PSEUDO SYMMETRIC MULTIFUNCTORS: COHERENCE AND EXAMPLES
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Date
2024-08-07
Authors
Manco, Diego
Journal Title
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Publisher
University of Oregon
Abstract
Donald Yau introduced pseudo symmetric Cat-multifunctors and proved that Mandell's inverse K-theory multifunctor is stably equivalent to a pseudo symmetric 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 E_n-algebras parameterized by free Cat-operads at the cost of changing the parameterizing Cat-operad by its product with the categorical Barrat-Eccles operad. Since Mandell's inverse K-theory is pseudo symmetric, we derive that E_n-algebras parameterized by free E_n Cat-operads in the symmetric monoidal category of $\Gamma$-categories can be realized, up to stable equivalence, as the K-theory of some E_n-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, and it implies a coherence result for pseudo commutative, strong 2-monads conjectured by Hyland and Power.
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Keywords
Algebraic Topology, Category Theory, K-theory, Multicategories, Pseudo symmetric multifuntors