟩nF j=1;Fc) ∥ ∏ ∏F× j F ∏ M(⟨b⟩; c)× nj=1 M(⟨aj⟩; bj) N (⟨Fb⟩;Fc)× n j=1 N (⟨Faj⟩;Fbj) σ×σ−1 σ×σ−1∏ Fσ×1 ∏ M(⟨b⟩σ; c)× n nj=1 M(⟨aσ(j)⟩; bσ(j)) ∏ N (⟨Fb⟩σ;Fc)× j=1 N (⟨Faσ(j)⟩;Fb× σ(j))F j F γ γ M(⟨⟨aσ(j)⟩⟩nj=1; c) N (⟨⟨Fa nF σ(j)⟩⟩j=1;Fc) Here σ⟨idk ⟩ = σ⟨idk , . . . , idkσ(n)⟩. This means that for 1-cells f ∈ Ob(M(⟨b⟩; c))σ(j) σ(1) and gj ∈ Ob(M(⟨aj⟩; bj)) for 1 ≤ j ≤ n,( ) F nσ⟨idk ⟩;γ(f ;⟨g ⟩) = γ Fσ;f ; ⟨1Fg ⟩ . (2.1.12)σ(j) j σ(j) j=1 The domains and codomains of these pasting diagrams are equal by top equiv- ariance in M and N , and the fact that F preserves γ implies the commuta- tivity of the empty rectangles, see [Yau24b]. • Bottom Equivariance: For every c ∈ Ob(M), n ≥ 0, ⟨b⟩ = ⟨b ⟩nj j=1 ∈ Ob(M)n k, kj ≥ 0 for 1 ≤ j ≤ n, and ⟨aj⟩ = ⟨a ⟩ j kjj,i i=1 ∈ Ob(M) for 1 ≤ j ≤ n and 1 ≤ i ≤ kj, and τj ∈ Σk , the following two pasting diagrams are equal.j 25 ∏ ∏F× j F ∏M(⟨b⟩; c)× n M(⟨a ⟩; b ) N (⟨Fb⟩;Fc)× nj=1 j j j=1(⟨Faj⟩;Fbj) γ γ M(⟨⟨aj⟩⟩n Fj=1; c) N (⟨⟨Fa ⟩⟩nj j=1;Fc) idn⟨τj⟩ idn⟨τj⟩ Fidn⟨τi⟩ M(⟨⟨a ⟩τ ⟩nj j j=1; c) N (⟨⟨Fa nj⟩τj⟩j=1;Fc)F ∥ ∏ ∏F× M ⟨ ⟩ j F ∏ ( b ; c)× nj=1M(⟨aj⟩; bj) N (⟨Fb⟩;Fc)× n j=1N (⟨Faj⟩;Fb )∏ ∏ j id× ∏j τj ∏ id×× j τj∏ 1 j Fτj ∏M(⟨b⟩; c)× n nj=1M(⟨aj⟩τj ; bj) N (⟨Fb⟩;Fc)× j=1N (⟨⟨Faj⟩τ× j⟩;Fbj)F j F γ γ M(⟨⟨aj⟩τj⟩nj=1; c) N (⟨⟨Faj⟩τ nF j⟩j=1;Fc) This means that for 1-cells f : ⟨b⟩ → c and gj : ⟨aj⟩ → bj for 1 ≤ j ≤ n, Fidn⟨τ ⟩;γ(f ;⟨g ⟩) = γ(1Ff ; ⟨Fτ ;g ⟩) (2.1.13)j j j j as 2-cells in N (⟨⟨Faj⟩τj⟩;Fc). The domain and codomain of these pasting di- agrams are equal by bottom equivariance for M and N , and the preservation of γ by F guarantees that the empty squares commute, see [Yau24b]. Next we describe the horizontal composition of 1-cells in the 2-category Cat- Multicatps. Definition 2.1.16. [Yau24b, Def. 4.1.1] Let F : M → N , and G : N → Q be pseudo symmetric Cat-multifunctors. We define the pseudo symmetric functor GF : M → Q. On objects GF is the composite function GF : Ob(M) → Ob(Q). The composite component functor is given for ⟨a⟩ ∈ Ob(M)n, and b ∈ Ob(M) by the pasting M(⟨a⟩; b) F N (⟨Fa⟩; b) G Q(⟨GFa⟩;GFb). The symmetry isomorphisms are given for each σ ∈ Σn, ⟨a⟩ ∈ Ob(M), and b ∈ Ob(M) by 26 M(⟨a⟩; b) F N (⟨Fa⟩;Fb) G Q(⟨GFa⟩;GFb) σ σ σ Fσ,⟨a⟩,b Gσ,⟨Fa⟩,Fb M(⟨a⟩σ; b) F N (⟨Fa⟩σ; fb) G Q(⟨GFa⟩σ;GFb). That is, for each 1-cell f : ⟨a⟩ → b, the f component of GFσ is given by the com- posite G((Ff)σ) G(Fσ;f ) Gσ;Ff (2.1.14) GF (fσ) (GFf)σ. (GF )σ;f Next we define the 2-cells of the category Cat-Multicatps. Definition 2.1.17. [Yau24b, Def. 4.2.1] Suppose that F,G : M → N are pseudo symmetric Cat-multifunctors. A pseudo symmetric Cat-multinatural transforma- tion θ : F ⇒ G is the data of a component 1-cell θa : Fa → Ga for each a ∈ Ob(M) subject to axioms (2.1.5), (2.1.6) and the following extra axiom. For each n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), object f ∈ Ob(M(⟨a⟩; b)), and permutation σ ∈ Σn, the following arrow equality holds in the category N (⟨Fa⟩σ; b), ( ) γ (1θ ;F ) = γ G ; ⟨1 ⟩ . (2.1.15)b σ;f σ;f θaσ(j) This can also be expressed diagrammatically as the equality of multicategorical pasting diagrams ⟨θa ⟩ ⟨θ ⟩ ⟨ ⟩ σ(j) a Fa σ G⟨a⟩σ ⟨ σ(j)Fa⟩σ ⟨Ga⟩σ Fσ;f Gσ;f F (fσ) (Ff)σ (Gf)σ = F (fσ) G(fσ) (Gf)σ θ Fb Gb Fb b Gb, θb where the diagrams are interpreted as in Remark 2.1.12, the squares commuting by (2.1.5) and top and bottom equivariance for N , see [Yau24b]. We define the vertical and horizontal composition of pseudo symmetric Cat-multinatural transformations in the same way that we did for symmetric ones, through diagrams (2.1.7) and (2.1.8). 27 It is a theorem of Yau [Yau24b] that the data we have just defined gives the structure of a 2-category, namely Cat-Multicatps. Definition 2.2.8 says that we can describe this 2-category solely in terms of symmetric Cat-multifunctors and symmetric Cat-multinatural transformations. 2.2 Equivalent definition of Pseudo Symmetry To prove our first result we use finite products in the category Cat-Multicat. Having just the 1-categorical structure in mind, the products in Cat-Multicat are given in the following way. If M and N are two Cat-multicategories, then M×N has objects Ob(M × N ) = Ob(M) × Ob(N ). Now, for n ≥ 0, ⟨a⟩ ∈ Ob(M)n, ⟨c⟩ ∈ Ob(N )n, b ∈ Ob(M), and d ∈ Ob(N ), we define M×N (⟨(a, c)⟩; (b, d)) = M(⟨a⟩; b)×N (⟨c⟩; d). The composition γ of M×N , as well as the Σ∗ action and the multicategorical units, are defined componentwise. Next we define the pseudo symmetric multifunc- tor ηM appearing in the statement of 1.0.1. Definition 2.2.1. Let M be a Cat-multicategory. We define the pseudo symmet- ric Cat-multifunctor ηM : M → M × EΣ∗ which, when there is no room for con- fusion, we will denote η. For an object a ∈ Ob(M) as η(a) = (a, ∗). We will abuse notation and denote the object (a, ∗) of M× EΣ∗ as a. For n ≥ 0, ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M) we need to define a functor η : M(⟨a⟩; b) → M(⟨a⟩; b)× EΣn. For a 1-cell f : ⟨a⟩ → b, we define η(f) = (f, idn) ∈ Ob(M(a; b)× EΣn). Similarly, for a 2-cell α : f → g in M(⟨a⟩; b), η(α) = (α, 1idn) ∈ M(⟨a⟩; b)× EΣn((f, idn)), (g, idn). Next, we define the components of the pseudo symmetry isomorphisms. For σ, τ ∈ Σn we will denote from here on by Eτσ the unique arrow σ → τ in EΣn. For σ ∈ Σn, ⟨a⟩ ∈ Ob(M)n, and b ∈ Ob(M) we need to define a natural isomorphism ησ,⟨a⟩,b : (η ◦ σ) → (σ ◦ η) that fits in the following diagram M(⟨ ηa⟩; b) M(⟨a⟩; b)× EΣn ∼= σ σ×σ ησ,⟨a⟩,b M(⟨a⟩σ, b) η M(⟨a⟩σ; b)× EΣn. 28 The isomorphism ησ,⟨a⟩,b is defined for every 1-cell f : ⟨a⟩ → b as the 2-cell η = (1 , Eσσ;f fσ id) : (fσ, idn) → (fσ, σ). Lemma 2.2.2. Let M be a Cat-multicategory, then ηM : M → M×EΣ∗ is pseudo symmetric. Proof. We start from a non symmetric multifunctor η : M → M × EΣ∗ that is the identity on the first coordinate and the multicategorical unit in the second co- ordinate. As a non symmetric multifunctor, η preserves units and γ composition. We need to show that η is a pseudo symmetric Cat-multifunctor. The natural- ity of ησ;f follows from the commutativity of the following diagram for any 2-cell α : f → g: (1fσ ,E σ id ) (fσ, id ) nn (fσ, σ) (ασ,1idn ) (ασ,1σ) (gσ, idn) (gσ, σ). (1gσ ,Eσid )n Next we focus on the coherence axioms. The unit permutation axiom (2.1.10) holds since, for all ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), and f : ⟨a⟩ → b, η idnidn;f = (1f idn , Eid ) = (1f , 1n idn) = 1(f,idn) = 1η(f). Let ⟨a⟩, b and f be as before, the product permutation axiom (2.1.11) holds again by definition. Indeed, for τ, σ ∈ Σn, we have η στ στ τστ ;f = (1fστ , Eid ) = (1fστ , Eτ ) ◦ (1fστ , Eid ) = (ηn σ;fτ) ◦ ητ ;fσ. For Top Equivariance (2.1.12), suppose that c ∈ Ob(M), n ≥ 1, ⟨b⟩ = ⟨b ⟩nj j=1 ∈ Ob(M k)n, kj ≥ 0 for 1 ≤ j ≤ n, ⟨aj⟩ = ⟨a jj,i⟩i=1 ∈ Ob(M)kj for 1 ≤ j ≤ n, σ ∈ Σn, f ∈ Ob(M(⟨b⟩; c)), and gj ∈ Ob(M(⟨aj⟩; bj)). We have that γ(η σσ;f ; ⟨1i(g )⟩) = (γ((1σ(j) fσ, Eid); ⟨(1g ( , 1id )⟩)σ(j) kσ(j)id ))k = ((γ(1fσ; 1 ), γ Eσ ;)E σ(j)gσ(j) id idkσ(j) σ⟨idk ⟩ = 1 σ(j)γ(f ;⟨g , Eσ(j)⟩) id⟨idk ⟩σ(j) 29 ( ) σ⟨idk ⟩ = 1 σ(j)γ(f ;⟨gj⟩)σ⟨idk ⟩, Eσ(j) idk = ησ⟨idk ⟩;γ(f ;⟨gj⟩).σ(j) For Bottom Equivariance, let c, n, ⟨b⟩, kj for 1 ≤ j ≤ n, ⟨aj⟩ for∑1 ≤ j ≤ n, f and gj be as above and let τj ∈ Σk for 1 ≤ j ≤ n. We also let k = n k . Bottomj j=1 j Equivariance (2.1.13)(for i is ) ( ) γ 1if ; ⟨ητ ;g ⟩ = ( τγ (1f , 1idn); ⟨(1 , E jj j gjτj id )⟩k)j τ = (γ(1f ; 1g ), 1 ⟨E j ⟩jτj idn )idkj id ⟨τ ⟩ = (1 , E n jγ(f ;⟨gjτj⟩) idk ) idn⟨τj⟩= 1γ(f ;⟨g ⟩)idn⟨τ , Ej j⟩ idk = ηid⟨τj⟩,γ(f ;⟨gj⟩). Thus, we conclude that η : M → M×EΣ∗ is a pseudo symmetric Cat-multifunctor. ■ Recall that j : Cat-Multicat → Cat-Multicatps denotes the inclusion func- tor. We are ready to present a proof of 1.0.1. Theorem 2.2.3. Let M and N be a Cat-multicategories and F : M → N a pseudo symmetric Cat-multifunctor. There exists a unique symmetric Cat-multifunctor ϕ(F ) : M× EΣ∗ → N such that the following diagram commutes: M× EΣ∗ ηM jϕ(F ) M N . F That is, F = jϕ(F ) ◦ η in Cat-MulticatpsM . Proof of Theorem 1.0.1. For uniqueness, suppose that ϕ(F ) : M × EΣ∗ → N is a symmetric Cat-multifunctor satisfying F = (jϕ(F ))◦η. We will abuse notation and write jϕ(F ) = ϕ(F ). We will prove there is a unique way of defining ϕ(F ). At the level of the objects of the multicategory we must have ϕ(F )(a, ∗) = ϕ(F ) ◦ η(a) = F (a) for each a ∈ Ob(M). Next, we show that there is a unique way of defining each component functor of ϕ(F ). For this let ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), and 30 consider the functor ϕ(F ) : M(⟨a⟩; b) × EΣn → N (⟨Fa⟩;Fb). If f : ⟨a⟩ → b is a 1-cell and σ ∈ Σn, we must have that ϕ(F )(f, σ) = ϕ(F )((fσ−1, idn)σ) = ϕ(F )((fσ−1, idn))σ = ϕ(F ) ◦ η(fσ−1)σ = F (fσ−1)σ, (2.2.1) where in the second equality we used that ϕ(F ) is symmetric. So the values of the component functors of ϕ(F ) on n-ary 1-cells are uniquely determined by F . In ex- actly the same fashion, for ⟨a⟩, b and σ as before, f, g : ⟨a⟩ → b, and α : f → g a 2-cell, ϕ(F )(α, 1 ) = F (ασ−1σ )σ. (2.2.2) Finally, if f, σ are as before and τ ∈ Σn, we get that ϕ(F )(1 , Eτ τσ −1 f σ) = ϕ(F )(1fσ−1σ,Eid σ) −1 = ϕ(F )((1 τσfσ−1 , Eid ))σ = ϕ(F )(ητσ−1;fτ−1)σ = (ϕ(F ) ◦ ητσ−1;fτ−1)σ = (Fτσ−1;fτ−1)σ. (2.2.3) We have used the definition of composition of pseudo symmetric Cat-multifunctors (2.1.14) where we see ϕ(F ) trivially as a pseudo symmetric functor. For ⟨a⟩, b, f, g, α, σ, and τ as before, we can write the morphism (α : f → g, Eτσ) in M(⟨a⟩; b) × Σn as (1y, Eτσ) ◦ (f, 1σ). Since both ϕ(F )(1 , Eτy σ) and ϕ(F )(f, 1σ) are uniquely deter- mined by F , we conclude that the component functors of ϕ(F ) are uniquely deter- mined. We have proven the uniqueness of ϕ(F ). Next we prove the existence of ϕ(F ). By uniqueness, we have no choice but to de- fine ϕ(F )(b, ∗) = Fb for any b ∈ Ob(M). Likewise, for ⟨a⟩ ∈ Ob(M)n and b ∈ Ob(M), uniqueness forces the definition of the component functor ϕ(F ) : M(⟨a⟩; b)× Σn → N (⟨Fa⟩; b). For f : ⟨a⟩ → b, a 1-cell in M(⟨a⟩; b) and σ ∈ Σn we define ϕ(F )(f, σ) = F (fσ−1)σ (2.2.4) as in (2.2.1). For a 2-cell α : f → g in M(⟨a⟩; b)(f, g), we define ϕ(F )(α, 1σ) = F (ασ −1)σ (2.2.5) 31 as in (2.2.2). For τ ∈ Σn we define ϕ(F )(1 , Eτf σ) = (Fτσ−1;fτ−1)σ (2.2.6) as in (2.2.3). We still have to prove that ϕ(F ) : M(⟨a⟩; b)×Σn is well defined and extend our def- inition to all 2-cells. Notice that for a 1-cell f : ⟨a⟩ → b our definition is ambiguous for the identity arrow (1f , 1σ) since both (2.2.5) and (2.2.6) apply. However, ϕ(F ) is well defined in this case since F is a functor componentwise and so, it preserves identities. Explicitly, F (1 σ−1f )σ = F (1fσ−1)σ = 1F (fσ−1)σ = 1F (fσ−1)σ, and (Fσσ−1,fσ−1)σ = Fid ,fσ−1σ = 1F (fσ−1)σ = 1n F (fσ−1)σ. So, our definition is so far unambiguous and ϕ(F ) preserves identities. We go on to extend the definition of ϕ(F ) to the rest of the arrows. For α : f → g 2-cell in M(⟨a⟩, b) and σ, τ in Σn, we define ϕ(F )(α,Eτσ) : F (fσ−1)σ → F (gτ−1)τ by ϕ(F )(α,Eτσ) =ϕ(F )(1g, E τ σ) ◦ ϕ(F )(α, 1σ) =ϕ(F )(α, 1 ττ ) ◦ ϕ(F )(1f , Eσ). (2.2.7) The last equality together with the preservation of identities already proven implies that our definition is unambiguous. This equality holds since, ϕ(F )(1 , Eτg σ) ◦ ϕ(F )(α, 1σ) = ((F −1τσ−1;gτ−1)σ ◦ F (ασ )σ = (F − )1τσ−1;gτ−1 ◦ F (ασ ) σ = F (ατ− ) 1)τσ−1 ◦ Fτσ−1;fτ−1 σ = F (ατ−1)τ ◦ (Fτσ−1;fτ−1)σ = ϕ(F )(α, 1τ ) ◦ ϕ(F )(1 τf , Eσ). The third equality holds since it is precisely the commutativity of the following dia- gram: F − −1 −1F (fτ 1τσ−1 τσ ;fτ ) F (fτ−1)τσ−1 F (ατ−1τσ−1) (Fατ−1)τσ−1 (2.2.8) F (gτ−1τσ−1) F (gτ−1)τσ−1. Fτσ−1;gτ−1 32 This diagram commutes since it is an instance of the pseudo symmetry naturality coherence axiom for F, (2.1.9). Next, we check that the defined assignments give a functor ϕ(F ) : M(⟨a⟩; b) × EΣn → N (⟨Fa⟩; b). The fact that ϕ(F ) preserves identities was already proven. We prove functoriality in the second variable first. For f : ⟨a⟩ → b 1-cell, σ, τ, and ρ in Σn, ϕ(F )(1f , E ρ τ τ ) ◦ ϕ(F )(1f , Eσ) = ((Fρτ−1;fρ−1τ) ◦ (Fτσ−1;fτ−1σ) ) = (F ) τσ−1ρτ−1;fρ−1 ◦ Fτσ−1;fτ−1 σ = (Fρσ−1;fρ−1)σ = ϕ(F )(1f , E ρ σ). (2.2.9) Here the third equality holds by (2.1.11), which implies the commutativity of the following diagram: F (fρ−1ρτ−1)τσ−1 Fτσ−1;fρ−1ρτ−1 (Fρτ−1;fρ−1 )τσ −1 (2.2.10) F (fρ−1ρτ−1τσ−1) F (fρ−1)ρτ−1τσ−1. Fρτ−1τσ−1;fρ−1 On the other hand, if α : f → g and β : g → h are 2-cells in M(⟨a⟩; b), and σ ∈ Σn we have that ϕ(F )(β, 1σ) ◦ ϕ(F )(α, 1σ) = ϕ(F )(βα, 1σ). (2.2.11) The functoriality of ϕ(F ) follows from a straightforward argument by eqs. (2.2.9) and (2.2.11) together with the exchange property (2.2.7). The next step is to prove that the component functors give rise to a symmetric Cat-multifunctor ϕ(F ) : M × EΣ∗ → N . First, notice that ϕ(F ) preserves units since, for a ∈ Ob(M) ϕ(F )(1a, id1) = F (1 id−1a 1 )id1 = F (1a) = 1Fa, since F it- self preserves units. Next we prove that ϕ(F ) preserves the Σn-action. For n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), and σ ∈ Σn, we show that the following diagram commutes in Cat : M ⟨ ϕ(F )( a⟩; b)× EΣ //n N (⟨Fa⟩;Fb) σ σ   M(⟨aj⟩σ; b)× EΣ //n N (⟨Fa⟩σ;Fb). ϕ(F ) 33 For this we don’t need any of the pseudo symmetry axioms for F. For 1-cells (f : ⟨a⟩ → b, τ) of M(⟨a⟩; b)× EΣn, ϕ(F )(f, τ)σ = (F (fτ−1)τ)σ = F (fτ−1)τσ = F (fσ(τσ)−1)τσ = ϕ(F )((fσ, τσ))) = ϕ(F )((f, τ)σ). A similar calculation works for 2-cells of the form (α : f → g, 1τ ) in M(⟨a⟩; b) × EΣn. For morphisms of the form (1f , Eρτ ) in M(⟨a⟩; b)× EΣn, (ϕ(F )(1f , E ρ τ ))σ = (Fρτ−1;fρ−1τ)σ = Fρτ−1;fρ−1(τσ) = Fρσ(τσ)−1;fσ(ρσ)−1(τσ) = ϕ(F )(1fσ, E ρσ τσ) = ϕ(F )((1f , E ρ τ )σ). By functoriality of ϕ(F ) and σ we conclude that ϕ(F ) preserves the action of the symmetric group. The only step we are missing to finish proving that ϕ(F ) defines a Cat-multifunctor is the preservation of γ. Let c ∈ Ob(M), n ≥ 0, ⟨b⟩ ∈ k ∑Ob(M)n, kj ≥ 0 for 1 ≤ j ≤ n, ⟨a ⟩ = ⟨a ⟩ jj j,i i=1 for 1 ≤ j ≤ n. Set k = n j=1 kj. As usual ⟨a⟩ = ⟨aj⟩ = ⟨⟨ k a j nj,i⟩i=1⟩j=1 denotes the concatenation of the aj’s. We will prove that the following square is commutative: ∏ ∏ϕ(F )× ϕ(F ) ∏ M(⟨b⟩; c)× EΣ nn × j=1 M(⟨aj⟩; bj)× EΣk N (⟨Fb⟩;Fc)× n j=1 N (⟨Faj⟩;Fbj j) γ γ (2.2.12) M(⟨a⟩; c)× E(Σk) N (⟨Fa⟩;Fc). ϕ(F ) The commutativity of this diagram at the level of 1-cells will follow from top and bottom equivariance for M and Σ∗, as well as the fact that F preserves γ. Let f : ⟨b⟩ → c, σ ∈ Σn, and gj : ⟨aj⟩ → bj and τj ∈ Σk for 1 ≤ j ≤ n. We have thatj γ(ϕ(F )(f, σ), ⟨ϕ(F )(gj, τj)⟩) = γ(F (fσ−1)σ, ⟨F (gjτ−1j )τj⟩) 34 ( 〈 ( )〉) = γ ((F((fσ−1),〈 F gσ−1(j)τ−〉1σ−)1()j) σ⟨τj⟩ = F (γ fσ−1, g −1σ−1(j)τσ)−1(j) σ⟨τj⟩ = F γ(f, ⟨gj⟩)(σ⟨τ ⟩)−1j σ⟨τj⟩ = ϕ(F )(γ(f, ⟨gj⟩), σ⟨τj⟩) = ϕ(F )(γ((f, σ), ⟨gj, τj⟩)). We have proven that our diagram is commutative at the level of 1-cells. For the morphisms we will consider again morphisms that change the first variable only and morphisms that change the second variable only separately. For 2-cells that change the first variable only, the commutativity of our diagram follows in the same way as it did for 1-cells. We consider two cases for 2-cells that change the second variable. For 2-cells of the form ((1f , Eτσ), ⟨1g , 1ρ ⟩) where f : ⟨b⟩ →j j c, σ, τ ∈ Σn, and gj ∈ Ob((M(⟨aj⟩; bj))〈and ρj ∈ Σk fo〉r)1 ≤ j ≤ n, we have thatj γ (ϕ(F )(1 τf , Eσ) 〈ϕ(F )(1g , 1j 〉ρj)) =γ ((Fτσ−1;fτ−1)〈σ, 1F (g ρ−1j j )ρj 〉) =γ Fτσ− ( )1;fτ−1 , 1F g ρ−1 σ⟨ρj⟩σ−1(j) σ−1(j) =F 〈 〉 ( 〈 〉)σ⟨ρj⟩ τσ−1 id −1 −1k ;γ fτ g −1 σ−1(j) τ (j) ρ τ−1(j) =Fτ⟨ρj⟩(σ⟨ρ ⟩)−1j ;γ(f,⟨gj⟩)(τ⟨ρ ⟩)−1σ⟨ρj j⟩ τ⟨ρ ⟩ =ϕ(F )(1 jγ(f,⟨g , E )j⟩) σ⟨ρj⟩ =ϕ(F )(γ(1f , ⟨1g ⟩), γ(Eτσ , ⟨1j ρ ⟩)).j The above equalities follow from our definitions, top and bottom equivariance in M,N , and EΣ∗ except the third equality which follows from top equivariance for F (2.1.12). Next, let’s consider two cells of the form ν((1f , 1σ), ⟨1g , E jj ρj ⟩) where f : ⟨b⟩ → c, σ ∈ Σn, and gj ∈ Ob(M(⟨aj⟩; bj)) and ρj, νj ∈ Σk for 1 ≤ j ≤ n.j We get that ( 〈 〉) γ (ϕ(F )(1f , 1(σ), ϕ(F ) (1g), E)νjρ )j j =γ (1F (fσ−1)σ,〈 Fν ρ−1;g ν−1 ρj jj j j 〉) =γ 1F (fσ−1), Fν −1 −1 σ⟨ρ−1 j⟩σ (j)ρ −1 ;g −1 νσ (j) σ (j) σ−1(j) 35 =F 〈 〉 ( 〈 〉)σ⟨ρ ⟩ id ν ρ−1 j n σ−1(j) −1 ;γ fσ −1, g ν−1σ−1σ (j) (j) σ−1(j) =Fσ⟨ν ⟩((σ⟨ρ ⟩)−1;γ(f〈,⟨g ⟩)(ρ⟨ν〉⟩))−1σ⟨ρ ⟩j j j j j =ϕ(F )(1 ( , E〈σ(⟨νj⟩γ(f,⟨gj⟩) σ⟨ρj⟩ )〉)) =ϕ(F ) γ (1 νjf , 1σ), 1g , Eρ .j j The third equality above follows from the bottom equivariance axiom for F (2.1.13) and the rest by our definitions as well as top and bottom equivariance for M,N , and EΣ∗. By functoriality of γ and ϕ(F ), and since every morphism in the source category can be written as a composite of arrows for which we already proved the commuta- tivity of (2.2.12), we can conclude that the square (2.2.12) is commutative. We are almost done, we just have to prove that our definition of ϕ(F ) gives us F = ϕ(F ) ◦ η in Cat-Multicatps. This is clear for objects of the multicategory M. For each n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ Ob(M), and f : ⟨a⟩ → b, ϕ(F ) ◦ η(f) = ϕ(F )(f, id ) = F (f id−1n n )idn = F (f). Similarly for α : f → g a 2-cell in M(⟨a⟩; b). Finally, we just need to prove that (ϕ(F ) ◦ η)σ,⟨a ⟩,b = Fσ,⟨a ⟩,b for any σ ∈ Σn. Let f : ⟨a⟩ → b be a 1-cell. Since ϕ(F ) isi i symmetric, (ϕ(F )η)σ;f = ϕ(F )(ησ;f ) = ϕ(F )(1fσ, Eidσ) = Fσ(id)−1;fσσ−1 = Fσ;f , where we have used the notation introduced just before (2.1.9). We have proven that jϕ(F ) ◦ η = F . This finishes our proof. ■ Similarly, pseudo symmetric Cat-multinatural transformations between F and G correspond to symmetric Cat-multinatural transformations between ϕ(F ) and ϕ(G). Lemma 2.2.4. Let M,N be Cat-multicategories with F,G : M → N pseudo sym- metric Cat-multifunctors and θ : F → G a pseudo symmetric Cat-multinatural transformation. There exists a unique symmetric Cat-multinatural transformation ϕ(θ) : ϕ(F ) → ϕ(G) such that ϕ(θ) ∗ 1 = θ in Cat-MulticatpsηM . That is, the following pasting diagram equality holds in Cat-Multicatps : 36 F M Fθ N M N G = ϕ(F ) ηM ϕ(G) ηM ϕ(θ) ϕ(G) M× EΣ∗ M× EΣ∗. Proof. We prove uniqueness first. Suppose ϕ(θ) is a symmetric Cat-multinatural transformation ϕ(θ) : ϕ(F ) → ϕ(G) such that ϕ(θ) ∗ 1η = θ. Any object of M×EΣ∗ takes the form (a, ∗) for some object a of M, with i(a) = (a, ∗). By definition, θa = γ(ϕ(θ)ηa, ϕ(F )((1η)a)) = γ(ϕ(θ)ηa, 1Fa)) = ϕ(θ)ηa. Since all objects of the Cat-multifunctor M× EΣ∗ are of the form ηa for some ob- ject a of M, this is the only possible way of defining such Cat-multinatural trans- formation ϕ(θ). Next, we check that by defining ϕ(θ)(a,∗) = θa for a ∈ Ob(M), we in fact get a symmetric Cat-multinatural transformation ϕ(θ) : ϕ(F ) → ϕ(G). Let n ≥ 0, ⟨a⟩ ∈ Ob(M)n, b ∈ (Ob(M)n), f : ⟨a⟩(→ b, and σ 〈∈ Σ〉n), then γ(ϕ(G)(f, σ); ⟨ϕ(θ) ⟩) = γ (G(fσ−1(a )σ; θj ,∗) 〈 aj 〉) = γ G(fσ−1); θa σσ−1(j) = γ(θ −1b;F (fσ ))σ = γ((θb;F (fσ−1)σ) ) = γ ϕ(θ)(b,∗), ϕ(F )(f, σ) Where we have used top and bottom equivariance, as well as the Cat-multinaturality of θ. Now we need to prove Cat-multinaturality of ϕ(θ) for 2-cells. As before, the case where the 2-cell changes just the first variable is similar to what was done for 1-cells. Now, if ⟨a⟩, b, f are as before and Eτσ is a morphism in EΣ τn, (1f , Eσ) is a morphism in M( (⟨a⟩; b)× EΣ〈n, and 〉) ( ) γ ϕ(G)(1f , E τ σ); 1ϕ(θ) ∗ = γ(aj, ) ((Gτσ−1;fτ−1)〈σ; ⟨1θa ⟩j 〉) = γ Gτσ−1;fτ−1 ; 1θa σ σ−1(j) = γ ((1θ ;Fτσ−1;fτ−1)σb ) = γ 1 τϕ(θ) ∗ ;ϕ(F )(1f , E(b, ) σ) . In the third equality we have used pseudo symmetric Cat-multinaturality for θ. In conclusion, by componentwise functoriality of γ, ϕ(F ) and ϕ(G) we conclude 37 that Cat-multinaturality holds for ϕ(θ) at the 2-cell level finishing the proof of the lemma. ■ Furthermore, Theorem 2.2.3 and Lemma 2.2.4 together give the following iso- morphism. Corollary 2.2.5. If M,N are Cat multicategories, then there is an isomorphism of small categories Cat-Multicatps(M,N ) ∼= Cat-Multicat(M× EΣ∗,N ). Proof. Recalling the definitions from the two previous results, we define ϕ : Cat-Multicatps(M,N ) → Cat-Multicat(M× EΣ∗,M) (2.2.13) for pseudo symmetric Cat-multifunctors as in Theorem 2.2.3 and for pseudo sym- metric Cat-multinatural transformations as in Lemma 2.2.4. It is immediate from the definitions that ϕ is a functor. Indeed, if α : F → G and β : G → H are pseudo symmetric Cat-multinatural transformations with F,G,H : M → N ϕ(β ∗ α)(c,∗) = (β ∗ α)c = γ(βc, αc) = γ(ϕ(β)(c,∗), ϕ(α)(c,∗)) = (ϕ(β) ∗ ϕ(α))(c,∗) We can define the inverse of ϕ, η∗, as the composite j Cat-Multicat(M× EΣ∗,N ) Cat-Multicatps(M× EΣ∗,N ) η∗ (2.2.14) ∗ Mη Cat-Multicatps(M,N ). Finally, the existence part of Theorem 2.2.3 and Lemma 2.2.4, implies that η∗ ◦ ϕ is the identity of Cat-Multicatps(M,N ), while the uniqueness part of both results implies that ϕ ◦ η∗ is the identity of Cat-Multicat(M× EΣ∗,N ). ■ The two previous results hint at the existence of a 2-adjunction between the 2-inclusion j : Cat-Multicat → Cat-Multicatps and the 2-functor which we define next. Definition 2.2.6. We define the 2-functor ψ : Cat-Multicatps → Cat-Multicat as follows. For a Cat-multicategory M, ψM = M×EΣ∗. For M,N Cat-multicategories, we define the component functor ψ as the composite 38 η Cat- NMulticatps(M,N ) ∗ Cat-Multicatps(M,N × EΣ∗) ϕ ψ Cat-Multicat(M× EΣ∗,N × EΣ∗). Thus, by Theorem 2.2.3 if F : M → N is a pseudo symmetric Cat-multifunctor, then ψF : M× EΣ∗ → N × EΣ∗ is the unique symmetric Cat-multifunctor which makes the diagram M ηM M× EΣ∗ F jψF (2.2.15) N η N × EΣN ∗ commute in Cat-Multicatps. Similarly, by Lemma 2.2.4, for θ : F → G a pseudo symmetric Cat-multinatural transformation between F,G : M → N pseudo sym- metric Cat-multifunctors, ψθ : ψF → ψG is the unique symmetric Cat-multinatural transformation such that the equality of pasting diagrams M ηM M× ηMEΣ∗ M M× EΣ∗ θ jψθ F G jψF = F jψF jψG (2.2.16) N η N × EΣ∗ N η N × EΣN N ∗ holds in Cat-Multicatps. Theorem 2.2.7. There is a 2-adjunction ψ Cat-Multicatps ⊥ Cat-Multicat j where j is the inclusion 2-functor. Proof. Following Corollary 2.2.5, we define the unit of the adjunction as the strict 2-natural transformation η : 1Cat-Multicatps → jψ having component ηM at a Cat- multicategory M. We also define the counit of the adjunction π : ψj → 1Cat-Multicat as having component at a Cat-multicategory M the projection πM : M × EΣ∗ → M. The fact that η defines a strict 2-natural transformation follows directly from (2.2.15) and (2.2.16). To prove that the data of π defines a strict 2-natural transformation 39 we need to prove that given F : M → N symmetric Cat-multifunctor, the following diagram commutes: M× πEΣ M∗ M ψjF F N × EΣ∗ π N .N Indeed, we prove that ψjF = F × 1EΣ∗ . By (2.2.15), it suffices to show that the following diagram commutes in Cat-Multicatps: M ηM M× EΣ∗ jF j(F×1) (2.2.17) N η N × EΣ .N ∗ It is clear that this diagram commutes at the level of objects, 1-cells, and 2-cells of the multicategory. The pseudo symmetry isomorphisms of both composites also agree. Indeed, for f : ⟨a⟩ → b a 1-cell of M and σ ∈ Σn, by (2.1.14), we get that (j(F × 1)ηM)σ;f =j(F × 1)σ;ηM(f) ◦ j(F × 1)(ηMσ;f ) = (1 σ(Ff)σ, 1σ) ◦ (1(Ff)σ, Eid) = (1(Ff)σ, E σ id) ◦ (1(Ff)σ, 1σ) = ηN σ;Ff ◦ ηN (jFσ;f ) = (ηN ◦ jF )σ;f . To finish proving the 2-naturality of πM, we need to prove that given M,N Cat- multicategories, F,G : M → N Cat-multifunctors and a Cat-multinatural transfor- mation θ : F → G, the following equality of pasting diagrams holds in Cat-Multicat: M× πEΣ M∗ M M× π EΣ M∗ M ψjθ j(F×1) j(G×1) G = j(F× θ1) F G N × EΣ∗ π N N × EΣ∗ π N .N N In turn, the last equality of pasting diagrams holds since ψjθ = j(θ × 1). To see this, by (2.2.16), we must show the following equality of pasting diagrams in Cat-Multicatps : M ηM M× EΣ∗ M ηM M× EΣ∗ jθ j(θ×1) jF jG j(G×1) = jF j(F×1) j(G×1) (2.2.18) N η N × EΣ∗ N ηN N × EΣ . N ∗ 40 To check that this equality holds l(et a ∈ Ob(M). W) e get, by (2.1.8), that (1ηN ∗ jθ)a = γ 1ηN (jGa); ηN (θa) = γ ((1Ga, 1id); (θa, 1id)) = γ (((θa, 1id); (1Fa, 1id)) ) = γ j(θ × 1)ηN (a); j(F × 1)(1ηM(a)) = (j(θ × 1) ∗ ηM)a. Thus, η and π are strict 2-natural transformations and we just need to prove that they satisfy the triangle identities. To prove that the identity (1j ∗ π)(η ∗ 1j) = 1j holds we need to prove that for M a Cat-multicategory the diagram M× EΣ∗ ηM jπM M M 1M commutes in Cat-Multicatps. This is clear at the level of objects, n-ary 1-cells and n-ary 2-cells. The pseudo symmetry isomorphisms of both pseudo symmetric Cat-multifunctors also agree since, for f : ⟨a⟩ → b an n-ary 1-cell of M and σ ∈ Σn, we obtain, by (2.1.14), ((jπM) ◦ ηM)σ;f = (jπM)σ;ηM(f) ◦ jπM(ηMσ;f ) = 1fσ = 1Mσ;f . The other triangle identity is (π ∗ 1ψ)(1ψ ∗ η) = 1ψ. To check it, we must prove that, given a Cat-multicategory M, the composite M× ψηMEΣ∗ M× EΣ∗ × πM×EΣ EΣ ∗∗ M× EΣ∗ agrees with 1M×EΣ∗ . This holds since, if ∆: EΣ∗ → EΣ∗ × EΣ∗ denotes the diago- nal map, then ψ(ηM) = 1M × ∆. To see this, notice that by (2.2.15) all we need is to prove that the following diagram is commutative: M ηM M× EΣ∗ ηM j(1×∆) (2.2.19) M× EΣ∗ η M× EΣ × EΣ .M×EΣ∗ ∗ ∗ Now, the previous diagram is evidently commutative at the level of objects, 1-cells, and 2-cells. The diagram also commutes at the level of pseudo symmetry isomor- phisms since, for f : ⟨a⟩ → b an n-ary 1-cell in M and σ ∈ Σn, (ηM×EΣ∗ ◦ ηM)σ;f = ηM×EΣ∗σ;ηM(f) ◦ ηM×EΣ∗(ηMσ;f ) 41 = (1fσ, 1σ, E σ id) ◦ (1 σfσ, Eid, 1id) = (1fσ, 1 , 1 ) ◦ (1 , Eσ , Eσσ σ fσ id id) = j(1×∆)σ;ηM(f) ◦ j(1×∆)(ηMσ;f ) = (j(1×∆) ◦ ηM)σ;f . We conclude that the triangle identities are satisfied and thus we get the desired 2-adjunction. ■ We can use this 2-adjunction to describe the 2-category Cat-Multicatps in terms of symmetric Cat-multifunctors and symmetric Cat-multinatural transfor- mations alone, thus upgrading the functors ϕ from Corollary 2.2.5 to an isomor- phism of 2-categories. Definition 2.2.8. The 2-category D has Cat-multicategories as objects. For M,N Cat-multicategories, the category of morphisms between M and N is D(M,N ) = Cat-Multicat(M× EΣ∗,N ). In particular, vertical composition of 2-cells is defined as in Cat-Multicat. For F : M × EΣ∗ → N and G : N × EΣ∗ → Q symmetric Cat-multifunctors, the composition G ◦ F is defined as the composite M× 1×∆EΣ∗ M× F×1 EΣ∗ × EΣ∗ N × EΣ G∗ Q in Cat-Multicat. Similarly, for F, J : M × EΣ∗ → N , G,K : N × EΣ∗ → Q symmetric Cat-multifunctors and θ : F → J, ζ : G → K Cat-multinatural transfor- mations, ζ ∗ θ is defined as the pasting F×1 G M× 1×∆EΣ∗ M× EΣ∗ × EΣ∗ θ×1 N × EΣ∗ ζ Q J×1 K in Cat-Multicat. The previous definition makes D into a 2-category and the functors ϕ, and η∗ from Corollary 2.2.5 into the components of isomorphisms of 2-categories. Theorem 2.2.9. The data of the previous definition defines a 2-category D iso- morphic to Cat-Multicatps. 42 Proof. The (horizontal) composition functors are defined so that ϕ and η∗ become the componentwise functors of a 2-category isomorphism between D and Cat- Multicatps. More precisely, for M,N and Q Cat-multicategories, we will prove that the D composition functor defined, ◦′ : D(N ,Q) × D(M,N ) → D(M,Q), makes the following diagram commute, where ◦ denotes the horizontal composition functor of Cat-Multicatps : ◦′ D(N ,Q)×D(M,N ) D(M,Q) η∗×η∗ ϕ Cat-Multicatps(N ,Q)×Cat-Multicatps(M,N ) ◦ Cat-Multicatps(M,Q). (2.2.20) Let G : N × Q and F : M × EΣ∗ → N be symmetric Cat-multifunctors. The commutativity of (2.2.20) for (G,F ) reduces to the commutativity of the following diagram by Theorem 2.2.3: M ηM M× EΣ∗ ηM j(1×∆) M× ηM×EΣEΣ ∗∗ M× EΣ∗ × EΣ∗ jF j(F×1) N ηN N × jGEΣ∗ Q. This diagram in turn is commutative by (2.2.17) and (2.2.19). Now, if F,G are as before, J : M× EΣ∗ and K : N × EΣ∗ → Q are symmetric Cat-multifunctors, and θ : F → J, ζ : G → K are Cat-multinatural transformations, by Lemma 2.2.4, the commutativity of (2.2.20) for (ζ, θ) reduces to the equality of pasting diagrams: M ηM M× ηMEΣ∗ M M× EΣ∗ ηM j(1×∆) ηM j(1×∆) M× ηM×EΣEΣ ∗∗ M× ηM×EΣ EΣ∗ × EΣ∗ = M× EΣ ∗∗ M× EΣ∗ × EΣ∗ j(θ×1) jθ jF j(F×1) j(J×1) jF jJ j(J×1) N ηN N × N ηNEΣ∗ N × EΣ∗ jζ jζ jG jK jG jK Q Q. 43 This equality holds by (2.2.18) and makes implicit use of (2.2.17) and (2.2.19). We can thus define ϕ : Cat-Multicatps → D in objects as the identity map, and do the same for η∗ : D → Cat-Multicatps, with the component functors given for M and N multicategories by (2.2.13) and (2.2.14) respectively. By (2.2.20) and the fact that ϕ and η∗ are componentwise isomorphisms, ϕ and η preserve vertical composition of 2-cells and horizontal composition of 1-cells and 2-cells. The fact that Cat-Multicatps is a 2-category implies that D is a 2-category. This further turns ϕ and η∗ into isomorphisms of 2-categories. ■ 2.3 Applications to inverse K-heory We use our understanding of pseudo symmetric Cat-multifunctors to show that they preserve certain En-algebras for n = 1, 2, 3, ...,∞. First we define En Cat-operads. Definition 2.3.1. For n = 1, ...,∞, an En Cat-operad is a Cat-operad that be- comes a topological En-operad (in the sense of [May72]) after applying the classify- ing space functor. A topological En-operad is one that has the same Σ-equivariant homotopy type as the little n-cubes operad. Example 2.3.2. An example of an E∞ Cat-operad is EΣ∗. There are also ex- amples of En Cat-operads for each n = 1, 2, . . . in [Ber96] and [BFSV03], which furthermore have a free action of the symmetric group. Importantly, symmetric algebras over topological En-operads are grouplike n-fold loop spaces. Symmetric algebras over the En Cat-operads in [BFSV03] are n-fold monoidal categories, with the group completion of the classifying space of an n-monoidal category being an example of an n-fold loop space. Definition 2.3.3. Let M be a Cat-multicategory and O a Cat-operad. A pseudo symmetric algebra in M over O is a pseudo symmetric Cat-multifunctor O → M. For n ∈ {1, 2, . . . ,∞}, a symmetric En-algebra (respectively a pseudo symmetric En-algebra) in M is a symmetric algebra (respectively a pseudo symmetric algebra) over an En-operad. Lemma 2.3.4. . 1. Let O be a Σ-free En Cat-operad. Then O × EΣ∗ is an En Cat-operad. 44 2. Pseudo symmetric En-algebras over Σ-free En Cat-operads are symmetric En-algebras for n = 1, 2, . . . ,∞. Proof. Let O be a Σ-free Cat-operad. We will show that O × EΣ∗ is componen- twise Σ-equivariantly homotopy equivalent to O (after taking nerves), that is, for each n ≥ 0, we will show that the projection O(n) × EΣn → O(n) induces a Σn equivariant homotopy equivalence on classifying spaces. Since B(O(n) × EΣn) and B(O(n)) are Σn-CW complexes we must show that for subgroups H ≤ Σn, the pro- jection induces homotopy equivalences B (O(n)× Σn)H → B (O(n)× Σ Hn) . Since the action of Σn on both O(n)×EΣn and O(n) is free, the fixed point map is either empty when H is non-trivial or the projection B (O(n)) × B (EΣn) → B (O(n)) , which is a homotopy equivalence since B (EΣn) is contractible. ■ Example 2.3.5. If O is Cat-operad and M is a Cat-multicategory, the pseudo symmetric algebras over O agree with symmetric algebras over the operad O×EΣ∗. For example, while algebras over the commutative operad {∗} in M are the com- mutative monoids in M, pseudo symmetric algebras over {∗} in M are precisely algebras over the Barratt-Eccles operad and thus, E∞-algebras. Similarly, pseudo symmetric algebras over the E∞ Cat-operad EΣ∗, which are defined in [Yau24b] as pseudo symmetric E∞-algebras in M, are algebras over EΣ∗ × EΣ∗ = E(Σ∗ × Σ∗) which is still an E∞ Cat-operad, and thus, they are still E∞-algebras in the sense defined above. Thus, we have the following result. Remark 2.3.6. We remind the reader that Σ-freedom is not a serious restriction since there are En-operads in Cat, like those in [Ber96] and [BFSV03] which are free. As a corollary, we conclude that pseudo symmetric Cat-multifunctors preserve certain En-algebras. Corollary 2.3.7. Let M and N be Cat-multicategories and F : M → N be a pseudo symmetric Cat-multifunctor, then: 1. F sends commutative monoids in M to E∞-algebras in N . 2. F preserves En-algebras parameterized by free Cat-operads. We conclude our paper by applying our understanding of pseudo symmetric Cat-multifunctors to multifunctorial inverse K-theory. In [JY22], Johnson and Yau define Mandell’s inverse K-theory multifunctor P as well as the Cat-multicategories 45 that are its domain (Γ-categories) and target (permutative categories). Yau proves in [Yau24b] that P is pseudo symmetric. We refer the interested reader [Yau24b] of which the following theorem is one of the main results. Theorem 2.3.8. [Yau24b] Mandell’s inverse K-theory functor is a pseudo symmet- ric Cat-multifunctor P : Γ-Cat → PermCatsg. As a consequence, P sends commutative monoids to E∞-algebras and preserves En-algebras parameterized by free En-operads, as was stated in Corollary 1.0.2. 46 CHAPTER 3 COHERENCE FOR SYMMETRIC PSEUDO COMMUTATIVE MONADS We will prove a coherence result for symmetric, pseudo commutative, strong 2-monads. In this Chapter, we assume that K is a 2-category with finite products which we will denote by ×, with 1 denoting the empty product in K. We will by ρ : 1 × − → 1K and λ : − × 1 → 1K the natural isomorphisms comming from the monoidal structure in K induced by products. As Hyland and Power, we believe what we do to work as well in a symmetric monoidal 2-category in general. In Section 3.1, we define pseudo commutative, strong 2-monads T : → K fol- lowing [HP02]. We also define the multicategory T -Alg when T is symmetric. This allows us to extend the free T -algebra 2-functor T : K → K to a non-symmetric multifunctor T : K → T -Alg in Section 3.2. We finish by proving that this mul- tifunctor is pseudo symmetric in Section 3.3. This implies that the free functor for each of the pseudo commutative operads defined in [GMMO23] and also considered in [Yau24a] is pseudo symmetric. 3.1 Symmetric pseudo commutative 2-monads Definition 3.1.1. [Koc70] Suppose that T : K → K is a 2-functor. A strength t on T is the data of a (strict) 2-natural transformation (see [YJ21]) with source K ×K 1K×T K ×K × K, and target K ×K × K T K. The component of t at (A,B) ∈ Ob(K × K), will be denoted by tA,B : A × TB → T (A× B) or just t when there is no room for confusion. These data are required to satisfy the following axioms: • Unity: the triangle t 1× 1,ATA T (1× A) Tλ λ TA. 47 commutes for all A ∈ Ob(K). • Associativity: the triangle 1 ×t A× × A B,CB TC A× T (B × C) tA,B×C tA×B,C T (A×B × C) commutes for every A,B ∈ Ob(K). In this case we say that T : K → K is strong with strength t. Remark 3.1.2. Suppose that T : K → K is a strong 2-functor. The following no- tation is introduced in [HP02]. For n ≥ 2, tni will denote the natural isomorphism having as component at (A1, . . . , An) ∈ Ob(Kn), the 1-cell tn A1 × · · · × Ai−1 × TAi × × · · · × i A1,...,AnAi+1 An T (A1 · · · × An) ∼= T∼= A1 × · · · × Ai−1 × Ai+1 × · · · × An × TAi T (A1 × · · · × At n × Ai). We will denote tni A ,...,A = ti when there is no room for confusion. Notice that1 n t = t22. In [HP02], t21 is also called t∗. We will write our arrows in terms of t21 and t22 when possible. We notice that the associativity axiom implies that tni can be writ- ten in many different ways using the tki for k < n. For example, one can prove by induction that the triangle 1A ×···×A ×t1 i−1 1 A1 × · · · ×Ai−1 × TAi ×Ai+1 × · · · ×An A1 × · · · ×Ai−1 × T (Ai × · · · ×An) t n 2ti A1,...,An T (A1 × · · · ×An) commutes, as well as the triangle t2×1Ai+1×···×An A1 × · · · ×Ai−1 × TAi ×Ai+1 × · · · ×An T (A1 × · · · ×Ai)×Ai+1 · · · ×An t1 tni A1,...,An T (A1 × · · · ×An). 48 Definition 3.1.3. Let (T : K → K, η : 1K → T, µ : T 2 → T ) be a 2-monad. That is, T is a strict 2-functor and η, µ are strict 2-natural transformations satisfying the usual triangle identities (see [YJ21]). We say that (T, η, µ, t) is a strong 2-monad with strength t, if T : K → K is strong with strength t as a 2-functor and η, µ and t are compatible in the sense that, for every A,B ∈ Ob(K), the diagram A× 1×ηB A× TB η t T (A×B) commutes, as well as the diagram 1×µ A× T 2B A× TB t t T (A× TB) T 2(A×B) µ T (A×B).Tt Definition 3.1.4. [Koc70] A strong 2-monad (T, η, µ) is called commutative when the diagram TA× t1 TtTB T (A× TB) 2 T 2(A×B) t2 µ T (TA×B) T 2(A×B) T (A×B) Tt µ1 commutes for every A,B ∈ Ob(K). Remark 3.1.5. Suppose that (T, η, µ, t) is a strong 2-monad. Then, T can be re- garded as a monoidal 2-functor in two different ways. In each case, the unitary component is given by η1 : 1 → T1. The binary components are given by the two 1-cells that form the boundary of the previous diagram. For each of these ways of seeing T as a monoidal 2-functor, η is a monoidal 2-natural transformation. It is proven in [Koc70] that T is commutative if and only if T is a monoidal 2-monad (i.e., µ is a monoidal 2-natural transformation). There are a lot of examples of strong 2-monads which are non-commutative, but that are commutative up to coherent natural isomorphism, these are called pseudo commutative monads and we will defined them next. The examples include the 2- monads T : Cat → Cat given by the free construction for symmetric stric monoidal 49 categories, symmetric monoidal categories, categories with finite products, cate- gories with finite coproducts, etc. A longer list is included in [HP02]. More ex- amples come from pseudo commutative operads as defined by Corner and Gurski [CG23]. These are operads whose associated monads are pseudo commutative. Guillou, Merling, May and Osorno [GMMO23] prove that chaotic operads are pseudo commutative. Definition 3.1.6. [HP02, Def. 5] A strong 2-monad (T, η, µ) is called pseudo- commutative with pseudocommutativity Γ if there exists an invertible modification with components × t T tTA TB 1 T (A× TB) 2 T 2(A×B) t2 µ ΓA,B Tt µ T (TA×B) 1 T 2(A×B) T (A×B) such that the following axioms are satisfied. We will write Γ instead of ΓA,B when A and B are clear from the context. 1. ΓA×B,C ◦ (t2A,B × 1TC) = t2A,B×C ◦ (1A × ΓB,C), i.e., the following pasting diagram equality holds: A t ×1 Γ A× TB × TC 2 T (A×B)× TC T (A×B × C) ω ∥ (3.1.1) A 1×Γ A× TB × tTC A× T (B × C) 2 T (A×B × C). A 2. ΓA,B×C ◦ (1TA × t2B,C) = ΓA×B,C ◦ (t1A,B × 1TC), i.e., the following equality holds: A 1×t Γ TA×B × TC 2 TA× T (B × C) T (A×B × C) ∥ ω (3.1.2)A t ×1 Γ TA×B × TC 1 T (A×B)× TC T (A×B × C). ω 50 3. ΓA,B×C ◦ (1TA × t1B,C) = t1A×B,C ◦ (ΓA,B × 1C), i.e., the following whiskering equality holds: A 1×t Γ TA× TB × C 1 TA× T (B × C) T (A×B × C) A ∥ ω (3.1.3) Γ×1 t TA× TB × C T (A×B)× C 1 T (A×B × C). A 4. ΓA,B ◦ (ηA × 1TB) is an identity 2-cell. That is, the following whiskering is an identity: A η×1 Γ A× TB TA× TB T (A×B). (3.1.4) ω 5. ΓA,B ◦ (1TA × ηB) is an identity 2-cell, that is, the following whiskering is an identity: A 1×η Γ TA×B TA× TB T (A×B). (3.1.5) ω 6. The whiskering A 2 × µ×1 Γ T A TB TA× TB T (A×B) ω is equal to the pasting 51 2 T 2A× tTB 1 T (TA× Tt T tTB) 1 T 2(A× TB) 2 T 3(A×B) t TΓ2 Tt2 Tµ 2 × Γ 2 × T 2t Tµ T (T A B) T (TA B) 1 T 3(A×B) T 2(A×B) Tt1 µ µ µ T 2(TA×B) µ T (TA×B) T 2(A×B) µ T (A×B).Tt1 (3.1.6) 7. The whiskering A 1×µ Γ TA× T 2B TA× TB T (A×B) ω is equal to the pasting t T t TA× T 2B 1 T (A× T 2B) 2 T 2(A× TB) t Γ2 µ T (TA× TB) T 2(A× TB) T (A× TB) Tt µ1 Tt T 2t T t (3.1.7)2 2 2 T 2(TA×B) T 3 µ(A×B) T 2(A×B) TΓ T 2t1 Tµ µ T 3(A×B) T 2(A×B) µ T (A×B)Tµ Remark 3.1.7. The fact that the source and target of the equal whiskering and pasting diagrams in the previous list of axioms are the same follows from the defi- nition of 2-strong monad. In other words, the pseudo commutativity axioms don’t introduce new relations among 1-cells. A modification is more than a mere collection of 2-cells (see [JY22]). For Γ to be a modification we need that given f : A → A′ and g : B → B′ in K, the following equality of pasting diagrams holds: 52 Tf×Tg Tf×Tg TA× TB TA′ × TB′ TA× TB TA′ × TB′ t1 t1 t2 t1 t2 t2 T (f×Tg) T (Tf×g) T (A× TB) T (A′ × TB′) T (TA′ ×B′) T (A× TB) T (TA×B) T (TA′ ×B′) Γ Γ Tt2 Tt2 Tt1 = Tt2 Tt1 Tt1 T 2(f×g) T 2(f×g) T 2(A×B) T 2(A′ ×B′) T 2(A′ ×B′) T 2(A×B) T 2(A×B) T 2(A′ ×B′) µ µ µ µ µ µ T (A×B) T (A′ ×B′) T (A×B) T (A′ ×B′) T (f×g) T (f×g) Following Blackwell, Kelly and Power [BKP02], we now define ,for any 2-monad T : K → K, the 2-category T -Alg of T -algebras and pseudo morphisms. Definition 3.1.8. [BKP02, Def. 1.2] Let T : K → K be a 2-monad. The 2-category T -Alg has strict T -algebras as 0-cells. A 1-cell f between T -algebras (A, a : TA → A) and (B, b : TB → B), also called a strong morphism of T -algebras in [JY22], consists of a 1-cell f : A → B in K, together with an invertible 2-cell Tf TA TB f a b A B, f subject to the following axioms. 1. The equality of pasting diagrams 2 2 T 2 T f T f A T 2B T 2A T 2B Tf µ µ Ta Tb Tf Tf TA TB = TA TB f f a b a b A B A B f f holds. 2. The following pasting diagram equals the identity of f : A → B : 53 f A B η η Tf TA Tb f a b A B. f A 2-cell in T -Alg between 1-cells (f, f̄), (g, ḡ) : A → B is a 2-cell α : f → g in K such that the following diagram commmutes: Tf Tf TA Tα TB TA TB f Tg a b = a b g f A B A α B. g g Hyland and Power [HP02] extend Blackwell, Kelly and Power’s 2-categorical construction to provide a non symmetric Cat-multicategory whose underlying 2- category is T -Alg. If T is a pseudo commutative 2-monad, the Cat-multicategory T -Alg is symmetric. When K = Cat and T is accesible, Bourke proves [Bou02] that the Cat-multicategory structure can be seen to arise from a monoidal bicate- gory structure on T -Alg. Guillou, May, Merling and Osorno [GMMO23] specialize this definition to define a multicategory O-Alg for O a pseudo commutative op- erad. To be able to define the multicategory T -Alg, we need to prove a coherence result. Definition 3.1.9. Suppose T : K → K is a pseudo-commutative 2-monad, n ≥ 2 and 1 ≤ i < j ≤ n. We define a modification from µ ◦ Ttj ◦ ti to µ ◦ Tti ◦ tj as follows. Suppose A1, . . . , An objects of K, we define the component 2-cell of our modificiation in K (A1 × · · · ×Ai−1 × TAi ×Ai+1 × · · · ×Aj−1 × TAj ×Aj+1 × · · · ×An, T (A1 × · · · ×An)) in the following way. In principle there are various ways of doing this. Consider a partition K of the symbols A1, . . . , TAi, . . . , TAj, . . . , An into 4 subsets K1, K2, K3, K4 obtained by placing 3 bars in between symbols such that: • K2 contains TAi, and 54 • K3 contains TAj. We will represent K in the following way: ·︸· · ︷×︷ · ·︸· | ︸· · · × T︷A︷ i × · ·︸· | ︸· · · × T︷A︷ j × · ·︸· | ︸· · · ︷×︷ · ·︸· . K1 K2 K K3 4 For such a partition K, we can define the 2-cell ΓKi,j as the whiskering A1 × · · · × TAi × · · · × TAj × · · · × An = ︸· · · ︷×︷ · ·︸· | ︸· · · × T︷A︷ i × · ·︸· | ︸· · · × T︷A︷ j × · ·︸· | ︸· · · ︷×︷ · ·︸· K1 K2 K K3 4 × |K2| × |K3|1 t i−| | t ×1K1 j−|K1|−|K2| · · · × · · · × T (· · · × Ai × · · · )× T (· · · × Aj × · · · )× · · · × · · · 1×Γ×1 a a ︸· · · ︷×︷ · ·︸· ×T (· · · × Ai × · · · × Aj × · · · )× ·︸· · ︷×︷ · ·︸· K1 K4 tn|K1|+1 T (A1 × · · · × An). Example 3.1.10. For n = 3, i = 2, and j = 3, we have 2 possible partitions: K = A1 | TA2 | TA3 |, and K ′ =| A1TA2 | TA3 | . By (3.1.1), we get that ΓK ′2,3 = ΓK2,3. For n = 3, i = 1 and j = 3, there are again two partitions: H =| TA1A2 | TA3 |, and H ′ =| TA1 | A2TA3 |, and they induce the same 2-cell ΓH = ΓH′1,3 1,2 by (3.1.2). Similarly for n = 3, i = 1 and j = 2, we have the two partitions J =| TA1 | TA2A3 |, and J ′ =| TA1 | TA2 | A3, with J J ′Γ1,2 = Γ1,2 by (3.1.3). In general, we have the following. Theorem 3.1.11. [HP02, Thm. 5] Suppose (T, η, µ, t,Γ) is a pseudo commutative strong 2-monad. The three strength axioms imply that given n ≥ 2, and 1 ≤ i < j ≤ 55 n, any two partitions K and K ′ as in Definition 3.1.9 induce the same 2-cell. That is, ′ ΓK Ki,j = Γi,j . Proof. The previous example generalizes and will allow us to change our partition without changing the induced 2-cell using three moves. Let K be a partition of A1, . . . , TAi, . . . , TAj, . . . , An as in Definition 3.1.9. The following hold: (i) If K1 ends by Ap, i.e. K = ·︸· · × ·︷·︷· × A︸p | ︸× · · · × ︷T︷Ai × · ·︸· | ︸· · · × T︷A︷ j × · ·︸· | ︸· · · ︷×︷ · ·︸·, K1 K2 K K3 4 and K ′ is obtained from K by moving the first bar one spot to the left, i.e. K ′ = ︸· · · ×︷︷· · · ×︸ | A︸ p × · · · ×︷︷TAi × · ·︸· | ︸· · · × T︷A︷ j × · ·︸· | ︸· · · ︷×︷ · ·︸·, K1 K′ K′ K42 3 then ΓK ′i,j = ΓKi,j by (3.1.1). (ii) If K2 ends by Ap, that is K = ·︸· · ︷×︷ · ·︸· | ︸· · · × TAi︷×︷ · · · × A︸p | ︸× · · · × ︷T︷Aj × · ·︸· | ︸· · · ︷×︷ · ·︸· . K1 K K K2 3 4 and K ′ is obtained from K by moving the second bar one spot to the left, i.e., K ′ = ·︸· · ︷×︷ · ·︸· | ︸· · · × TA︷︷i × · · ·×︸ | A︸ p × · · · ×︷︷TAj × · ·︸· | ︸· · · ︷×︷ · ·︸·, K1 K′ ′ K2 K 43 then ′ΓKi,j = ΓKi,j by (3.1.2). (iii) If K3 ends by Ap, i.e., K = ·︸· · ︷×︷ · ·︸· | ︸· · · × T︷A︷ i × · ·︸· | ︸· · · × TAj︷×︷ · · · × A︸p | ︸× · ·︷· ︷× · ·︸· . K1 K2 K K3 4 and K ′ is obtained from K by moving the third bar one spot to the left, i.e. K ′ = ·︸· · ︷×︷ · ·︸· | ︸· · · × T︷A︷ i × · ·︸· | ︸· · · × TA︷︷j × · · ·×︸ | ︸Ap ︷×︷ · ·︸·, K1 K2 K′ K′3 4 then ΓK = ΓK′i,j i,j by (3.1.3). 56 Finally, we notice that any partition K ′ can be obtained from the partition K = A1 × · · · × Ai−1 | ×TAi × · · ·× | TAj × · · · × An | by making some number of moves (i), (ii) and (iii), and so K K′Γi,j = Γi,j . ■ Definition 3.1.12. Let (T, η, µ, t,Γ) be a pseudo commutative, strong 2-monad, n ≥ 2, 1 ≤ i < j ≤ n and A1, . . . , An objects of K we define the unique 2-cell in the previous theorem as Γi,j. That is, if K is a partition as in Definition 3.1.9, then Γ = ΓKi,j i,j Remark 3.1.13. To save some space in the following definitions we will denote the product A1 × · · · × Ai−1 as Ai = Ai+1 × · · · × An. The 2-cell Γi,j defined in the previous theorem fits in the following diagram by the µ axiom for strong monads in Definition 3.1.3: Ttj T (Aj) T (A1 × · · · ×An) ti µ Γi,j Aj T (A1 × · · · ×An). t µj T (A × TA ×A ) T 2i (A1 × · · · ×An) Tti Next, we define the Cat-multicategory T -Alg, whose underlying 2-category is T -Alg from Definition 3.1.8. In Definition 3.1.14 we define the 2-cells of T -Alg, in Definition 3.1.15 we define the 2-cells in T -Alg, and in Definition 3.1.16 we define the composition in T -Alg. Definition 3.1.14. [HP02, Def. 10] Let (T, η, µ, t,Γ) be a pseudo commutative, strong 2-monad. The n-ary 1-cells of the Cat-multicategory T -Alg are defined as follows. When n = 0, and B is a T -algebra, we define the category T -Alg(−;B) as K(1, B). Suppose that (Ai, ai : TAi → Ai) for 1 ≤ i ≤ n and (B, b : TB → B) are T - algebras. An n-ary 1-cell of T -Alg, ⟨A1 × · · · × An⟩ → B is the data of a 1-cell h : A1 × · · · × An → B in K, together with 2-cells hi for 1 ≤ i ≤ n fitting in the square: 57 Ai T (A1 × · · · × An) TB 1×ai×1 bhi A1 × · · · × An B.h These data have to satisfy the following axioms. • η axiom: The following pasting diagram is the identity of h : A1 × · · · × An → B. Ai B 1×η×1 η η × × tAi T (A (3.1.8)1 × · · · × An) TBTh 1×ai×1 bhi A1 × · · · × An B.h • µ axiom: The pasting diagrams ti Tti T 2h Ai T (Ai) T 2(A1 × · · · ×An) T 2B 1×µ×1 µ µ ti Th A × TA ×A T (A × · · · ×A ) TB (3.1.9)i T (Ai) T (A1 × · · · ×An) T B 1×Tai×1 T (1×ai×1) Th Tbi ti A × TA ×A T (A × · · · ×A ) TB (3.1.10)i 1 n Th 1×ai×1 h bi A1 × · · · ×An B h are equal. • Coherence: For i < j, the pasting diagrams 58 ti Ttj µ Aj T (Aj) T (A1 × · · · ×An) T (A1 × · · · ×An) 1×aj×1 T (1×aj×1) T 2h Th Thj ti µ Ai T (A1 × · · · ×An) T B TB Tb × × Th1 ai 1 TB b hi b A1 × · · · ×An B h (3.1.11) and, Ttj T (Aj) T (A1 × · · · ×An) ti µΓi,j tj µ Aj T (Ai) T (A1 × · · · ×An) T (A1 × · · · ×An) Tti 1×ai×1 T (1×ai×1) T 2h Th Th t ij µ Aj T (A1 × · · · ×A ) T 2n B TB Tb × × Th1 aj 1 TB b hj b A1 × · · · ×An B h (3.1.12) are equal. Definition 3.1.15. [HP02, Def. 10] Let (T, η, µ, t,Γ) be a pseudo commutative, strong 2-monad. We define the 2-cells of T -Alg as follows. Suppose that (Ai, ai : TAi → Ai) and (B, b : TB → B) are T -algebras for 1 ≤ i ≤ n, and that (f, ⟨fi⟩) and (g, ⟨gi⟩) are 1-cells in T -Alg(⟨A1, . . . , An⟩, B) . A 2-cell α : f → g in T -Alg is the datum of a 2-cell in K f A1 × · · · × An α B, g subject to the equality, for i < n, of the pasting diagrams 59 Ai T (A1 × · · · × At n) TBi fi 1×a ×1 b (3.1.13)i f A1 × · · · × An α B. g and, Tf Ai T (At 1 × · · · × An) Tα TBi 1×a (3.1.14)i×1 g bi Tg A1 × · · · × An Bh are equal. Vertical composition of 2-cells in T -Alg is given by vertical composition in K. Next we define the γ composition in T -Alg. Definition 3.1.16. Let (T, η, µ, t,Γ) be a pseudo commutative, strong 2-monad. For C ∈ Ob(T -Alg), n ≥ 0, ⟨B⟩ = ⟨B nj⟩j=1 ∈ Ob(T -Alg)n, kj ≥ 0 for 1 ≤ j ≤ n, and ⟨ ⟩ ⟨ ⟩kAj = A jj,i i=1 ∈ Ob(T -Alg)kj for 1 ≤ j ≤ n, we define∏n γ T -Alg(⟨B⟩;C)× T -Alg(⟨Aj⟩;Bj) T -Alg(⟨A⟩;C) j=1 as follows. Let (f, fj) : B1 × · · · × Bn → C and (gj, gji) : Aj,1 × · · · × Aj,k → Bj j 1-cells of T -algebras. We define their γ composition as the K 1-cell ∏ × · · · × gi × · · · × f∏A1 An B1 Bn C ∑ where Aj denotes kj i=1Aj,i. A∑ny number between s with 1 ≤ s ≤ nj=1 kj can be uniquely written as s = d + t 1. (c) σiσi+1σi = σi+1σiσi+1. The relations between the different Tσ will follow from relations between 2-cells ini T -Alg which can be proven in K. The relations in K can be proven even when T is not symmetric, except for the relation induced by σiσi = id. The following follows (in fact, it is equivalent to) symmetry for T. Lemma 3.3.6. Suppose that T is a symmetric, pseudo commutative, strong 2- monad. Then the following pasting diagram is the identity: K(A1 × · · · × × TAn, C) T -Alg(TA1, . . . , TAn;TC) Tσ σ ii σi K(A1 × · · · ×Ai+1 ×Ai × · · · ×An, C) T -Alg(TA1, . . . , TAi+1, TAi, . . . , TAn;C)T σi σi Tσi K(A1 × · · · ×An, C) T -Alg(TA1, . . . , TAn;TC).T 81 The following holds even in the absence of symmetry. Lemma 3.3.7. Suppose that (T, η, µ, t,Γ) is a pseudo commutative, strong 2-monad. Then the pasting diagram 1×ω ω TA1 × TA2 × TA3 × TA4 TA1 × TA2 × T (A3 ×A4) T (A1 ×A2 ×A3 ×A4) 1×∼= 1×T∼= T∼= 1×Γ TA1 × × × 1×ω × × ωTA2 TA4 TA3 TA1 TA2 T (A4 ×A3) T (A1 ×A2 ×A4 ×A3) ω×1 ω ∼× T (A1 ×A2)× TA4 × TA= 1 3 T=∼ T∼=×1 Γ×1 ω TA2 × TA1 × TA4 × TA3 × T (A2 ×A1)× TA4 × TA3 T (A2 ×A1 ×Aω 1 4 ×A3), equals the pasting TA1 × TA2 × TA3 × ω×1 ω TA4 T (A1 × TA2)× TA3 × TA4 T (A1 ×A2 ×A3 ×A4) =∼×1 T=∼×1 T∼= Γ×1 TA2 × ωTA1 × TA3 × TA4 × T (A2 ×A1)× TA3 × TA4 T (A ×A ×A ×A )ω 1 2 1 3 4 1×ω ω ×∼ TA2 × TA1 × T (A3 ×A4)1 = T∼= 1×T∼= 1×Γ TA2 × TA1 × TA4 × TA3 × TA1 ω 2 × TA1 × T (A4 ×A3) ω T (A2 ×A1 ×A4 ×A3). Proof. Both pastings are equal to the pasting × × × ω×ω ωTA1 TA2 TA3 TA4 T (A1 ×A2)× T (A3 ×A4) T (A1 ×A2 ×A3 ×A4) Γ ×Γ ∼=×∼ A1,A2 A3,A4 = T∼=×T=∼ T∼= TA2 × TA1 × TA4 × TA3 × T (A2 ×A1)× T (A4 ×A3) ω T (A1 ×A2 ×A3 ×A4).ω ω ■ When T is symmetric, a slight generalization of the previous lemma can be interpreted as follows. To save space we will write TA = TA1 × · · · × TAn and TAσ = TAσ(1) × · · · × TAσ(n) when σ ∈ Σn and A1, . . . , An are objects of K 82 Lemma 3.3.8. Supposet T is a symmetric, pseudo commutative, strong 2-monad, n ≥ 3 and 1 ≤ i < i + 2 ≤ j ≤ n − 1. Let A1, . . . , An, C be objects of K. Then, the pasting K(TA,C) T T -Alg(⟨TA⟩;C) Tσ σ ii σi K(TAσi, C) T -Alg(⟨TA⟩σ ;C)T i Tσj σj σj K(TAσiσj, C) T -Alg(⟨TA⟩σiσj;C),T equals the pasting K(TA,C) T T -Alg(⟨TA⟩;C) Tσj σj σj K(TAσj, C) T -Alg(⟨TA⟩σj;C)T Tσ σ ii σi K(TAσjσi, C) T -Alg(⟨TA⟩σjσi;C),T Next, we focus on the Yang-Banxter equation. First we prove the following lemma that we will also need later. It doesn’t require symmetry. Lemma 3.3.9. Let (T, η, µ, t,Γ) be a pseudo comutative, strong 2-monad, and A1, A2, A3 objects of K. Then, 1. The pasting diagram ω×1 ω TA1 × TA2 × TA3 T (A1 ×A2)× TA3 T (A1 ×A2 ×A3) ∼× Γ×1= 1 T (∼=×1) T∼=×1 TA2 × TA1 × TA3 × T (A2 ×A1)× TAω 1 3 ω T (A2 ×A1 ×A3) (3.3.1) 1×ω ω ×∼ TA2 × T (A1 ×A )1 = 3 T (1×∼=) 1×Γ 1×T=∼ TA2 × TA3 × TA1 × TA2 × T (A3 ×A1) ω T (A1 ω 2 ×A3 ×A1) equals the whiskering 83 ω Γ × × 1×ωTA1 TA2 TA3 TA1 × T (A2 × A3) T (A1 × A2 × A3). ω′ 2. The pasting diagram TA1 × 1×ω ω TA2 × TA3 TA1 × T (A2 ×A3) T (A1 ×A2 ×A3) 1×∼ 1×Γ= 1×T∼= T∼= TA1 × TA3 × TA2 × TA1 × T (A3 ×A2) ω T (A1 ×A3 ×A )1 ω 2 (3.3.2) ω×1 ω ∼× T (A= 1 1 ×A3)× TA2 T∼= Γ×1 T∼=×1 TA3 × TA1 × TA2 × T (A3 ×A1)× TA2 ω T (A3 ×A1 ×A )ω 1 2 equals the whiskering ω Γ ω×1 TA1 × TA2 × TA3 T (A1 × A2)× TA3 T (A1 × A2 × A3). ω′ Proof. For part (1) we start from 1×t TA1 × TA 12 × TA3 TA1 × T (A2 × TA3) 1×Tt ω2 Γ TA1 × T 2(A2 × A3) × TA1 × T (A2 × A3) T (A1 × A2 × A1 µ 3). ω′ By (7) in Definition 3.1.6, the previous whiskering equals the pasting diagram 84 × 1×t1TA1 TA2 × TA3 TA1 × T (A2 × TA3) 1×Tt2 × 2 t1 Tt2TA T (A ×A ) T (A × T 2(A ×A )) T 21 2 3 1 2 3 (A1 × T (A2 ×A3)) Γ t2 µ T (TA × T (A × TA )) T 21 2 3 (A1 × T (A2 ×A3)) µ T (A1 × T (A ×A ))Tt 2 31 Tt T 22 t2 Tt2 2 × × TΓ µT (TA1 A2 A ) T 33 (A1 ×A2 ×A3) T 2(A1 ×A2 ×A3) T 2t1 Tµ µ T 3(A1 ×A2 ×A ) T 23 (A1 ×A2 ×A3) µ T (A ×A ×A ).Tµ 1 2 3 (3.3.3) First we will prove that the whiskering TA1 × TA2 × 1×t TA 13 TA1 × T (A2 × TA3) ω 1×Tt2 Γ TA × T 21 (A2 × A3) T (A1 × T (A2 × A3)) Tt2 ω′ T 2(A1 × A2 × A3) µ T (A1 × A2 × A3) (3.3.4) coming from the previous diagram equals the whiskering ω×1 Γ×1 t1 Tt3 TA1 × TA2 × TA3 T (A1 ×A2)× TA3 T (A1 ×A2 × TA3) T 2(A1 ×A2 ×A3) µ ω′×1 T (A1 ×A2 ×A3) coming from diagram (3.3.1). By (3) in Definition 3.1.6, the previous whiskering equals 85 ω 1×t Γ1 TA1 × TA2 × TA3 TA1 × T (A2 × TA3) T (A1 ×A2 × TA3) Tt3 1×Tt2 ω′ T (1×t2) TA1 × T 2(A2 ×A3) ′ T (A1 × T (A2 ×A3)) T 2(A1 ×A2 ×A3) ω Tt2 µ T (A1 ×A2 ×A3). Since Γ is a modification, the previous diagram equals (3.3.4). To finish part (1), we will prove that the whiskering TA1 × TA2 × 1×t1 1×Tt2 TA3 TA1 × T (A2 × TA3) TA1 × T 2(A2 ×A3) t2 Tω (3.3.5) T (TA1 × T (A2 ×A3)) TΓ T 2(A1 ×A2 ×A3) µ Tω′ T (A1 ×A2 ×A3) coming from (3.3.3) equals the whiskering TA1 × TA2 × TA3 ∼=×1 1×ω TA2 × TA1 × TA3 1×Γ TA2 × T (A1 ×A3) t1 1×ω′ t1 T (A2 × TA1 × TA3) ′ T (A2 × T (A1 ×A3))T (1×ω ) Tt2 T 2 µ (A2 ×A1 ×A3) T (A2 ×A1 ×A3) T (∼=×1) T (A1 ×A2 ×A3) coming from (3.3.1). By 2-naturality of t1, the previous diagram equals TA1 × TA2 × TA3 T (1×ω) =∼×1 T (1×Γ) TA2 × TA1 × TA3 T (A2 × TA1 × TA3) T (A2 × T (A1 ×A3))t1 Tt2 T (1×ω′) T 2(A2 ×A1 ×A3) T (∼=×1)◦µ T (A1 ×A2 ×A3). 86 By (1) in Definition 3.1.6, this whiskering equals t2 T (t1×1) Tω TA1 × TA2 × TA3 T (A1 ×A2 × TA3) T (T (A1 ×A2)× TA ) T 23 (A1 ×A2 ×A3) ∼=×1 T (=∼×1) T (T∼=×1) Tω T 2(=∼×1) TΓ TA2 × TA 21 × TA3 T (A2 × TAt 1 × TA3) T (T (A2 ×A1)× TA3) T (A× 2 ×A1 ×A3) 1 T (t2 1) µ Tω′ T (A2 ×A1 ×A3) T (∼=×1) T (A1 ×A2 ×A3). Since Γ is a modificiation we can write the previous whiskering as Tω t T (t ×1) TΓ2 1 TA1 × TA2 × TA3 T (TA1 ×A2 × TA3) T (T (A1 ×A2)× TA ) T 23 (A1 ×A2 ×A3) 1×t1 µ t2 Tω′ TA1 × T (A2 × TA3) T (A1 ×A2 ×A3). By (2) in Definition 3.1.6, we get 1×t1 t2 TA1 × TA2 × TA3 TA1 × T (A2 × TA3) T (TA1 ×A2 × TA3) Tω 1×Tt2 T (1×t2) TA 21 × T (A2 ×A3) T (TA1 × T (A2 ×A3)) TΓ T 2(A1 ×A2 ×A3)t2 µ Tω′ T (A1 ×A2 ×A3), which is precisely (3.3.5). We have proven part (1). Part (2) is proven in a similar fashion. ■ The next Lemma is the Yang-Baxter equation for non-symmetryc pseudo com- mutative, strong 2-monads. Part (3) is called the Associativity Equation in [HP02]. Lemma 3.3.10. Let T be a pseudo commutative, strong, 2-monad. Then: 1. The pasting 87 TA1 × TA2 × 1×ω ω TA3 TA1 × T (A2 ×A3) T (A1 ×A2 ×A3) ×∼ 1×Γ1 = 1×T=∼ T (1×∼=) TA1 × TA3 × TA2 × TA1 × T (A3 ×A2)1 ω ω T (A1 ×A3 ×A2) ω×1 ω ∼× T (A1 ×A3)× TA= 1 2 T (∼=×1) Γ×1 T=∼×1 (3.3.6) TA3 × TA1 × TA2 × T (A3 ×A1)× TA2 ω T (A3 ×A1 ×A2)ω 1 1×ω ω ×∼ TA3 × T (A ×A )1 = 1 2 T (1×∼=) 1×Γ 1×T∼= ω TA3 × TA2 × TA1 × TA3 × T (A2 ×A1) T (A3 ×A2 ×A1)1 ω equals the horizontal composite 1×ω ω 1×Γ Γ TA1 × TA2 × TA3 TA1 × T (A2 × A3) T (A1 × A2 × A3). 1×ω′ ω′ 2. The pasting TA1 × ω×1 TA2 × TA3 T (A1 ×A3)× TA2 ω T (A1 ×A2 ×A3) ∼ Γ×1=×1 T∼=×1 T (=∼×1) TA2 × TA1 × TA3 × T (A2 ×A1)× TA3 ω T (Aω 1 2 ×A1 ×A3) 1×ω ω ×∼ TA2 × T (A1 ×A3)1 = T (1×=∼) 1×Γ 1×T∼= (3.3.7) TA2 × TA3 × ωTA1 × TA2 × T (A3 ×A1) T (A2 ×A3 ×A1)1 ω ω×1 ω ∼× T (A ×A )= 1 2 3 T (=∼×1) Γ×1 T=∼×1 TA3 × TA2 × TA1 × T (A3 ×A2)× TA1 ω T (A3 ×A2 ×Aω 1 1) 88 equals the whiskering ω×1 ω Γ×1 Γ TA1 × TA2 × TA3 T (A1 × A2)× TA3 T (A1 × A2 × A3). ω′×1 ω′ (3.3.8) 3. The pastings and horizontal composites in (1) and (2) are equal. Proof. For (1), notice that by the Lemma 3.3.9, the pasting diagram TA1 × TA2 × TA3 T (A1 ×A2 ×A3) 1×∼= T (1×=∼) ω×1 ω TA1 × TA3 × TA2 T (A1 ×A3)× TA2 T (A1 ×A3 ×A2) Γ×1 =∼×1 T=∼×1 T (=∼×1) ω×1 ω TA3 × TA1 × TA2 T (A3 ×A1)× TA2 T (A3 ×A1 ×A2) 1×ω ω ∼ TA3 × T (A1 ×A2)1×= 1×Γ T (1×∼=) 1×T∼= TA3 × TA2 × TA1 × TA3 × T (A2 ×A1) ω T (A3 ×A2 ×A1 ω 1) equals the whiskering 1×ω′ TA ω1 × TA2 × TA3 TA1 × T (A2 × A3) T (A1 × A2 × A3) ω 1×∼= 1×T=∼ T (1×=∼) Γ TA1 × TA3 × TA2 × TA1 × T (A3 × A2) T (A3 × A2 × A1).1 ω ω′ Since Γ is a modification the last whiskering equals ω ′ Γ TA1 × TA2 × 1×ω TA3 TA1 × T (A2 × A3) T (A1 × A2 × A3). ω′ Part (1) follows from this and part (2) is proven similarly. To prove part (3) we will prove that diagrams (3.3.6), and (3.3.8) are equal. By (2) in Lemma 3.3.9, we are reduced to proving that the whiskerings 89 TA1 × TA2 × TA3 T (A1 × A2 × A3) 1×∼= T (1×=∼) TA1 × TA3 × TA2 T (A1 × A3 × A2) 1×ω =∼×1 T (=∼×1) 1×Γ TA3 × TA1 × TA2 TA3 × T (A1 × A2) ω T (A3 × A1 × A2) 1×ω′ and, ω×1 Γ×1 ′ TA1 × TA2 × TA3 T (A1 × A2)× TA ω3 T (A1 × A2 × A3) ω′×1 are equal. This holds since both whiskerings are equal to ω×1 ′ TA1 × TA2 × TA3 T (A1 × A2)× TA ω3 T (A1 × A2 × A3) ∼ 1×ω= ∼= T∼= 1×Γ TA3 × TA1 × TA2 TA3 × T (A1 × A2) ω T (A3 × A1 × A2). 1×ω′ ■ In the presence of symmetry, we can give (a slight generalization of) the previ- ous lemma the following interpretation. Lemma 3.3.11. Suppose that (T, η, µ, t,Γ) is a symmetric, pseudo commutative, strong 2-monad. Then the pasting diagram T K(A1 × · · · ×An, C) T -Alg(TA1, . . . , TAn;TC) Tσ σ ii σi K(A1 × · · · ×Ai+1 ×Ai × · · · ×An, C) T -Alg(TA1, . . . , TAi+1, TAi, . . . , TAn;TC) T σi+1 σi+1 Tσi+1 K(A1 × · · · ×Ai+1 ×Ai+2 ×Ai × · · · ×An, C) T -Alg(TA1, . . . , TAi+1, TAi+2, TAi, . . . , TAn;TC) T σi σi Tσi K(A1 × · · · ×Ai+2 ×Ai+1 ×Ai × · · · ×An, C) T -Alg(TA1, . . . , TAi+2, TAi+1, TAi, . . . , TAn;TC), T equals the pasting diagram 90 T K(A1 × · · · ×An, C) T -Alg(TA1, . . . , TAn;TC) Tσi+1 σi+1 σi+1 K(A1 × · · · ×Ai+2 ×Ai+1 × · · · ×An, C) T -Alg(TA1, . . . , TAi+2, TAi+1, . . . , TAn;TC) T σi σi Tσi K(A1 × · · · ×Ai+2 ×Ai ×Ai+1 × · · · ×An, C) T -Alg(TA1, . . . , TAi+2, TAi, TAi+1, . . . , TAn;TC) T σi+1 σi+1 Tσi+1 K(A1 × · · · ×Ai+2 ×Ai+1 ×Ai × · · · ×An, C) T -Alg(TA1, . . . , TAi+2, TAi+1, TAi, . . . , TAn;TC), T The three previous lemmas give us the following. Theorem 3.3.12. Suppose that (T, η, µ, t,Γ) is a symmetric, pseudo commutative strong to monad and let A1, . . . , An, C be objects of K. The transformations Tσ fori 1 ≤ i ≤ n− 1 assemble together to give, for σ ∈ Σn, a unique transformation K(A1 × · · · × An, C) T T -Alg(TA1, . . . , TAn;TC) σ σ Tσ K(Aσ(1) × · · · × Aσ(n)) T -Alg(TAT σ(1), . . . , TAσ(n);TC). These satisfy the unit and the product permutation axiom in Definition 2.1.15. We are just missing the top and bottom equivariance axioms to prove that our functor T : K → T -Alg is pseudo symmetric. When T is a pseudo commutative, strong 2-monad that fails to be symmetric, we can still give Lemma 3.3.10 an in- terpretation using the Bruhat order of the symmetric group Σn on generators σi for 1 ≤ i ≤ n− 1. Definition 3.3.13. Let Σn be the symmetric group with generators {σi}1≤ii+1;C T -Alg (⟨TAji+1⟩;TC) T equals the pasting diagram ( ) ∏ ( ) ∏T× T ∏ K B;C × j K Aj ;Bj T -Alg(⟨FB⟩;FC)× j T -Alg (⟨FAj⟩;FBj) σi×σ−1i ∏ ( Tσi)×1 K (B −1i+1;C)× j K Aσ (j);Bi σi(j) σi×σi ∏ T× T ∏ γ T -Alg(⟨TB⟩σi;TC)× j T -Alg(⟨TAσ (j)⟩;TBi σ )i(j) ( ) γ K Ai+1;C T -Alg (⟨TAji+1⟩;TC) . T Proof. The lemma follows at once from Lemma 3.3.15 Definition 3.3.5, and (1) in Lemma 3.3.9. ■ Lemma 3.3.17. Suppose (T, η, µ, t,Γ) is a symmetric, pseudo commutative, strong 2-monad. Let n ≥ 1 and 1 ≤ i ≤ n − 1, and consider the Cat-multifunctor T : K → T -Alg. Then, the bottom equivariance axiom in Definition 2.1.15 holds for idn⟨idk1 , . . . , σi, . . . , idkn⟩ that is, For every C ∈ Ob(K), ⟨B⟩ = ⟨Bj⟩nj=1 ∈ Ob(K)n, k kj ≥ 0 for 1 ≤ j ≤ n, and ⟨Aj⟩ = ⟨Aj,l⟩ jl=1 ∈ Ob(K)kj for 1 ≤ j ≤ n, the pasting diagram ( ) ∏ ( ) ∏T ∏ K B,C × K Aj , Bj T -Alg (⟨FB⟩;FC)× j T -Alg (⟨TAj⟩;TBj) γ (∏ ) γ T K j Aj , C T -Alg (⟨⟨TAj⟩⟩j ;TC) idn⟨idk ,...,σi,...,idkn ⟩ idn⟨idk ,...,σi,...,id1 1 kn ⟩ ( ∏ ) Tidn⟨idk ,...,σi,(...,id1 k ⟩n ) K Ai, C T -Alg ⟨TAi⟩;TCi T i is equal to the pasting 95 ( ) ∏ ( )∏T ∏ K B,C × K Aj , Bj T -Alg (⟨FB⟩;FC)× j T -Alg (⟨TAj⟩;TBj) id×idk ×···×σi×···×idk 1×Tσi×1 ( ) 1 ( ) (n∏ ) ( ) K B,C ×K A1 × · · · × K j Aj,σ (j) × · · · × K ∏A ,B id×idk ×···×σi×···×idn n 1 kni T ( ) γ ∏ ∏T -Alg (⟨TB⟩;TC)× ji T -Alg (⟨TAi j⟩;TBj) γ ( ∏ ) ( ) K Ai, C T -Alg ⟨TAi⟩;TCi T i Proof. The lemma follows at once from Definition 3.3.5, and (2) in Lemma 3.3.9. ■ Finally we arrive at the proof of our main theorem. Theorem 3.3.18. Let (T, η, µ, t,Γ) be a symmetric, pseudo commutative, strong 2-monad. The free algebra Cat-multifunctor T : K → T -Alg is pseudo symmetric. Proof. We just need to prove that the bottom and top equivariance axioms hold for T. For the top equivariance axiom we notice that given σ, τ ∈ Σn, and k1, . . . , kn, we can write στ⟨idk , . . . , id ⟩ as the compositionστ(1) kστ(n) Aστ(1) × · · · × Aστ(n) τ⟨idk ⟩στ(i) Aσ(1) × · · · × Aσ(n) σ⟨idkσ(i)⟩ A1 × · · · × An. By an application of the product axiom, if σ⟨idk ⟩ and τ⟨idk ⟩ satisfy the topσ(i) στ(i) invariance axiom, then so does στ⟨idk , . . . , idk ⟩. We are done by Lemmaστ(1) στ(n) Lemma 3.3.16. Similarly, for the bottom equivariance axiom. Given n, k1, . . . kn and σ, τ ∈ Σk . Ifi the bottom equivariance axiom holds for the permutations idn⟨idk1 , . . . , τ, . . . idkn⟩ and idn⟨idk1 , . . . , σ, . . . , idkn⟩, then it also holds for idn⟨idk1 , . . . , στ, . . . , idkn⟩ by an application of the product axiom. By Lemma 3.3.17, we get the bottom equivari- ance axiom for idn⟨idk1 , . . . , σ, . . . , idkn⟩ for any σ ∈ Σk . On the other hand, ifi 96 the bottom equivariance axiom holds for idn⟨σ1, . . . , σn⟩ and idn⟨τ1, . . . , τn⟩, where σi, τi ∈ Σk , then it also holds for idn⟨σi 1τ1, . . . , σnτn⟩ by another application of the product axiom. We conclude that T satisfies bottom equivariance and this con- cludes the proof that T is pseudo symmetric. ■ Since the free functor associated to a pseudo commutative operad is a symmet- ric, pseudo commutative strong 2-monad, the free functor of the pseudo commu- tative operads defined in [GMMO23] and considered as well in [Yau24a] is pseudo symmetric. 97 REFERENCES CITED [BFSV03] C. Balteanu, Z. Fiedorowicz, R. Schwänzl, and R. Vogt, Iterated monoidal categories. Adv. Math. 176 no. 2 (2003), 277–349. [BGS20] C. Barwick, S. Glasman, and J. Shah, Spectral Mackey functors and equivariant algebraic K-theory, II, Tunis. J. Math. 2 (2020), no. 1, 97–146. 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