THE HOMOTOPY CALCULUS OF CATEGORIES AND GRAPHS
by
DEBORAH A. VICINSKY
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2015
DISSERTATION APPROVAL PAGE
Student: Deborah A. Vicinsky
Title: The Homotopy Calculus of Categories and Graphs
This dissertation has been accepted and approved in partial fulfillment of
the requirements for the Doctor of Philosophy degree in the Department of
Mathematics by:
Dr. Hal Sadofsky Chair
Dr. Boris Botvinnik Core Member
Dr. Dev Sinha Core Member
Dr. Peng Lu Core Member
Dr. Eric Wiltshire Institutional Representative
and
Scott Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2015
ii
© 2015 Deborah A. Vicinsky
iii
DISSERTATION ABSTRACT
Deborah A. Vicinsky
Doctor of Philosophy
Department of Mathematics
June 2015
Title: The Homotopy Calculus of Categories and Graphs
We construct categories of spectra for two model categories. The first is the
category of small categories with the canonical model structure, and the second
is the category of directed graphs with the Bisson-Tsemo model structure. In
both cases, the category of spectra is homotopically trivial. This implies that the
Goodwillie derivatives of the identity functor in each category, if they exist, are
weakly equivalent to the zero spectrum. Finally, we give an infinite family of
model structures on the category of small categories.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Deborah A. Vicinsky
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Bucknell University, Lewisburg, PA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2015, University of Oregon
Master of Science, Mathematics, 2011, University of Oregon
Bachelor of Science, Mathematics, 2009, Bucknell University
AREAS OF SPECIAL INTEREST:
Algebraic Topology
Model Categories
Homotopy Calculus
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Mathematics, University of
Oregon, Eugene, 2009–2015
v
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor, Hal Sadofsky, for always
being patient and encouraging. I could not have done this without your guidance.
Thank you to my friends and family for all of your support over the years. I
would also like to thank my husband, John Jasper, for motivating me to persevere
in this endeavor.
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TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. HOMOTOPY CALCULUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1. Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2. The Taylor Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3. Operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
III. DIFFERENTIAL GRADED LIE ALGEBRAS . . . . . . . . . . . . . . . . . 29
IV. THE CATEGORY OF SMALL CATEGORIES . . . . . . . . . . . . . . . . 37
4.1. Suspensions of Categories . . . . . . . . . . . . . . . . . . . . . . . 42
4.2. Categories of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 46
V. THE CATEGORY OF DIRECTED GRAPHS . . . . . . . . . . . . . . . . . 55
VI. OTHER MODEL STRUCTURES ON CAT . . . . . . . . . . . . . . . . . . 62
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
vii
CHAPTER I
INTRODUCTION
A model category is essentially a category in which there is a notion of
homotopy. The standard example is T op, the category of topological spaces.
Another example is the category Ch(R) of bounded-below chain complexes
of modules over an associative unital ring R, where homotopy means chain
homotopy. More formally, a model category is a category that has all small limits
and colimits together with a model structure, which consists of three classes
of morphisms called weak equivalences, fibrations, and cofibrations satisfying
certain axioms [Hov99].
In this dissertation, we consider model categories of graphs and categories.
Bisson and Tsemo [BT+09] defined a model structure on a category of directed
graphs, denoted Gph∗, in which a graph map is a weak equivalence if it is bijective
on n-cycles for all n ≥ 1. A similar but more complex category is Cat∗, in which
the objects are pointed small categories and the morphisms are functors [Rez96].
In Cat∗, the weak equivalences are equivalences of categories.
In the early 1990’s, Goodwillie invented a method for analyzing certain
functors F ∶ T op → T op in a way that mimics the method of Taylor series from
calculus [Goo03]. The Taylor series of a function f ∶ R → R gives us a way
to approximate the value of f using polynomial functions. Furthermore, under
the right circumstances, we can make our approximation arbitrarily close to the
correct answer, and the series converges to the value of the function. The goal
of Goodwillie calculus, also known as homotopy calculus, is to approximate
F with other functors that are homotopically simpler so that the values of the
1
approximations approach the value of the functor, at least for some inputs. These
approximating functors are called polynomial functors and are denoted PnF.
There are natural maps between these polynomial functors, which results in a
“Taylor tower”
. . . PnF → Pn−1F → . . . → P0F = ∗.
Ideally, the inverse limit of this tower is F. We define DnF to be the fiber of the
map PnF → Pn−1F. Each DnF corresponds to a spectrum, which is called the nth
derivative of F.
Goodwillie’s methods have been generalized to other model categories.
Although the domain and codomain of a functor need not be the same, the
derivatives of the identity functor 1 ∶ C → C on a model category C are of particular
interest. Here the analogy to regular calculus becomes more tenuous as 1 is rarely
polynomial itself. The complexity of 1 in terms of homotopy calculus is related to
the complexity of the homotopy theory of C.
The derivatives of 1 often have the structure of an operad, which is an object
that models a certain algebraic property. For example, there is an operad that
encodes commutativity and another that encodes associativity. There is one
operad, called the Lie operad, that determines if a vector space has the structure
of a Lie algebra. Walter showed that the derivatives of the identity functor on
DGLr, the category of differential graded Lie algebras over Q that are zero below
level r, have the structure of the Lie operad [Wal06].
In all known examples, the derivatives of a functor F ∶ C → C are objects
in the category of spectra on C [Hov01], which is denoted SpN(C,Σ) and is
called the stabilization of C. The objects of SpN(C,Σ) are sequences {Xi}i≥0 of
objects of C together with structure maps ΣXi → Xi+1, where Σ ∶ C → C is
2
the suspension functor. Morphisms in SpN(C,Σ) act level-wise and respect the
structure maps. We show that Σ2X is weakly equivalent to the zero object in
Cat∗ for all X ∈ Ob(Cat∗) and, moreover, that any object in SpN(Cat∗,Σ) is weakly
equivalent to the zero spectrum {0, 0, . . .}. Therefore we expect the derivatives of
the identity functor 1 ∶ Cat∗ → Cat∗ to be homotopically trivial. In other words,
the derivatives would have the structure of the trivial operad.
We apply similar methods to the category Gph∗. We construct the suspension
functor Σ ∶ Gph∗ → Gph∗ and show that ΣX is weakly equivalent to the zero object
of the category for all X ∈ Ob(Gph∗). Although this is not enough to conclude
that there is a category of spectra SpN(Gph∗,Σ) having the properties necessary
to apply Goodwillie’s work, we show that if the derivatives of the identity functor
in this category exist, then they are homotopically trivial.
Overview
In chapter 2, we explain Goodwillie’s construction and give many of the
definitions related to model categories that we will use throughout this work. In
chapter 3, we review Walter’s result in detail. Chapter 4 contains a description of
the model category structure on Cat∗, and we show that Cat∗ has the properties
necessary to apply Goodwillie’s method. Then we construct the stabilization of
Cat∗ and prove that the stabilization is homotopically trivial. We discuss a version
of the Bisson-Tsemo category of directed graphs in chapter 5 and show that the
stabilization of this category, if it exists, must be homotopically trivial as well.
Finally, in chapter 6 we present a model structure on Cat∗ for each positive integer
n and make a conjecture about how the stabilization of Cat∗ depends on the choice
of model structure.
3
CHAPTER II
HOMOTOPY CALCULUS
In calculus, one studies functions f ∶ R → R by studying the derivatives of
f . In particular, given a function f that is infinitely differentiable at a point x = a,
one may approximate f using its nth Taylor polynomial Tn(x) = n∑
i=0
f (i)(a)
i!
(x − a)i.
Then there is some interval on which the Taylor series converges to the value of
the function, that is,
f (x) = ∞∑
i=0
f (i)(a)
i!
(x − a)i.
In the 1990’s, Goodwillie [Goo90], [Goo91], [Goo03] invented an analogous
method for analyzing functors on T op, the category of topological spaces, or
from T op to Sp, the category of spectra. This method, which is now known as
Goodwillie calculus or homotopy calculus, has since been applied to functors on
other appropriately nice categories. Essentially, one needs a category in which
there is some notion of homotopy. Such categories are called model categories.
2.1. Model Categories
Before we can define a model category, we will need some preliminary
definitions.
Definition 2.1. [Hov99, 1.1.1] Let C be a category.
1. A map f in C is a retract of a map g in C if and only if there is a commutative
diagram of the form
4
A C A
B D B
f g f
where the horizontal composites are identities.
2. A functorial factorization is an ordered pair (α, β) of functors Mor C →Mor C
such that f = β( f ) ○ α( f ) for all f ∈ Mor C.
Definition 2.2. [Hov99, 1.1.2] Suppose i ∶ A → B and p ∶ X → Y are maps in a
category C. Then i has the left lifting property with respect to p and p has the right
lifting property with respect to i if, for every commutative diagram
A X
B Y
f
i
g
p
there is a lift h ∶ B → X such that hi = f and ph = g.
Definition 2.3. [Hov99, 1.1.3] A model structure on a category C consists of three
subcategories of C called weak equivalences, cofibrations, and fibrations, and two
functorial factorizations (α, β) and (γ, δ) satisfying the following properties:
1. (2-out-of-3) If f and g are morphisms of C such that g f is defined and two
of f , g and g f are weak equivalences, then so is the third.
2. (Retracts) If f and g are morphisms of C such that f is a retract of g and g is
a weak equivalence, cofibration, or fibration, then so is f .
5
3. (Lifting) Define a map to be a trivial cofibration if it is both a cofibration and
a weak equivalence. Similarly, define a map to be a trivial fibration if it is
both a fibration and a weak equivalence. Then trivial cofibrations have the
left lifting property with respect to fibrations, and cofibrations have the left
lifting property with respect to trivial fibrations.
4. (Factorization) For any morphism f , α( f ) is a cofibration, β( f ) is a trivial
fibration, γ( f ) is a trivial cofibration, and δ( f ) is a fibration.
Definition 2.4. [Hov99, 1.1.4] A model category is a category C with all small limits
and colimits together with a model structure on C.
We will denote weak equivalences by ≃Ð→, cofibrations by ↪, and fibrations by
↠ .
Since a model category C has all small limits, C contains the limit of the empty
diagram, which is a terminal object T of C. Similarly, C having small colimits
means that C contains the colimit of the empty diagram, which is an initial object
I of C. Therefore any model category contains an initial object and a terminal
object, though these are not necessarily the same.
An object X is called cofibrant if the unique map I → X is a cofibration and
fibrant if the unique map X → T is a fibration. It is always possible to replace an
object X by a cofibrant or fibrant object by part (4) of Definition 2.3. That is, we
may factor the map I → X as I ↪ Q(X) ≃Ð→ X, and we may factor the map X → T
as X ≃Ð→ R(X) ↠ T. We call Q(X) the cofibrant replacement of X and R(X) the
fibrant replacement of X.
For any model category, we can construct an associated model category
which is pointed, that is, which has a zero object 0.
6
Definition 2.5. [Hir03, 7.6.1] If C is a category and A is an object of C, then the
category of objects of C under A, denoted A ↓ C, is the category in which
● an object is a map A → X in C,
● a map from g ∶ A → X to h ∶ A → Y is a map f ∶ X → Y in C such that f ○ g = h,
and
● composition of maps is defined by composition of maps in C.
This is also called the coslice category of C with respect to A.
Theorem 2.6. [Hir03, 7.6.4] If M is a model category, then the category A ↓ M is a
model category in which a map is a weak equivalence, fibration, or cofibration if it is one
in M.
Given a model category M with terminal object T, the coslice category T ↓M
is isomorphic to the category of pointed objects of M and basepoint-preserving
morphisms. That is, we may force T to be an initial object of M as well as
a terminal object. The weak equivalences, fibrations, and cofibrations are then
precisely the largest possible subsets of those classes in the unpointed case.
The archetypal example of a model category (and the motivation for its
definition) is the category of topological spaces.
Example 2.7. [DS95, 8.3] Call a map in T op
● a weak equivalence if it is a weak homotopy equivalence,
● a fibration if it is a Serre fibration, and
● a cofibration if it has the left lifting property with respect to trivial fibrations.
7
With these choices, T op is a model category.
Example 2.8. [DS95, 7.2] Another example of a model category is Ch(R), the
category of non-negatively graded chain complexes of R-modules, where R is
an associative ring with unit. There is a model structure on Ch(R) in which a map
f ∶ M → N is
● a weak equivalence if f induces isomorphisms on homology Hk M → HkN
for all k ≥ 0,
● a cofibration if for each k ≥ 0 the map fk ∶ Mk → Nk is injective with a
projective R-module as its cokernel, and
● a fibration if for each k > 0 the map fk ∶ Mk → Nk is surjective.
Many constructions in T op have analogs in other model categories. For
example, one may construct suspension objects and loop objects in any pointed
model category.
Definition 2.9. [Qui67, I.2.9] Given a cofibrant object A in a pointed model
category M, the suspension ΣA is the pushout of the diagram
A ∨ A Cyl(A)
0
where a cylinder object Cyl(A) is an object of M that satisfies the following
diagram:
8
AL
AL ∨ AR Cyl(A) A
AR
≃
id
id
Here the maps AL → AL ∨ AR and AR → AL ∨ AR are inclusions, and the
composition AL ∨ AR ↪ Cyl(A)→ A is the coproduct map.
Definition 2.10. [Qui67, I.2.9] Given a fibrant object B in a pointed model category
M, ΩB (“loops on B”) is the pullback of the diagram
BI
0 B × B
where a path object BI is an object of M that satisfies the following diagram:
BL
B BI BL × BR
BR
≃
id
id
Here the composition B → BI ↠ BL × BR is the diagonal map, and the maps
BL × BR → BL, BL × BR → BR are projections.
Note that cylinder objects and path objects exist in any model category M
because M has functorial factorizations.
9
Proposition 2.11. ΩB ≃ X, where X is the homotopy pullback of the diagram
0
0 B
Proof. Let i ∶ B → BI and p ∶ BI → B × B be the maps from Definition 2.10. Let
piL ∶ B × B → B and piR ∶ B × B → B be the projections onto the left and right factors
of the product, respectively. Note that by the two-out-of-three property, piR ○ p is
a weak equivalence. Also, piR ○ p is a fibration since both piR and p are fibrations.
Similarly, piL ○ p is a trivial fibration.
Let Y be the pullback of the diagram
BI
0 B
piR ○ p
Let γ ∶ Y → BI and 0 ∶ Y → 0 be the maps defined by this pullback. The class of
trivial fibrations in M is closed under pullbacks [Hir03, 7.2.12], so 0 ∶ Y → 0 is a
trivial fibration. Note piR ○ p ○γ = 0.
Let Y be the pullback of the diagram
BI
B B × B
p
(id, 0)
Let γ ∶ Y → BI and δ ∶ Y → B be the maps defined by this pullback. Note that
p ○γ = (id, 0) ○ δ, so piL ○ p ○γ = δ and piR ○ p ○γ = 0.
10
We claim that Y ≅ Y. To see this, consider the following diagrams:
Y
Y BI
0 B
piR ○ p
γ
α
γ
Y
Y BI
B B × B
p
γ
(id, 0)
β
piL ○ p ○ γ
γ
By the above relations, both diagrams commute, and so α and β are unique.
Consider the commutative diagram
Y
Y
Y BI
0 B
piR ○ p
β
γ
γ
α
γ
Since the square is a pullback, the map α ○ β ∶ Y → Y is unique. However, idY also
makes the diagram commute, and so α ○ β = idY. A similar argument shows that
β ○ α = idY. Therefore Y and Y′ are isomorphic.
11
Since Y ≃ 0 and piL ○ p ○ γ ∶ Y → B is a fibration, X is weakly equivalent to the
pullback X of the diagram
Y
0 B
piL ○ p ○ γ
Thus we have two pullback squares
X Y BI
0 B B × B
γ
ppiL ○ p ○ γ
(id, 0)
It follows that the outer square is also a pullback square. Therefore X is weakly
equivalent to the pullback ΩB of the diagram
BI
0 B × B
p
An important class of model categories are those which are cofibrantly
generated. We describe this below.
Definition 2.12. [Hov99, 2.1.9] Let C be a category containing all small colimits.
Let I be a set of maps in C. A relative I-cell complex is a transfinite composition of
pushouts of elements of I. The collection of relative I-cell complexes is denoted
by I-cell.
12
Definition 2.13. [Hov99, 2.1.3] Let C be a category containing all small colimits,
D a collection of morphisms in C, A an object of C, and κ a cardinal. We say that
A is κ-small relative to D if, for all κ-filtered ordinals λ and all λ-sequences
X0 → X1 → . . . → Xβ → . . .
such that each map Xβ → Xβ+1 is in D for β + 1 < λ, the map of sets
colimβ<λ C(A, Xβ) → C(A, colimβ<λ Xβ) is an isomorphism. We say that A is small
relative to D if it is κ-small relative to D for some κ. We say that A is small if A is
small relative to C.
Definition 2.14. [Hov99, 2.1.17] A model category M is cofibrantly generated if
there are sets of maps I and J in M such that
● the domains of the maps of I are small relative to I-cell;
● the domains of the maps of J are small relative to J-cell;
● the fibrations are exactly those maps that have the right lifting property with
respect to J;
● the trivial fibrations are exactly those maps that have the right lifting
property with respect to I.
We call I the set of generating cofibrations and J the set of generating trivial
cofibrations.
It turns out to be very important that a model category have enough small
objects. A related notion is that of a compact object. We say that an object X
is compact if mapping out of it commutes with filtered colimits. That is, given a
13
functor F ∶ D → C, X is compact if colimC(X, F(D))→ C(X, colim F(D)) whenever
D is a filtered category.
Definition 2.15. [Wal06, 2.4.3] A nonempty category D is filtered if
● for every pair of objects A, B in D there exists an object D in D and maps
A → D, B → D, and
● for every pair of maps f , g ∶ A → B, there exists an object D in D and a
morphism h ∶ B → D such that h f = hg.
The following definitions will help us understand especially nice model
categories, i.e. those that are cellular (2.17), combinatorial (2.19), accessible (2.20),
proper (2.22), or simplicial (2.24).
Definition 2.16. [Hir03, 10.9.1] Let C be a category that is closed under pushouts.
The map f ∶ A → B in C is an effective monomorphism if f is the equalizer of the pair
of natural inclusions B ⇉ B ⊔A B, i.e. if
A B
B B ⊔A B
f
f
is a pullback diagram.
Definition 2.17. [Hir03, 12.1.1] A model category M is cellular if there is a set
of cofibrations I and a set of trivial cofibrations J making M into a cofibrantly
generated model category and also satisfying the following conditions:
● The domains and codomains of I are compact relative to I.
14
● The domains of J are small relative to I.
● Cofibrations are effective monomorphisms.
Definition 2.18. [Dug01, 2.2] A category C is locally presentable if it has all small
colimits and if there is a regular cardinal λ and a set of objects A in C such that
● every object in A is small with respect to λ-filtered colimits, and
● every object of C can be expressed as a λ-filtered colimit of elements of A.
Definition 2.19. [Dug01, 2.1] A model category M is combinatorial if it is
cofibrantly generated and the underlying category is locally presentable.
Definition 2.20. [AR94, 2.1] A category C is accessible if there is a regular cardinal
λ such that
● C has λ-directed colimits, and
● C has a set A of λ-presentable objects such that every object is a λ-directed
colimit of objects from A.
Proposition 2.21. [Cen04] The coslice category of an accessible category is
accessible.
Definition 2.22. [Hir03, 13.1.1] A model category M is left proper if every pushout
of a weak equivalence along a cofibration is a weak equivalence. M is right proper
if every pullback of a weak equivalence along a fibration is a weak equivalence.
If M is both left proper and right proper, we say that M is proper.
Proposition 2.23. [Hir03, 13.1.3] If every object of M is cofibrant, then M is left
proper. If every object of M is fibrant, then M is right proper.
15
Definition 2.24. [Hir03, 9.1.6] A simplicial model category is a model category M
that is enriched over simplicial sets. That is, for every two objects X and Y of M,
Map(X, Y) is a simplicial set, and these sets satisfy composition, associativity, and
unital conditions. Also:
● For every two objects X and Y of M and every simplicial set K, there are
objects X ⊗K and YK of M such that there are isomorphisms of simplicial
sets
Map(X⊗K, Y) ≅ Map(K, Map(X, Y)) ≅ Map(X, YK)
that are natural in X, Y, and K.
● If i ∶ A → B is a cofibration in M and p ∶ X → Y is a fibration in M, then the
map of simplicial sets
Map(B, X) i∗×p∗ÐÐÐ→Map(A, X)×Map(A,Y) Map(B, Y)
is a fibration and is a trivial fibration if either i or p is a weak equivalence.
This condition is known as the pushout-product axiom.
The “right” notion of equivalence of two model categories is that of a Quillen
equivalence.
Definition 2.25. [Hov99, 1.3.1] Let C and D be model categories. A pair of adjoint
functors F ∶ C ⇆ D ∶ U is a Quillen adjunction if the following equivalent definitions
are satisfied:
● F preserves cofibrations and trivial cofibrations.
● U preserves fibrations and trivial fibrations.
16
We call F a left Quillen functor and U a right Quillen functor.
Definition 2.26. [Hov99, 1.3.12] A Quillen adjunction is a Quillen equivalence if
whenever X is a cofibrant object of C and Y is a fibrant object of D, a morphism
X → U(Y) is a weak equivalence in C if and only if the adjunct morphism
F(X)→ Y is a weak equivalence in D.
A Quillen equivalence preserves the model structure on a category. In other
words, if two model categories are Quillen equivalent, then they have the same
homotopy theory. For example, the geometric realization and singular functors
∣− ∣ ∶ sSet ⇆ T op ∶ Sing provide a Quillen equivalence between the category sSet
of simplicial sets and T op. The advantage is that one can transport combinatorial
results from sSet to T op, where calculations tend to be significantly more difficult.
2.2. The Taylor Tower
The goal of this section is to explain Goodwillie’s main theorem [Goo03, 1.13],
which is stated as follows.
Theorem 2.27. A homotopy functor between two appropriate model categories F ∶ C → D
determines a tower of functors PnF ∶ C → D with maps from F
⋮
P2F(X) D2F(X)
P1F(X) D1F(X)
F(X) P0F(X)
q2F
q1F
p2F
p1F
p0F
17
where the functors PnF are n-excisive, pnF is universal among maps from F to n-excisive
functors, the maps qnF are fibrations, the functors DnF = hofib(qnF) are n-homogeneous,
and all the maps are natural.
The sequence
. . . → P2F → P1F → P0F
is called the Taylor tower of F. Ideally, but not necessarily, the inverse limit of the
tower is F. In this section, we explain when and how we construct this tower.
Hypothesis 2.28. The tools of homotopy calculus are applicable in model
categories that are pointed, left proper, and simplicial [Kuh07]. Let C and D
be categories that satisfy these conditions. Additionally, we assume that in
D, the sequential homotopy colimit of homotopy cartesian cubes is homotopy
cartesian. Let F ∶ C → D be a homotopy functor, i.e. a functor that preserves weak
equivalences. We also require that F commutes with filtered homotopy colimits.
Let S be a finite set. Denote the power set of S by P(S) = {T ⊆ S}, which is
partially ordered by inclusion. Let P0 = P(S) ∖ ∅ and P1 = P(S) ∖ S. An n-cube
in C is a functor χ ∶ P(S) → C with ∣S∣ = n. We say χ is (homotopy) cartesian if
χ(∅) → holimT∈P0(S)χ(T) is a weak equivalence, and χ is (homotopy) cocartesian if
hocolimT∈P1(S) → χ(S) is a weak equivalence.
Example 2.29. A 0-cube χ(∅) is cartesian if and only if it is cocartesian if and only
if χ(∅) ≃ 0.
A 1-cube f ∶ χ(∅) → χ({1}) is cartesian if and only if it is cocartesian if and
only if it is a weak equivalence.
18
A 2-cube
χ(∅) χ({1})
χ({2}) χ({1, 2})
is cartesian if it is a homotopy pullback square and cocartesian if it is a homotopy
pushout square.
We say that χ is strongly cocartesian if every two-dimensional face of χ is
cocartesian.
Definition 2.30. [Kuh07, 4.10] A functor F ∶ C → D is n-excisive or polynomial of
degree at most n if, whenever χ is a strongly cocartesian (n + 1)-cube in C, then
F(χ) is a cartesian cube in D.
For example, the identity functor in Sp is 1-excisive because homotopy
pushout squares and homotopy pullback squares coincide in this category.
Another example of a 1-excisive functor is the spectrification functor
Σ∞ ∶ T op → Sp, which preserves homotopy pushouts. In fact, both of these
functors are n-excisive for all n ≥ 1 by the following lemma. For this reason,
we simply refer to 1-excisive functors as excisive or linear.
Lemma 2.31. [Kuh07, 4.16] If F is n-excisive, then F is k-excisive for all k ≥ n.
We wish to construct an n-excisive functor PnF for each n ≥ 0. The idea will be
to construct particular strongly cocartesian (n + 1)-cubes for which this condition
is satisfied and then to show that the condition must be satisfied for all strongly
cocartesian (n + 1)-cubes.
19
If T is a finite set and X is an object in C, let X∗T (“X join T”) be the homotopy
cofiber of the folding map ⊔TX → X. That is, X ∗ T is the homotopy pushout of
the diagram
⊔TX X
0
For example, in T op∗:
● X ∗∅ ≃ X
● X ∗ {1} ≃ CX
● X ∗ {1, 2} ≃ ΣX.
In general, if T is a k-element set, then X ∗T is equivalent to k cones identified
at the open ends.
Let X ∈ Ob(C), let ∣S∣ = n + 1, and let T ⊆ P(S). It turns out that the map
T → X ∗ T defines a strongly cocartesian (n + 1)-cube. The following functor is an
intermediary step in defining the n-excisive functors.
Definition 2.32. [Kuh07, 5.1] Let TnF ∶ C → D be defined by
TnF(X) = holimT∈P0(S)F(X ∗ T).
Note that there is a natural transformation tn(F) ∶ F → TnF since the initial
object of the cube is F(X ∗∅) ≃ F(X). Furthermore, tn(F) is a weak equivalence if
F is n-excisive.
20
For example, T1F(X) is the homotopy pullback of
F(X ∗ {1})
F(X ∗ {2}) F(X ∗ {1, 2})
or, equivalently,
F(CX)
F(CX) F(ΣX)
If F(0) ≃ 0, we say that F is reduced. In this case, F(CX) ≃ 0, so T1F(X) is the
homotopy pullback of
0
0 F(ΣX)
which is weakly equivalent to ΩF(ΣX) by Proposition 2.11.
This process can be repeated. That is, T21 F(X) ≃ T1(T1F)(X) is the homotopy
pullback of the diagram
T1F(X ∗ {1})
T1F(X ∗ {2}) T1F(X ∗ {1, 2})
But T1F(X ∗ {1}) ≃ T1F(0) ≃ ΩF(Σ(0)) ≃ ΩF(0) ≃ Ω(0) ≃ 0, and
T1F(ΣX) ≃ ΩF(ΣΣX) ≃ ΩF(Σ2F). Finally, the homotopy pullback of
21
00 T1ΩF(Σ2X)
is T21 F(X) ≃ Ω2F(Σ2X).
Definition 2.33. [Kuh07, 5.2] Let PnF ∶ C → D be defined by
PnF(X) = hocolim{F(X) tn(F)ÐÐÐ→ TnF(X) tn(TnF)ÐÐÐÐ→ T2n F(X)→ . . .} .
So in our example,
P1F(X) = hocolim{F(X)→ ΩF(ΣX)→ Ω2F(Σ2X)→ . . .}
≃ hocolimn→∞ΩnF(ΣnX).
The composition of the tn’s gives a natural transformation pnF ∶ F → PnF. If F
is n-excisive, then pnF is the homotopy colimit of weak equivalences, so pnF is a
weak equivalence.
The following results are proven in [Goo03] when C = T op∗ and D is T op∗
or Sp, but the proofs apply equally well for any functor F and categories C,D
satisfying Hypothesis 2.28.
Proposition 2.34. 1. PnF is a homotopy functor.
2. PnF is n-excisive.
3. PnF is universal among n-excisive functors G such that F → G.
22
Next, we show that there are natural maps qnF ∶ PnF → Pn−1F for all n ≥ 1.
Note that there is a natural map qn,1 ∶ TnF → Tn−1F because TnF comes from the
homotopy limit of an n + 1-cube, and Tn−1F comes from the homotopy limit of
one of the n-dimensional faces of this cube. Similarly, the inclusion of n-cubes
into (n + 1)-cubes gives natural maps qn,i ∶ TinF → Tin−1F for all i ≥ 1. Moreover,
there is a commutative diagram
F TnF T2n F . . .
F Tn−1F T2n−1F . . .
tn F tnTn F tnT
2
n F
tn−1F tn−1Tn−1F tn−1T2n−1F
qn,1 qn,2=
We define qn as the induced map of the horizontal homotopy colimits.
The homotopy fiber of the map qnF ∶ PnF → Pn−1F is denoted DnF.
Equivalently, for each X ∈ Ob(C), DnF(X) ≃ holim(0 → Pn−1F(X) qn←Ð PnF(X)). In
the analogy with calculus, PnF is like the nth Taylor polynomial of a function. The
functor DnF behaves like the nth term in the Taylor series. In particular, one can
recover PnF if one knows Pn−1F and DnF. Pictured as in Theorem 2.27, we call PnF
the nth stage of the Taylor tower of F and DnF the nth layer. Each functor DnF
is homogeneous of degree n, which means PnDnF ≃ DnF (DnF is n-excisive) and
Pn−1DnF ≃ ∗ (DnF is n-reduced) [Goo03, 1.17]. That is, homogeneous functors
are n-excisive with trivial (n − 1)-excisive part. This terminology is inspired by
homogeneous polynomials, which are concentrated in a single degree.
The following two results are often helpful in proving that DnF is n-excisive
and n-reduced. Again, although Goodwillie assumed that C = T op∗ and D is T op∗
or Sp, the proofs hold for any functor F and categories C,D satisfying Hypothesis
2.28.
23
Proposition 2.35. [Goo91, 3.4] Let ∆ ∶ C → Cn be the diagonal inclusion. If
F ∶ Cn → D is ki-excisive in the ith variable for all 1 ≤ i ≤ n, then the composition
F ○∆ is k-excisive, where k = k1 + . . . + kn.
Proposition 2.36. [Goo03, 3.1] Let ∆ ∶ C → Cn be the diagonal inclusion. If
F ∶ Cn → D is 1-reduced in the ith variable for all 1 ≤ i ≤ n, then the composition
F ○∆ is n-reduced.
If F ∶ T op∗ → T op∗, then
DnF(X) ≃ Ω∞(A ∧ (Σ∞X)∧n)hΣn
where A is a spectrum with an action of the nth symmetric group Σn [Goo03]. This
spectrum A is called the nth derivative of F and is denoted ∂nF. In cases where C is
a sufficiently nice simplicial model category, given a homotopy functor F ∶ C → C,
the layers of the Taylor tower have the form
DnF(X) ≃ Ω∞C (∂nF⊗ (Σ∞C X)⊗n)hΣn
where ∂nF is a C-spectrum with Σn-action, ⊗ is the symmetric monoidal product
in the category of spectra on C induced by the smash product of simplicial sets,
and Σ∞C and Ω∞C are the canonical Quillen adjoint pair between C and the category
of spectra on C [Wal06, p. 5]. We will explain what we mean by the category of
spectra on C in section 4.2.
The identity functor is especially interesting to study because the derivatives
of the identity functor provide a measure of how complicated the homotopy
theory of a category is. There are several categories in which the derivatives of
24
the identity functor are known to have extra structure, specifically, the structure
of an operad.
2.3. Operads
Definition 2.37. [MSS02, I.1.4] Let V be a symmetric monoidal category. An
operad O in V consists of a set of objects O(n), n ≥ 1 of V equipped with
● an action of the symmetric group Σn on O(n),
● an element e ∈ O(1) called the identity, and
● structure maps, also known as composition operations,
γk;n1,...,nk ∶ O(k)⊗O(n1)⊗O(n2)⊗ . . .⊗O(nk)→O(n1 + n2 + . . . + nk).
These data must also satisfy compatibility and associativity conditions, but
we do not list those here. Operads are useful for bookkeeping because they are
an abstraction of a family of composable functions of n variables for various n.
Thus operads encode algebraic properties.
Definition 2.38. We say that an object A in V is an algebra over the operad O if
there is a family of Σn-equivariant morphisms µn ∶ O(n)⊗ A⊗n → A which are
compatible with the structure morphisms of O.
That is, we interpret O(n) as objects of actual n-ary operations on an object
A. We discuss a few common examples.
Example 2.39. Let Ass denote the associative operad, so-named because a vector
space V over a field k is an algebra over the associative operad if and only if V
25
is an associative algebra. Here A(n) = k⟨Σn⟩, the vector space over k with basis
σ1, . . . ,σn! ∈ Σn. The action of Σn on A(n) is the composition of permutations, so
( n!∑
i=1 kiσi) g =
n!∑
i=1 ki(σig).
The identity is the unit in A(1). Finally, the maps
γn;m1,...,mn ∶ A(n)⊗ A(m1)⊗ . . . A(mn)→ A(m1 + . . . +mn)
simply multiply elements.
Suppose that a vector space V is an algebra over Ass. The map
µn ∶ A(n)⊗V⊗n → V is given by µn(σ, v1, . . . vn) = vσ−1(1) ⋅ . . . ⋅ vσ−1(n). For example,(123) is a basis element of A(3), and (123)⊗ v ⊗ w ⊗ z = zvw. The existence of
the map A(2)⊗V ⊗V → V shows that V has multiplication, so V is an algebra,
and the associativity condition from the definition of an operad implies that V is
associative.
Example 2.40. Let Com denote the commutative operad, so-named because a vector
space V over k is a commutative algebra if and only if V is an algebra over the
commutative operad. Here C(n) = k for all n ≥ 1. The action of Σn on C(n) is the
trivial action, i.e. g ⋅ x = x for all g ∈ Σn and x ∈ C(n) = k . The identity is the unit
in C(1). Again, the structure maps multiply elements.
Suppose that a vector space V is an algebra over the commutative operad.
The map µn ∶ C(n)⊗V⊗n → V is given by k1 ⋅ (v1, . . . , vn) ↦ k1v1 . . . vn. As before,
the algebra structure comes from µ2. By the equivariance condition, given σ ∈ Σn,
µn(σ(k1),σ(v1, . . . , vn)) = µn(k1, v1, . . . , vn),
26
or
k1vσ(1) . . . vσ(n) = k1v1 . . . vn,
so the algebra must be commutative.
Example 2.41. Let Lie denote the Lie operad. A vector space over k is a Lie algebra
if and only if it is an algebra over the Lie operad. Here Lie(n) is the differential
graded vector space concentrated in degree 0 generated by all abstract bracket
expressions of n elements (e.g. [x1, [x2, . . . , [xn−1, xn]⋯]]) modulo anti-symmetry
and the Jacobi identity. Note that the dimension of Lie(n) as a k-vector space is
(n−1)! [GK+94, 1.3.9]. The action of Σn on Lie(n) is generated by the permutation
of the elements in a bracket expression with no negative signs. The identity is the
unit in Lie(1) ≅ k. The maps µn “plug in” elements. For example, given a Lie
algebra g, µ3 ∶ Lie(3)⊗ g⊗3 → g behaves as
[x1, [x2, x3]]⊗ x⊗ y⊗ z ↦ [x, [y, z]][x2, [x3, x1]]⊗ x⊗ y⊗ z ↦ [y, [z, x]][x3, [x1, x2]]⊗ x⊗ y⊗ z ↦ [z, [x, y]]
The link between Goodwillie calculus and operads is that there are several
categories in which the derivatives of the identity functor are known to be the
objects of an operad. Such categories include
● T op∗ [Chi05],
● Sp [AC11],
● the category of algebras over an operad in the category of symmetric spectra
[HH13], and
27
● DGLr, the category of differential graded Lie algebras over Q that are 0
below grading r [Wal06].
In fact, the derivatives of the identity functor in DGLr are the objects of theLie operad, though the grading is shifted. We prove this result in the next chapter.
28
CHAPTER III
DIFFERENTIAL GRADED LIE ALGEBRAS
A differential graded vector space (DG) is V = (V●, dV), where V● is a graded
vector space and dV = {di ∣ di ∶ Vi → Vi−1} is a differential. If v ∈ Vn, we say
that the degree of v is n and write ∣v∣ = n. Given differential graded vector spaces
V = (V●, dV) and W = (W●, dW), a DG map f ∶ V → W is a family of vector space
maps fk ∶ Vn →Wn+k for all n. The integer k is called the degree of f . Let DG denote
the category of differential graded Q-vector spaces with degree 0 maps. A quasi-
isomorphism of DGs is a map that induces an isomorphism on homology.
A DG V is called r-reduced if Vk = 0 for all k < r. We write DGr for the full
subcategory of DG consisting of all r-reduced DGs. The r-reduction functor
redr ∶ DG → DGr acts by (redrV)k = Vk for k > r, (redrV)k = 0 for k < r, and(redrV)r = ker(dr ∶ Vr → Vr−1). Note that Hk(redr(V)) = Hk(V) for all k ≥ r, and
Hk(redr(V)) = 0 for all k < r. There are also shift functors s, s−1 ∶ DG → DG given
by (s(V))i = Vi−1 and (s−1(V))i = Vi+1. The category of graded vector spaces is
denoted G, and the category of r-reduced graded vector spaces is denoted Gr.
We are interested in differential graded Lie algebras, which are DGs with
extra structure.
Definition 3.1. [Wal06, 4.1] A differential graded Lie algebra (DGL) L = (L●, dL, [−,−]L)
consists of a differential graded vector space (L●, dL) equipped with a linear,
degree zero, graded DG map
[−,−] ∶ (L●, dL)⊗ (L●, dL)→ (L●, dL)
29
satisfying
● [x, y] = −(−1)∣x∣⋅∣y∣[y, x] (graded anti-symmetry) and
● [x, [y, z]] + (−1)∣x∣(∣y∣+∣z∣)[y, [z, x]] + (−1)∣z∣(∣x∣+∣y∣)[z, [x, y]] = 0 (graded Jacobi
identity).
We call [−,−] the Lie bracket map. Because the Lie bracket is a DG map,
dL[x, y] = [dLx, y]+ (−1)∣x∣[x, dLy]. Maps of DGLs are degree 0 graded vector space
maps f ∶ V● → W● such that f (dV x) = dW f (x) and f ([x, y]) = [ f (x), f (y)]. We
denote the category of DGLs with these maps by DGL.
A DGL L is r-reduced if Lk = 0 for all k < r. The category of all r − reduced
DGLs and DGL maps is denoted DGLr. The reduction functor redr ∶ DGL→ DGLr
is the restriction of the reduction functor on DG. The category of graded Lie
algebras and degree 0 Lie algebra maps is denoted GL. Given V ∈ Ob(DG), we
write [V]DGL to represent the DGL that has V as the underlying DG and a trivial
Lie bracket, i.e. [v, w] = 0 for all v, w ∈ V. Similarly, we write [L]DG when we want
to consider an object L of DGL as an object of DG and [L]GL when we want to
forget the differential of L.
We define (−)ab ∶ DGLr → DGr by (L)ab = L/[−,−], the graded abelianization
of L. By (L)ab, we really mean [(L)ab]DG since (L)ab is actually an object of DGL
with a trivial Lie bracket. Note that (−)ab ∶ DGLr ⇄ DG ∶ [−]DGL ○ redr are adjoint
functors.
Quillen showed that DGLr is a model category for each r ≥ 1.
Theorem 3.2. [Qui69, 5.1] There is a model structure on DGLr, r ≥ 1 in which
the weak equivalences are quasi-isomorphisms, the fibrations are degree-wise
30
surjections in degrees k > r, and the cofibrations are maps which have the left
lifting property with respect to trivial fibrations.
We need to describe the cofibrant objects in this category. Given V ∈ Ob(Gr),
we construct LV ∈ Ob(DGLr) by taking the tensor algebra TV =⊕
k
V⊗k, defining
the Lie bracket as [x, y] = x⊗ y − (−1)∣x∣⋅∣y∣y⊗ x, and taking LV to be the sub-Lie
algebra of TV generated by V. We say that L ∈ Ob(DGLr) is free if [L]GL ≅ LV for
some graded vector space V. The cofibrant objects in DGLr are these free DGLs
[Wal06, 4.2.3]. We write fDGLr to denote the full subcategory of free DGLs inDGLr.
Note that the functor L(−) ∶ Gr → GLr defined by V ↦ LV is the left adjoint
of the forgetful functor [−]Gr ∶ GLr → Gr. We point out that L(−) is not left adjoint
to the forgetful functor [−]DGr ∶ DGLr → DGr, as one might expect from the name
“free DGL.” That is, LV is free in GLr but not in DGLr. The forgetful functor[−]DGr does have a left adjoint L(−) ∶ DGr → DGL, which is defined as above but
on differential graded vector spaces rather than graded vector spaces, thus with
consequences for the differential on the resulting DGL. DGLs which are free in
this sense are known in the literature as “truly free” DGLs.
Homotopy theorists are interested in differential graded Lie algebras because
DGL1 and the category of rational simply connected topological spaces, T opQ,
have equivalent homotopy categories. In particular, Quillen [Q69] constructed
a chain of Quillen equivalences from T opQ to DGL1 in which the homology of
the DGL corresponding to a rational simply-connected space XQ is equal to the
shifted homotopy of XQ.
Because of this relationship, it is reasonable to ask what the category of
spectra on DGL1 is, and more generally, what the category of spectra on DGLr−1
31
is for r ≥ 2. One may construct rational spectra by considering the stabilization of
DGLr−1. In particular, we have the following definition.
Definition 3.3. [Wal06, 6.2.1] A DGL-spectrum E is a sequence of DGLs {Li}i≥0
equipped with maps Li → ΩLi+1 = [redr(s−1[Li+1]DG)]DGL.
In fact, the category of DGL-spectra is isomorphic to DG [Wal06, 6.2.3].
Therefore DG is to DGL as Sp is to T op. Differential graded rational vector spaces
behave like rational spectra, and differential graded rational vector spaces that
are r-reduced behave like (r − 1)− connected rational spectra. Moreover, there is a
spectrification functor analogous to Σ∞ ∶ T op → Sp.
Definition 3.4. [Wal06, 6.2.5] Define Σ∞DGL ∶ DGLr−1 → DGr by Σ∞DGL(L) = s(L)ab
and Ω∞DGL ∶ DGr → DGLr−1 by Ω∞DGL(V) = [s−1redrV]DGL.
Lemma 3.5. [Wal06, 6.2.6] Σ∞DGL and Ω∞DGL are adjoint functors.
Walter’s main theorem is stated as follows.
Theorem 3.6. [Wal06, 8.1.1] The nth derivative of the identity functor
1DGL ∶ DGLr−1 → DGLr−1, r ≥ 2 is Lie(n) graded in degree (1 − n) with Σn-action
twisted by the sign of permutations.
The goal of this chapter is to present Walter’s proof. First, Walter shows that
Theorem 2.27 applies in DGLr−1. In particular, since 1DGL is a homotopy functor,
there is a universal approximating tower of fibrations
. . . → Pn1DGL → Pn−11DGL → . . . → P11DGL → P01DGL
with homotopy fibers Dn1DGL → Pn1DGL → Pn−11DGL.
32
Let L ∈ Ob( fDGLr−1). It suffices to consider free DGLs because the cofibrant
replacement of any DGL is free, and Dn1DGL is a homotopy functor. Because of
our analogy with T op, we expect these homotopy fibers to have the form
Dn1DGL(L) ≃ Ω∞DGL(∂n1DGL ⊗ (Σ∞DGLL)⊗n)Σn≃ [s−1(∂n1DGL ⊗ (s(L)ab)⊗n)Σn]DGL≅ [(sn−1∂n1DGL ⊗ ((L)ab)⊗n)Σn]DGL
where ∂n1DGL is a DG with Σn-action. Here Σn acts on [s(L)ab]⊗nDG and[(L)ab]⊗nDG by permutation of elements with signs according to the Koszul
convention and on sn−1 by multiplication by (−1)sgn(σ). That is, the Σn-equivariant
isomorphism on the last line is given by applying s−1 to the Σn-equivariant map(sx1 ⊗⋯⊗ sxn)↦ (−1)ksn ⊗ x1 ⊗⋯⊗ xn, where k = n−1∑
j=1
j∑
i=1 ∣xi∣ is the sign incurred by
moving all of the s’s to the beginning of the expression.
We will show that the Taylor tower of 1DGL is the same as a tower of quotients
of the lower central series of L. Recall that the lower central series of L is
L ⊃ [L, L] ⊃ [L, [L, L]] ⊃ . . .
Let L = Γ1(L), [L, L] = Γ2(L), [L[L, L]] = Γ3(L), etc. Then this chain of inclusions
induces the following tower by taking quotients:
33
⋮
L/Γ3(L) =∶ B2(L)
L/Γ2(L) =∶ B1(L)
L/L = 0
Each map is degree-wise surjective, so each map is a fibration in DGL. The
limit of this tower is L. This is because L being (r − 1)-reduced means that Γn(L)
is n(r − 1) reduced, so Lk = (L/Γn(L))k for n sufficiently large. When L is free, this
tower is called the bracket-length filtration of L since, for L = (LV , d), we have
Bn(L) = ((T≤nV) ∩LV , d = d0 + . . . + dn). That is, Bn(L) consists of the elements of
LV with bracket-length at most n. Here di is a differential on LV that increases
bracket length by i.
Let Hn = hofib(Bn fÐ→ Bn−1). By [Wal06, 4.2.13], Hn = redr((s−1Bn−1 × Bn), d, [−,−]),
where
● d(s−1a, b) = (s−1( f (b)− d(a)), db)
● [s−1a, b] = 12 s−1[a, f (b)]Bn−1
● [b1, b2] = [b1, b2]Bn
● [s−1a1, s−1a2] = 0
Note Hn(L)/(Tn ∩LV) ≃ hofib(Bn−1 idÐ→ Bn−1), and this homotopy fiber has
zero homology. Hence Hn(L)/(Tn ∩LV) has zero homology, meaning the
34
inclusion (Tn ∩LV)↪ Hn(L) = (s−1Bn−1 × Bn) given by x ↦ (0, [x]) is a quasi-
isomorphism. Therefore Hn(L) ≃ Tn ∩LV , i.e. Hn(L) is all elements of L of
bracket length n.
However, [(Lie(n)⊗ ((L)ab)⊗n)Σn]DGL also represents the elements of L with
bracket length n. This is because Lie(n) is the nth space of the Lie operad, and
[(Lie(n)⊗ ((L)ab)⊗n)Σn]DGL is a free Lie algebra concentrated in bracket length
n. Since L = LV and the free object of the Lie operad on V both give a free Lie
algebra on (L)ab ≅ V, we must have Hn(L) ≃ [(Lie(n)⊗ ((L)ab)⊗n)Σn]DGL.
Next, we show that Hn ∶ fDGLr−1 → DGLn(r−1) is an n-homogeneous functor.
Consider the case when n = 1. We observe that H1(L) ≃ T1 ∩LV ≃ (L)ab. We must
show that P1H1 = H1 and P0H1 ≃ ∗. First, consider the homotopy pushout square
of free DGLs
LV LW
LX LX⊕sV⊕W
Applying H1 to this square, we get
V W
X X⊕ sV ⊕W
Since V ≃ X ⊕ s−1(X ⊕ sV ⊕ W) ⊕ W, this is a homotopy pullback square,
and hence H1 is 1-excisive. Certainly if V ≃ 0, then LV ≃ 0. Also, if LV ≃ 0,
then H1(LV) = (LV)ab ≃ V ≃ 0. Therefore H1 is 1-reduced. By Proposition 2.35,
since the functor (−)ab is 1-excisive, the functor ((−)ab)⊗n is n-excisive. Similarly,
35
by Proposition 2.36, (−)ab being 1-reduced implies that ((−)ab)⊗n is n-reduced.
Cartesian cubes of dimension n + 1 are preserved by tensoring with Lie(n). It
follows that Hn is n-excisive and n-reduced.
Now we show that Bn is n-excisive for all n by induction. Note that
B0 = L/L = 0, so the fiber sequence H1 → B1 → B0 implies H1 ≃ B1. Since H1 is
1-homogenous, so is B1.
Assume that for all 1 ≤ i ≤ n − 1, Bi is i-excisive. Note this implies that
B0, . . . , Bn−1 are n-excisive. Consider the following diagram.
Hn(L) Bn(L) Bn−1(L)
PnHn(L) PnBn(L) PnBn−1(L)
Since Hn and Bn−1 are n-excisive, the outside vertical maps are equivalences.
Each of the horizontal sequences induces a long exact sequence in homology.
By the Five Lemma, H∗(Bn(L)) → H∗(PnBn(L)) is an isomorphism. Thus
Bn(L)→ PnBn(L) is a weak equivalence, so Bn is n-excisive.
Therefore this tower is an approximating tower of fibrations of n-excisive
functors converging to 1DGL in the sense that the maps L → Bn(L) are vector
space isomorphisms up to degree n(r − 1). By the universality of Pn1DGL, we
know Pn1DGL ≃ PnBn, and PnBn ≃ Bn since Bn is n-excisive. Therefore the bracket-
length filtration is the rational Taylor tower of 1DGL evaluated on L. By defining
∂n1DGL to be Lie(n) graded in degree (1 − n) with Σn-action twisted by the sign
of permutations, we find that [(Lie(n)⊗ ((L)ab)⊗n)Σn]DGL matches the expected
Dn1DGL(L) ≃ [(sn−1∂n1DGL ⊗ ((L)ab)⊗n)Σn]DGL .
36
CHAPTER IV
THE CATEGORY OF SMALL CATEGORIES
In this chapter, we consider Cat∗, the category of pointed small categories
and functors between them. Our goal is to calculate the derivatives of the identity
functor on Cat∗. As we have previously noted, the derivatives should be objects
in the category of spectra on Cat∗. In order to construct this category of spectra,
we must first determine how the suspension functor behaves in Cat∗.
The zero object 0 in Cat∗ is the category with one object and only the identity
morphism. Then any category C has a chosen morphism 0 → C. We call the image
of this morphism the basepoint of C and denote it by *. We use the canonical
model structure on Cat, which was first explicitly described by Rezk [Rez96, 3.1].
That is, we define a morphism to be
● a weak equivalence if it is an equivalence of categories (equivalently, if it is
fully faithful and essentially surjective),
● a cofibration if it is injective on objects, and
● a fibration if it is an isofibration. An isofibration is a functor F ∶ C → D such
that for any object C ∈ Ob(C) and any isomorphism φ ∶ F(C) ≅Ð→ D, there is
an isomorphism ψ ∶ C ≅Ð→ C′ such that F(ψ) = φ.
Let T be the category with two objects and only identity morphisms:
∗ A
37
Let I be the category with a disjoint basement, two other objects, and unique
isomorphisms between them:
∗ A B≅
Let 1 be the category with a disjoint basepoint, two other objects, and a map
α between them:
∗ A Bα
Let 1˙ be the maximal subcategory of 1 not containing α:
∗ A B
Finally, let P = 1 ⊔1˙ 1 be the category consisting of a disjoint basepoint, two
other objects, and a pair of parallel arrows:
∗ A B
That is, P is the pushout of the diagram (1 ← 1˙ → 1), where both of the maps are
inclusions.
38
Proposition 4.1. The canonical model structure on Cat∗ is cofibrantly generated with
generating trivial cofibration J = {j}, where j = T ↪ I, and generating cofibrations
I = {u, v, w}, where u ∶ 0 → T and v ∶ 1˙ → 1 are inclusions and w ∶ P → 1 identifies
the parallel arrows.
Proof. This is the pointed version of the structure given by Rezk, and the proof is
analogous. First, consider a commutative square
T C
I D
Fj
Note that a lift exists precisely when F is an isofibration, so J determines the
fibrations.
Next, we show that a functor F is a trivial fibration if and only if F has the
right lifting property with respect to every map in I. Each map in I is a cofibration,
and hence every trivial fibration has the right lifting property with respect to each.
Conversely, suppose F ∶ C → D has the right lifting property with respect to each
map in I. Since F has the right lifting property with respect to u, F is surjective
on objects. Since F has the right lifting property with respect to v, F is surjective
on Hom sets. Finally, since F has the right lifting property with respect to w, F
is injective on Hom sets. Therefore F is a weak equivalence. Note that F also has
the right lifting property with respect to j, and so F is a trivial fibration.
Proposition 4.2. Cat∗ is a proper model category.
39
Proof. Clearly, all objects in Cat∗ are cofibrant. It is also true that all objects in Cat∗
are fibrant. It suffices to show that every map C → 0 has the right lifting property
with respect to j ∶ T ↪ I. Indeed, given a commutative square
T C
I 0
we define a lift by sending both non-basepoint objects of I to the image of the
non-basepoint object of T in C and the isomorphism of I to the identity morphism
on that object. By Proposition 2.23, Cat∗ is proper.
Proposition 4.3. Cat∗ is a simplicial model category.
Proof. Rezk shows that Cat satisfies the pushout-product axiom [Rez96, 5.1], and
the proof for the pointed case is analogous. We define a pair of adjoint functors
pi ∶ sSet∗ ⇄ Cat∗ ∶ µ
as follows. Let µ take a category C to the simplicial nerve of the subcategory C′ ⊆ C
having Ob C′ = Ob C and having as morphisms the isomorphisms of C. Let pi take
a simplicial set K to the category piK, where Ob(piK) = K0, and where there is
a generating isomorphism k ∶ d1k → d0k for each k ∈ K1 subject to the relation
d0l ⋅ d2l = d1l for each l ∈ K2. The category piK is called the fundamental groupoid of
K.
We show that (pi,µ) is a Quillen adjunction by showing that pi preserves
cofibrations and trivial cofibrations. Since the cofibrations in sSet∗ are injective on
40
n-simplices for all n, it is immediate that pi preserves cofibrations. Consider the
generating trivial cofibrations in sSet∗
ιn,k ∶ Λk[n]→ ∆[n], n ≥ 1, 0 ≤ k ≤ n.
For n > 1, the map piιn,k is an isomorphism because both the source and target
are mapped to the category with n + 1 objects and trivial automorphism groups.
Also, pi∆[1] is the category with two objects and unique isomorphisms between
them, so piι1,k ∶ 0 → pi∆[1] is an equivalence of categories for k = 0, 1. Therefore
each piιn,k is a trivial cofibration in Cat∗. Since pi preserves the generating trivial
cofibrations, pi preserves all trivial cofibrations.
Let DC = Hom(C,D) denote the category whose objects are functors F ∶ C → D
and whose morphisms are natural transformations. Define the tensoring
⊗ ∶ Cat∗ × sSet∗ → Cat∗ by C ⊗K = (C ×piK)/(∗×piK). By ∗ × piK, we mean the
category where Ob(∗×piK) ≅ Ob(piK) and Mor(∗×piK) ≅ Mor(piK). Define the
powering Cat∗ × sSetop∗ → Cat∗ by CK = CpiK and the enrichment Catop∗ × Cat∗ → sSet∗
by Map∗(C,D) = µ(DC).
We must show that these definitions satisfy the isomorphisms of Definition
2.24. Let C,D ∈ Ob(Cat∗) and K ∈ Ob(sSet∗). Then
41
Map∗(C ⊗K,D) = µ(Hom(C ⊗K,D)) by definition of the enrichment≅ µ(Hom(piK, Hom(C,D)))
≅ µ(Hom(piK,DC))
≅ Map∗(K,µ(DC))≅ Map∗(K, Map∗(C,D)) by definition of the powering
and
Map∗(C ⊗K,D) = µ(Hom(C ⊗K,D)) by definition of the enrichment≅ µ(Hom(C, Hom(piK,D)))
= µ(Hom(C,DK)) by definition of the powering
≅ Map∗(C,DK) by definition of the enrichment.
4.1. Suspensions of Categories
We will identify the cylinder objects in Cat∗ and use this to construct
suspensions (see Definition 2.9). Fortunately, there is a natural choice of cylinder
object in any simplicial model category [Pel11, 1.6.6].
Proposition 4.4. Given C ∈ Ob(Cat∗), a cylinder object for C is Cyl(C) = C ⊗∆1.
Proof. Label the objects in pi∆1 by ∗ and C, and let f denote the unique
isomorphism ∗ → C. The objects of Cyl(C) are pairs of the form (A,∗) or
(A, C), where A ∈ Ob(C). Note that (∗,∗) = (∗, C), so Cyl(C) has the same
42
objects as C ∨ C. In particular, C ∨ C ↪ Cyl(C). Furthermore, there is a unique
isomorphism fA = (idA, f ) between (A,∗) and (A, C) for all A ∈ Ob(C), so
MorCyl(C)((A, C), (A′, C′)) ≅ MorC(A, A′). Therefore the map given by projection
onto the first factor Cyl(C)→ C is a weak equivalence.
Definition 4.5. Given two objects a and b in a category C, if there is a zig-zag of
morphisms from a to b, then we say that a and b are in the same component of C.
Proposition 4.6. Given a category C, ΣC has one object A, and End(A) = F(n − 1), the
free group on n − 1 generators, where n is the number of components of C.
Proof. In taking the pushout of the diagram
C ∨ C Cyl(C)
0
we identify all the objects of Cyl(C) with the unique object of 0 and all the
morphisms of C ∨ C with the unique morphism of 0. If there is a morphism
α ∶ ∗ → a in C, then fa ○ α = α implies that im( fa) ○ id0 ≃ id0, or equivalently,
im( fa) ≃ id0 in the pushout. In turn, the uniqueness of the isomorphisms fa and
fb implies that all morphisms in the same component as the basepoint of Cyl(C)
get identified to id0 in the pushout. If a and b are in a component of C that does
not contain the basepoint and there is a morphism α ∶ a → b, then fb ○ α = α ○ fa
implies that im( fb) ○ id0 ≃ id0 ○ fa, or equivalently, im( fa) ≃ im( fb) in the pushout.
Therefore we are left with one isomorphism for each component of C that does
not contain the basepoint.
43
We have shown that the category F(n − 1) completes the diagram. In order
to prove that F(n − 1) is the pushout, we must show that for any other category
D that makes the diagram commute, there is a unique map p ∶ F(n − 1) → D that
makes both triangles commute.
C ∨ C Cyl(C)
0 F(n − 1)
D
g
h
p
Since the outer square commutes, h sends all objects of Cyl(C) to the
basepoint of D, which we denote by ∗D, and all morphisms of Cyl(C) to
isomorphisms of ∗D. In particular, h maps all morphisms of C to the identity
morphism of ∗D, and the same argument as above shows that if a and b are in the
same component of C, then h( fa) = h( fb).
Define p ∶ F(n− 1)→ D by p(∗Fn−1) = ∗D and p(g( fa)) = h( fa) for all a ∈ Ob(C).
Since p is determined by g and h, p is unique. We remark that p defines an
isomorphism between F(n − 1) and some subcategory of D.
Corollary 4.7. For any category C, Σ2C ≃ 0.
Proof. For any category C, ΣC has one component. Thus Σ2C has one object A and
End(A) ≃ F(0) ≃ idA.
Similarly, there is a standard path object in any simplicial model category
[Pel11, 1.6.7].
44
Proposition 4.8. Given C ∈ Ob(Cat∗), a path object for C is C I = Hom(pi∆1,C).
Proof. The objects in C I are isomorphisms g ∶ A0 → A1 in C. Morphisms u ∶ g → h
between g and h ∶ B0 → B1 are pairs u = (u0, u1) so that u1 ○ g = h ○u0. The basepoint
of C I is the morphism pi∆1 → ∗.
Note that there is a functor f ∶ C → C I given by A ↦ idA and that f is a weak
equivalence. Let (s, t) ∶ C I → C × C, where s is the source map and t is the target
map. Then the composition C → C I → C × C is the diagonal map. Also, (s, t) is a
fibration. Let a ∶ A0 → A1 be an object of C I , and let (u0, u1) ∶ (A0, A1)→ (B0, B1) be
an isomorphism in C × C. Then there is a unique isomorphism b ∶ B0 → B1 making
the square commute. Thus u = (u0, u1) defines an isomorphism a → b in C I such
that (s, t)(u) = (u0, u1).
Proposition 4.9. For any category C, Ω2C ≃ 0.
Proof. The pullback ΩC of the diagram
C I
0 C × C
(s, t)
has the set of objects {a ∶ A0 → A1 ∣ s(a) = t(a) = ∗}. That is, ΩC has one object
for each isomorphism of the basepoint of C. Let f , g ∈ Ob(ΩC), so f , g ∶ ∗ → ∗.
A morphism u ∶ f → g in ΩC is a pair of morphisms u = (u0, u1) satisfying the
commutative diagram
45
∗ ∗
∗ ∗
u1
g
f
u0
and such that (s, t)(u) = (u0, u1) = (id, id). Thus Mor( f , g) = (id, id) if f = g. If f ≠ g,
then there are no such commutative diagrams, and so Mor( f , g) = ∅. Finally, Ω2C
is the pullback of the diagram
ΩC I
0 ΩC ×ΩC
which has one object for each isomorphism of the basepoint of ΩC. The basepoint
of ΩC corresponds to the identity morphism of the basepoint of C, and the only
morphism of the basepoint of ΩC is (id, id). Therefore Ω2C ≃ 0.
4.2. Categories of Spectra
Now we are ready to define the category of spectra on Cat∗. Hovey [Hov01]
gives a method for constructing a category of spectra SpN(C, G) for certain model
categories C and functors G ∶ C → C.
Definition 4.10. [Hov01, 1.1] Let G be a left Quillen endofunctor of a left proper
cellular model category C. Define SpN(C, G), the category of spectra on C, as follows.
A spectrum X is a sequence {Xn}n≥0 of objects of C together with structure maps
σ ∶ GXn → Xn+1 for all n. A map of spectra from X to Y is a collection of maps
fn ∶ Xn → Yn so that the following diagram commutes for all n:
46
GXn Xn+1
GYn Yn+1
σX
G fn
σY
fn+1
We are interested in the category SpN(Cat∗,Σ). By Proposition 2.23, since all
objects of Cat∗ are cofibrant, Cat∗ is left proper. However, we still cannot use
Hovey’s construction as it is given.
Proposition 4.11. Cat∗ is not cellular.
Proof. Consider the generating cofibration w ∶ P → 1. Note 1 ⊔P 1 = 1, so
eq(1 ⇉ 1 ⊔P 1) = 1, not P. Therefore w is not an effective monomorphism.
Fortunately, Hovey only requires cellularity in order to guarantee that a
particular Bousfield localization exists. We will take advantage of the fact that
localizations also exist in left proper combinatorial model categories. We will say
more about this later, but for now, we proceed with the construction.
Given n ≥ 0, the evaluation functor Evn ∶ SpN(Cat∗,Σ) → Cat∗ takes X to
Xn. The evaluation functor has a left adjoint Fn ∶ Cat∗ → SpN(Cat∗,Σ) defined by(FnX)m = Σm−nX if m ≥ n and (FnX)m = 0 for m < n with the obvious structure
maps. By results in [Hov01], SpN(Cat∗,Σ) is bicomplete and can be given what is
called the projective model structure. This is a preliminary step toward making
the model structure that we actually wish to use, the stable model structure.
Definition 4.12. In the projective model structure on SpN(Cat∗,Σ), a map of spectra
f ∶ X → Y is
● a weak equivalence if fn ∶ Xn → Yn is a weak equivalence in Cat∗ for all n ≥ 0.
47
● a fibration if fn ∶ Xn → Yn is a fibration in Cat∗ for all n ≥ 0.
● a cofibration if f has the left lifting property with respect to all trivial
fibrations in SpN(Cat∗,Σ).
Furthermore, this model structure is cofibrantly generated with generating
cofibrations IΣ = ⋃n Fn I and generating trivial cofibrations JΣ = ⋃n Fn J, where I
and J are the generating cofibrations and generating trivial cofibrations of Cat∗.
Note that the zero object in SpN(Cat∗,Σ) is the spectrum 0, where 0n = 0 for all
n ≥ 0.
The functor Σ extends to a functor on SpN(Cat∗,Σ) called the prolongation
of Σ. The prolongation of Σ is given by Σ ∶ SpN(Cat∗,Σ)→ SpN(Cat∗,Σ), where(ΣX)n = ΣXn with the obvious structure maps. The functor Σ is a left Quillen
functor with respect to the projective model structure. However, Σ is not a Quillen
equivalence. For example, ΣF(1) ≃ 0, but F(1) is not weakly equivalent to Ω0 ≃ 0.
Our goal is to define a model structure on SpN(Cat∗,Σ), called the stable model
structure, with respect to which Σ is a Quillen equivalence. In order to do so, we
need SpN(Cat∗,Σ) to be a combinatorial model category. We begin by showing
that Cat∗ is combinatorial.
Proposition 4.13. Cat∗ is a proper combinatorial simplicial model category.
Proof. We have already seen that Cat∗ is cofibrantly generated (Proposition 4.1),
proper (Proposition 4.2), and simplicial (Proposition 4.3). Note that Cat is locally
(finitely) presentable because it is equivalent to the category of models of a finite
limit sketch [BW08]. In particular, this category is accessible. By Proposition
2.21, Cat∗ is accessible. Therefore Cat∗ is also locally presentable and hence
combinatorial.
48
Proposition 4.14. [Hov01, 1.15] A map i ∶ A → B in SpN(Cat∗,Σ) is a (trivial)
cofibration in the projective model structure if and only if the maps A0 → B0 and
An ⊔ΣAn−1 ΣBn−1 → Bn for n ≥ 1 are (trivial) cofibrations in Cat∗.
Proposition 4.15. Every object in SpN(Cat∗,Σ) is cofibrant with respect to the projective
model structure.
Proof. Let B ∈ Ob(SpN(Cat∗,Σ)). Certainly 0 → B0 is a cofibration in Cat∗. Also,
0 ⊔0 ΣBn−1 ≃ ΣBn−1 → Bn is a cofibration in Cat∗ for all n ≥ 1 since ΣBn−1 has one
object.
The above proposition together with Proposition 2.23 implies that
SpN(Cat∗,Σ) is left proper. We also could have used the following lemma.
Lemma 4.16. [Hov01, 1.14] With the projective model structure, SpN(C,Σ) is left
proper (resp. right proper, proper) if C is left proper (resp. right proper, proper).
Lemma 4.17. [Sch97, 2.1.5] Let C be a pointed proper simplicial model category
which admits the small object argument. Then SpN(C,Σ) is a simplicial model
category.
In particular, if X is a spectrum, define X ⊗ K by (X ⊗ K)n = Xn ⊗ K. The
powering and enrichment over simplicial sets is also defined level-wise.
Proposition 4.18. SpN(Cat∗,Σ) is a left proper combinatorial simplicial model category.
Proof. By Lemmas 4.16 and 4.17, since Cat∗ is proper and simplicial, so is
SpN(Cat∗,Σ). Also, we showed in the proof of Proposition 4.13 that Cat∗ is locally
49
presentable. Thus there is a set of morphisms K that generate all morphisms
in Cat∗ over colimits. Let KΣ = ⋃n FnK. Then KΣ is a set, and KΣ generates the
morphisms of SpN(Cat∗,Σ) over colimits because colimits in SpN(Cat∗,Σ) are
computed level-wise. Since SpN(Cat∗,Σ) is cofibrantly generated, we conclude
that SpN(Cat∗,Σ) is combinatorial.
Now we describe Bousfield localization. Bousfield localization is a procedure
that takes a model structure and produces a new one with the same cofibrations
and more weak equivalences. It allows us to turn the maps of a set S into weak
equivalences while keeping the model category axioms satisfied.
Let A be an object in a simplicial model category M. By [Hir03, 16.1.3]), there
is a functorial cosimplicial resolution of A induced by the functorial factorizations
of M. This cosimplicial resolution A● is given by An = QA⊗∆[n]. In general, A●
is a cofibrant replacement for the constant cosimplicial object cc∗A in the Reedy
model structure on M∆. By mapping out of this cosimplicial resolution, we get a
simplicial set Mapl(A●, RX). The face and degeneracy maps of this simplicial set
are induced by the coface and codegeneracy maps of A●.
Dually, there is a functorial simplicial resolution X● of X, where X● is a fibrant
replacement of the constant simplicial object cs∗X in the Reedy model category
structure on M∆op . By mapping into it, we get a simplicial set Mapr(QA, X●). We
define the homotopy function complex
map(A, X) ∶= Mapr(QA, RX) ≅ Mapl(QA, RX),
50
where these two sets are naturally isomorphic in the homotopy category of
simplicial sets. In fact, map(A, X) ≅ Map(QA, RX).
Definition 4.19. [Hov01, 2.1] Let S be a set of maps in a simplicial model category
M.
1. An S-local object of M is a fibrant object W such that, for every f ∶ A → B
in S , the induced map map(B, W) → map(A, W) is a weak equivalence of
simplicial sets.
2. An S-local equivalence is a map g ∶ A → B in M such that the induced map
map(B, W)→map(A, W) is a weak equivalence of simplicial sets for all
S-local objects W.
The stable model structure on SpN(Cat∗,Σ) is a Bousfield localization of the
projective model structure. This process does not necessarily produce a model
structure given any model category M and any set of maps S , but it does when M
is combinatorial. For the following theorem, the reader should take the universe
X to be Set, the category of sets. Then, for example, “X-combinatorial” matches
our definition of combinatorial.
Theorem 4.20. [Bar10, 4.7] If M is left proper and X-combinatorial (X some
universe), and S is an X-small set of homotopy classes of morphisms of M, the left
Bousfield localization LSM of M along any set representing S exists and satisfies
the following conditions.
● The model category LSM is left proper and X-combinatorial.
● As a category, LSM is simply M.
51
● The cofibrations of LSM are exactly those of M.
● The fibrant objects of LSM are the fibrant S-local objects of M.
● The weak equivalences of LSM are the S-local equivalences.
Definition 4.21. Define a set of maps in SpN(Cat∗,Σ) by S = {Fn+1ΣC sCnÐ→ FnC}, asC runs through the set of domains and codomains of the maps of I and n ≥ 0. Here
sCn is adjoint to the identity map of ΣC. The stable model structure on SpN(Cat∗,Σ)
is the localization of the projective model structure with respect to S . We call
the S-local weak equivalences stable equivalences and the S-local fibrations stable
fibrations.
Theorem 4.22. [Hov01, 3.8] The functor Σ ∶ SpN(Cat∗,Σ) → SpN(Cat∗,Σ) is a
Quillen equivalence with respect to the stable model structure.
We now have a model structure on our category of spectra that makes
SpN(Cat∗,Σ) analogous to Sp. However, it turns out that SpN(Cat∗,Σ) is actually
very simple homotopically.
Proposition 4.23. Every object in SpN(Cat∗,Σ) is cofibrant with respect to the stable
model structure.
Proof. By Proposition 4.15, every object in SpN(Cat∗,Σ) is cofibrant with respect
to the projective model structure, and cofibrations in the stable model structure
are the same as in the projective model structure.
52
Proposition 4.24. Every object in SpN(Cat∗,Σ) is stably equivalent to 0.
Proof. Certainly 0 is fibrant. Also, Ω2(0) ≃ 0 for all n ≥ 0. This is because Ω is
a right adjoint, 0 is the limit of the empty diagram, and right adjoints preserve
limits. Finally, for any (cofibrant) object X ∈ Ob(SpN(Cat∗,Σ)), Σ2X ≃ 0 in the
projective model structure, and all level-wise weak equivalences are still weak
equivalences after Bousfield localization. Since Σ is a Quillen equivalence, having
a weak equivalence Σ2X ≃Ð→ 0 implies that the adjunct morphism X → Ω2(0) ≃ 0 is
a weak equivalence.
Note that we could repeat this argument in any similar category in which
there is some number of suspensions after which any object becomes equivalent
to the zero object. This leads to the following theorem.
Theorem 4.25. If M is a pointed, left proper, simplicial, cellular or combinatorial model
category, and if there is n ∈ Z≥0 such that ΣnX ≃ 0 for all X ∈ Ob(M), then every object
in SpN(M,Σ) is stably equivalent to 0.
Since Cat∗ satisfies Hypothesis 2.28 and the identity functor 1 is a homotopy
functor, Goodwillie’s construction applies. In particular, we expect the nth layer
of the Taylor tower of 1Cat∗ to have the form
Dn1Cat∗(C) ≃ Ω∞(∂n1Cat∗ ⊗ (Σ∞C)⊗n)hΣn ,
where Ω∞ = Ev0, Σ∞ = F0, and the nth derivative ∂n1Cat∗ is an object of
SpN(Cat∗,Σ). Hence ∂n1Cat∗ ≃ 0 for all n, which implies that Dn1Cat∗(C) ≃ 0 for
all n and for all C ∈ Ob(Cat∗). Again, we may generalize this result to any similar
category.
53
Corollary 4.26. In any category M satisfying the conditions of Theorem 4.25, the
derivatives of the identity functor ∂∗1M exist and ∂∗1M ≃ 0.
Remark 4.27. On the surface, this example seems analogous to the example in
calculus of the function f ∶ R → R defined by f (x) = e−1/x2 for x ≠ 0 and f (0) = 0.
For all i, the ith derivative at a = 0 is f (i)(0) = 0. Hence the Taylor series of f about
a = 0 is T(x) = ∞∑
i=0
f i(0)
i!
xi = 0. Thus the series converges for all x, but T(x) = f (x)
if and only if x = 0. Here Dn1Cat∗(C) ≃ 0 for all C ∈ Ob(Cat∗), so Pn1Cat∗(C) ≃ 0
for all C ∈ Ob(Cat∗). Therefore the inverse limit of the Taylor tower of 1Cat∗ exists
and is equivalent to 0 for all C, and so the value of the limit of the Taylor tower
is equivalent to the value of the identity functor if and only if C ≃ 0. However,
the analogy would be better if the only linear function ψ ∶ R → R were ψ(x) = 0.
In this case, the only possible derivatives of a function f would be f (i) = 0, just
as we determined that ∂∗1Cat∗ ≃ 0 by showing that this was the only option.
Furthermore, we would expect the derivatives of any functor F ∶ Cat∗ → Cat∗ to
be objects in SpN(Cat∗,Σ), and so ∂∗F would also be weakly equivalent to 0. This
means that no endofunctors of Cat∗ are analytic.
In short, the world of categories is very different from the usual setting of
calculus.
54
CHAPTER V
THE CATEGORY OF DIRECTED GRAPHS
Definition 5.1. A directed graph X = (V(X), E(X), s, t), where V(X) is a set of
vertices, E(X) is a set of edges, and s, t ∶ E(X) → V(X) are functions that specify
the source and target vertices of each edge. If e is an edge with s(e) = v and
t(e) = w, we write e ∶ v → w. Denote the set of edges in X with source vertex x by
X(x,−).
A morphism of directed graphs f ∶ X → Y is a pair of functions
fV ∶ V(X)→ V(Y) and fE ∶ E(X)→ E(Y) such that s ○ fE = fV ○ s and t ○ fE = fV ○ t.
Let Gph be the category of directed graphs and graph morphisms. Note that
the terminal object in Gph is the graph with one vertex and one edge. We force this
graph to be the initial object in our category as well. Thus every graph X comes
with a chosen graph morphism 0 → X, which means that X has a designated
looped basepoint, denoted ∗. Let Gph∗ be the category of pointed graphs and
basepoint-preserving graph morphisms.
The categorical product of two graphs A × B has vertex set V(A)×V(B) and
edge set E(A) × E(B), where there is an edge e = (e1, e2) ∶ (v1, w1) → (v2, w2)
in A × B if e1 ∶ v1 → v2 is an edge in A and e2 ∶ w1 → w2 is an edge in B. The
basepoint of A × B is the vertex (∗,∗) together with the loop at this vertex. The
categorical coproduct is the wedge, so V(A ∨ B) = (V(A)⊔V(B))/(∗A ∼ ∗B) and
E(A ∨ B) = E(A)⊔ E(B), where the loops at the basepoints of A and B have also
been identified.
Let Cn be the unpointed graph with n vertices labeled 0, 1, . . . , n − 1 and n
edges labeled ei for i = 0, . . . , n − 1, where s(ei) = i and t(ei) = i + 1 mod n. The
55
image of Cn in a graph is called an n-cycle. We denote the cycle graph with a
disjoint basepoint by Cn,∗.
A tree T is an unpointed graph with a unique vertex r called the root of T
such that there are no edges with target r and such that, for every other vertex x
in T, there is a unique path in T from r to x. Let Ti1 , . . . , Tik be trees. Let Tn,k be
a graph obtained by taking Cn,∗ ⊔ Ti1 ⊔ . . . ⊔ Tik and identifying the root of Tij with
the vertex vij of Cn,∗. Note that Cn,∗ can also be considered as a graph of the type
Tn,k, where each Tij is a tree with one vertex.
Bisson and Tsemo [BT+09] showed that there is a model structure on Gph in
which the
● weak equivalences are acyclic graph morphisms. A graph morphism
f ∶ X → Y is acyclic when f is a bijection on n-cycles for all n ≥ 1.
● fibrations are surjectings. A graph morphism f ∶ X → Y is a surjecting when
the induced function f ∶ X(x,−)→ Y( f (x),−) is surjective for all x ∈ V(X).
● cofibrations are those maps that have the left lifting property with respect
to trivial fibrations.
We use the Bisson-Tsemo model structure and Theorem 2.6 to put a model
structure on Gph∗.
Let G be the following graph:
∗ v we
Let G′ be the subgraph
56
∗ v
Let s ∶ G′ ↪ G and in ∶ 0 ↪ Cn,∗ be inclusions. Let jn ∶ Cn,∗ ∨ Cn,∗ → Cn,∗ be the
coproduct graph morphism. Finally, set K = {in, jn ∶ n ≥ 1}.
The following statements are based on results in [BT11]. The proofs for Gph∗
are analogous to the proofs for Gph.
Proposition 5.2. 1. The above model structure on Gph∗ is cofibrantly generated with
generating trivial cofibration J = {s} and generating cofibrations I = J ∪K.
2. A graph X is cofibrant if and only if X is a wedge of graphs of the form Tn,k.
3. A graph X is fibrant if and only if every vertex of X is the source of some edge.
As before, we wish to construct the suspension functor, and so we must first
construct cylinder objects.
Proposition 5.3. Given any graph A, Cyl(A) = (AL ∨ AR)/ ∼, where vL ∼ vR for all
vertices v ∈ V(A) and eL ∼ eR if e ∈ E(A) is part of a cycle.
Proof. Note that Cyl(A) has the same cycles as A because we have identified the
two copies of each cycle in AL ∨ AR. Thus Cyl(A) ≃Ð→ A, where the map from
Cyl(A) → A collapses any remaining duplicate edges. Clearly the composition
A ∨ A → Cyl(A)→ A is the identity on each copy of A. It remains to show that
the gluing map q ∶ AL ∨ AR → Cyl(A) is a cofibration. Let f ∶ X → Y be a trivial
fibration, and consider a commutative square
57
A ∨ A X
Cyl(A) Y
j
q
g
f
Suppose e ∈ E(Cyl(A)) is part of a cycle C. Then g(e) is part of a cycle g(C)
in Y, which lifts to a unique cycle C′ in X since f is a trivial fibration. Define
a lift h on e by h(e) = e′, where e′ is the edge of C′ satisfying f (e′) = g(e). Since
e is part of a cycle, there are two edges (which are also parts of cycles) which q
maps to e. Denote these by eL and eR. By the commutativity of the square, since
gq(eL) = gq(eR), we have f j(eL) = f j(eR). Since f is a weak equivalence and j(eL)
and j(eR) are each part of a cycle in X, we conclude j(eL) = j(eR). Therefore h
makes both triangles commute.
Now suppose e ∈ E(Cyl(A)) is not part of a cycle. Then there is a unique
edge e′′ in A ∨ A that is also not part of a cycle satisfying q(e′′) = e. Define a
lift by h(e) = j(e′′). By the commutativity of the diagram, gq(e′′) = f j(e′′). Thus
gq(e′′) = f hq(e′′), or g(e) = f h(e), so g = f h.
Theorem 5.4. For any cofibrant graph A, ΣA ≃ 0.
Proof. In taking the pushout of the diagram below, we identify all the vertices of
Cyl(A) with the unique vertex of 0 and all the edges of Cyl(A) with the unique
edge of 0 because both of these maps are surjective.
A ∨ A Cyl(A)
0
58
Theorem 5.5. For any fibrant graph B, ΩB ≃ 0.
Proof. Let B have vertex set V and edge set E. We define BI as well as the map
BI → B × B using induction. Let B0 = B, and let f0 ∶ B0 → B × B be the diagonal
map. For each vertex v ∈ V, and for each edge e ∈ E × E such that s(e) = (v, v) and
e ∉ im( f0), add an edge e′ and vertex v′ to B0 so that s(e′) = (v, v) and t(e′) = v′.
Call this new graph B1. There is a map f1 ∶ B1 → B × B such that f1∣B0 = f0 and
f1(e′) = e. Thus f1 surjects onto the vertices of im( f0).
Suppose we have constructed B1, . . . , Bn together with maps fi ∶ Bi → B × B
so that fi∣Bi−1 = fi−1 and fi surjects onto the vertices of im( fi−1) for each 1 ≤ i ≤ n.
For each vertex v ∈ V(Bn) and for each edge e ∈ E × E such that s(e) = fn(v) and
e ∉ im( fn), add an edge e′ and vertex v′ to Bn so that s(e′) = v and t(e′) = v′. Call
this new graph Bn+1. Define fn+1 by fn+1∣Bn = fn and fn+1(e′) = e. Note fn+1 surjects
onto the vertices of im( fn).
Define BI = lim→ Bi, and define f ∶ BI → B × B by f = lim→ fi. By construction,
f is a surjecting. Also, BI is weakly equivalent to B because BI was formed by
attaching trees to B.
Recall that ΩB is the pullback of the diagram
BI
0 B × B
f
Thus the graph ΩB has vertices V = {(v,∗) ∈ V(BI) ×V(0) ∣ f (v) = ∗B×B} and
edges E = {(e,∗) ∈ E(BI)× E(0) ∣ f (e) is the 1-cycle at ∗B×B}. By construction, the
subgraph of BI that maps to the basepoint of B × B contains the basepoint of
59
BI and a (possibly infinite) disjoint union of edges that are not part of a cycle.
Therefore ΩB ≃ 0.
One would hope that Gph∗ is left proper and either cellular or combinatorial
so that we may repeat the argument we used to construct a category of spectra
for Cat∗. However, only one of these properties holds.
Proposition 5.6. Gph∗ is combinatorial.
Proof. An unpointed directed graph is equivalent to a presheaf over the unpointed
category with two objects and two parallel non-identity morphisms. Here the
objects correspond to sets of vertices and edges, and the two morphisms specify
the source and target vertex for each edge. A category of presheaves is locally
presentable; indeed, another definition of a locally presentable category is as
a localization of a category of presheaves [AR94]. Therefore Gph is locally
presentable, and hence Gph∗ is locally presentable.
Proposition 5.7. Gph∗ is not cellular.
Proof. Consider the generating cofibration X = Cn,∗ ∨ Cn,∗ jnÐ→ Cn,∗ = Y. Note
Y ⊔X Y = Y, so eq(Y ⇉ Y ⊔X Y) = Y, not X. Therefore jn is not an effective
monomorphism.
Proposition 5.8. Gph∗ is not left proper.
Proof. We provide an example of a weak equivalence f that is not preserved
by pushout along a cofibration. Consider the generating cofibration
j3 ∶ C3,∗ ∨C3,∗ → C3,∗ and the pushout diagram
60
C3,∗ ∨C3,∗ X
C3,∗
f
j3
in which X is the graph
∗ a b
c
a′ b′
c′
e
and f is the inclusion. In the pushout, we glue together the vertices a and a′, b
and b′, and c and c′ along with the corresponding pairs of edges. However, the
edge e is not in the image of f . Hence the pushout is
∗
im(e)
This graph is not weakly equivalent to C3,∗ because it contains a 2-cycle.
This means that the construction we used to define a category of spectra on
Cat∗ does not apply to Gph∗. It may still be the case that SpN(Gph∗,Σ) exists.
However, since ΣX ≃ 0 for all X ∈ Ob(Gph∗), we would expect any category of
spectra on Gph∗ to be homotopically trivial.
61
CHAPTER VI
OTHER MODEL STRUCTURES ON CAT
The canonical model structure is not the only model structure on Cat∗ that
we could have used. In this chapter, we describe an infinite family of model
structures on Cat, the category of unpointed small categories, which we believe
induce homotopically trivial categories of spectra on Cat∗. We obtain these model
structures by transporting them from other model categories across Quillen
adjunctions. Recall that by the remarks following Theorem 2.6, each model
structure on Cat determines a model structure on Cat∗.
Definition 6.1. [Hir03, 10.5.15] Let I be a set of morphisms in a model category
M. We say that I permits the small object argument if the domains of the maps in I
are small with respect to I-cell.
Theorem 6.2. [Hir03, 11.3.2] Let M be a cofibrantly generated model category
with generating cofibrations I and generating trivial cofibrations J. Let N be a
category that is closed under small limits and colimits, and let F ∶M ⇆ N ∶ U be
a pair of adjoint functors. If we let FI = {Fu ∣ u ∈ I} and FJ = {Fv ∣ v ∈ J}, and if
● both of the sets FI and FJ permit the small object argument, and
● U takes relative FJ-cell complexes to weak equivalences,
then there is a cofibrantly generated model category structure on N in which FI
is a set of generating cofibrations, FJ is a set of generating trivial cofibrations,
and the weak equivalences are the maps that U takes to weak equivalences in M.
Furthermore, with respect to this model category structure, (F, U) is a Quillen
adjunction.
62
This theorem is due to Kan and is often called the transfer theorem. We note
that the new model structure is not necessarily Quillen equivalent to the original
model structure.
Proposition 6.3. [MP12, 16.2.3] If the pair (F, U) satisfies the hypotheses of the
transfer theorem, then (F, U) is a Quillen equivalence if and only if the unit of the
adjunction ηX ∶ X → UFX is a weak equivalence in M for all cofibrant X ∈ Ob(M).
Proof. (⇐) Suppose that ηX ∶ X → UFX is a weak equivalence for all cofibrant
X ∈ Ob(M). Fix a particular X, and suppose that f ∶ FX → Y is a weak equivalence
in N . Consider the commutative diagram
X UY
UFX
ηX U f
Since f is a weak equivalence, U f is a weak equivalence. Therefore the
composition U f ○ ηX ∶ X → UY is a weak equivalence.
Now suppose that the adjunct morphism X → UY is a weak equivalence.
By the two-out-of-three property, U f is also a weak equivalence. By the transfer
theorem, f is a weak equivalence.
(⇒) Let X ∈ Ob(M) be cofibrant, and suppose (F, U) is a Quillen equivalence.
Let FX RÐ→ Y be a fibrant replacement of FX in N . Then
FX idÐ→ FX RÐ→ Y
is a weak equivalence, and thus the adjunct composition
X UFX UY
ηX UR
≃
63
is a weak equivalence. Also, UR is a weak equivalence because U preserves weak
equivalences. By the two-out-of-three property, X ≃Ð→ UFX.
Now we consider the “n-equivalence” model structures on T op, which we
define in the next paragraph. We will use these and Theorem 6.2 to put a model
structure on Cat∗ for each n ≥ 1.
Definition 6.4. [EDHP95, 2.1] A map of topological spaces f ∶ X → Y is a
weak n-equivalence if for all k = 0, . . . , n and for each x ∈ X, the induced map
pik( f ) ∶ pik(X, x)→ pik(Y, f (x)) is an isomorphism. The identity of the empty space
is also a weak n-equivalence.
A map of simplicial sets f ∶ X → Y is a weak n-equivalence if ∣ f ∣ ∶ ∣X∣→ ∣Y∣ is a
weak n-equivalence in T op.
For all n ≥ 1, there is a model structure on sSet [EDHP95] in which
● the weak equivalences are weak n-equivalences.
● the fibrations are n-fibrations. We say f ∶ X → Y is an n-fibration if f has the
right lifting property with respect to Λpk → ∆[p] for 0 < p ≤ n + 1, 0 ≤ k ≤ p
and with respect to Λn+2k → ∆˙[n + 2], 0 ≤ k ≤ n + 2.
● the cofibrations are n-cofibrations, i.e. maps that have the left lifting property
with respect to all trivial n-fibrations.
This model structure is cofibrantly generated with generating cofibrations
I = {∆˙[q]→ ∆[q] ∣ 0 ≤ q ≤ n + 1}
64
and generating trivial cofibrations
J = {Λpk → ∆[p] ∣ 0 < p ≤ n + 1, 0 ≤ k ≤ p}∪ {Λn+2k → ∆˙[n + 2] ∣ 0 ≤ k ≤ n + 2}.
We call sSet with this model structure sSetn. If n ≠ m, then sSetn and sSetm
are not equivalent. In addition, each of these categories is different from sSet
with the Quillen model structure, in which the weak equivalences are those
morphisms whose geometric realization is a weak homotopy equivalence of
topological spaces, the cofibrations are level-wise injections, and the fibrations
are Kan fibrations [Hir03, 7.10.8].
In [Tan13], Tanaka puts a cofibrantly generated model structure on Cat using
the pair of adjoint functors c ∶ sSet1 ⇆ Cat ∶ N. Here c is the categorization
functor, which is defined as follows. The set of objects in cX is X0, and morphisms
in cX are freely generated by the set X1 subject to relations given by elements
of X2, namely, x1 = x2x0 in cX if there exists a 2-simplex x such that d2x = x2,
d0x = x0, and d1x = x1. The right adjoint N is the nerve functor, which is defined
by NnC = Cat([n],C) and maps
di( f1, . . . , fn) = ( f1, . . . , fi−1, fi+1 ○ fi, fi+1, . . . , fn)
sj( f1, . . . , fn) = ( f1, . . . , f j, 1, f j+1, . . . , fn).
We remark that (c, N) are the same functors as (pi,µ) from Chapter IV. We
adopt different notation because we are now considering unpointed categories.
65
Tanaka shows that (c, N) is a Quillen equivalence. He calls Cat with the
model structure obtained using the transfer theorem and this adjunction Cat1.
This model structure is also related to the canonical model structure on Cat.
Proposition 6.5. [Tan13, 3.20] Let I be the unpointed category with two objects
and one morphism between them. Let S∞ = c∆1 be the unpointed category
with two objects and a unique isomorphism between them. The Tanaka model
structure on Cat is the left Bousfield localization of the canonical model structure
on Cat with respect to the inclusion φ ∶ I → S∞.
More generally, we consider the adjunction c ∶ sSetn ⇆ Cat ∶ N.
Proposition 6.6. For n ≥ 2, Cat with the model structure obtained by using the transfer
theorem with (c, N) is not Quillen equivalent to sSetn.
Proof. Since the fibrations in sSetn have the right lifting property with respect to
maps of the form Λpk → ∆[p] for 0 < p ≤ n + 1, 0 ≤ k ≤ p, in particular, the trivial
fibrations of sSetn are surjective on k-simplices for all 0 ≤ k ≤ n + 1. Since the
inclusion ∅ → ∆˙[3] has the left lifting property with respect to all such maps,
∆˙[3] is cofibrant. However, Nc∆˙[3] ≃ ∆[3] /≃ ∆˙[3]. Applying Proposition 6.3, we
conclude that the adjunction is not a Quillen equivalence.
Upon reflection, it is unsurprising that c ∶ sSet ⇆ Cat ∶ N is only a Quillen
equivalence when n = 1 because c only uses the information of a simplicial set up
to the 2-simplices. Now we consider another pair of adjoint functors between sSet
and Cat.
Theorem 6.7. [Tho80] Let Sd be the subdivision endofunctor of sSet, and let Ex be
its right adjoint. Then Cat with the model structure obtained by using the transfer
66
theorem and the adjunction cSd2 ∶ sSet ⇄ Cat ∶ Ex2N is Quillen equivalent to sSet
with the Quillen model structure.
Proposition 6.8. There is a model structure on Cat where
● the weak equivalences are those maps f such that Ex2N f is a weak n-equivalence.
● the fibrations are those maps f such that Ex2N f is an n-fibration.
● the cofibrations have the left lifting property with respect to all trivial fibrations.
We call Cat with this model structure Catn.
Proof. We must show that Catn satisfies the hypothesis of the transfer theorem.
First, Catn is closed under small limits and colimits because Cat is. Next, cSd2 I and
cSd2 J permit the small object argument because the domains of these sets contain
finitely many objects and morphisms. Finally, we must show that Ex2N takes
relative cSd2 J-cell complexes to weak equivalences. Note that every cSd2 J − cell
complex is cSd2 applied to a relative J-cell complex. Let f ∈ J-cell, so f is a
weak equivalence [Hov99, 2.1.19]. By [FL81], Id → Ex2NcSd2 is a weak homotopy
equivalence, and so f ≃ Ex2NcSd2 f . Thus Ex2NcSd2 f is a weak equivalence, i.e.
Ex2N takes relative cSd2 J-cell complexes to weak equivalences.
Proposition 6.9. Catn is Quillen equivalent to sSetn.
Proof. By the transfer theorem, (cSd2, Ex2N) is a Quillen adjunction. Since
X ≃Ð→ Ex2NcSd2(X) for all X ∈ Ob(sSetn), in particular, this holds for all cofibrant
objects of sSetn. By Proposition 6.3, (cSd2, Ex2N) is a Quillen equivalence.
67
Let ∅ ≠ X ∈ Ob(T op). Note that pik(ΣnX) = 0 for all k < n. Therefore
in the (n − 1)-equivalence model structure on T op, ΣnX ≃ ∗. If T opn satisfies
the hypotheses of Theorem 4.25, then this theorem implies that every object in
SpN((T opn)∗,Σ) is stably equivalent to 0. Hovey [Hov01, 5.5] shows that under
certain conditions, a Quillen equivalence Φ ∶ C → D induces a Quillen equivalence
SpN(Φ) ∶ SpN(C,Σ)→ SpN(D,Σ). In light of these results, we make the following
conjecture.
Conjecture 6.10. The category of spectra SpN((Catn)∗,Σ) exists, and every object
in SpN((Catn)∗,Σ) is stably equivalent to 0.
68
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