Download Spectra of Small Categories and Infinite Loop Space Machines

K-Theory (2006) 37:249–261
DOI 10.1007/s10977-006-0017-0
Spectra of Small Categories and Infinite Loop Space
Machines
Elias Gabriel Minian
Received: February 2006
© Springer Science+Business Media B.V. 2006
Abstract We show how to construct a categorical spectrum of small categories, and hence
a cohomology theory, starting from a -category. This extends Segal’s infinite loop space
machine for topological spaces to small categories. Our results form a part of Bak’s program
for delooping global actions, global categories, and related objects.
Keywords
Small categories · Simplicial objects · Infinite loop spaces
Mathematics Subject Classification (2000) 55U10 · 55P40 · 55P48 · 18G30
1 Introduction
-spaces were introduced by Segal [15] in order to construct spectra, and hence cohomology theories of spaces, out of simpler space level data. He showed that -spaces arise naturally from -categories via the classifying space functor B: C at → T op which is the
composition of the nerve functor N : C at → S.Sets with the geometric realization functor
| |: S.Sets → T op.
In the current paper, we work solely with small categories and thereby avoid passing from
categories to topological spaces. This is accomplished by replacing the homotopy theory
of topological spaces by that of small categories developed in previous papers [9–12]. This
theory includes the notions of loop, suspension, and spectrum for small categories. Using
these notions and ideas of Segal, we show in the current paper how to construct a categorical
spectrum, starting from a -category. That one should look for a result of this kind was communicated to me by Tony Bak. The result is the first milestone in his program for delooping
global actions, global categories, and related objects.
Our main theorems are as follows.
E. G. Minian (B)
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina
e-mail: [email protected]
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Theorem 1.1 Let A: F → C at∗ denote a -category. Let B A denote the classifying category of A. Then the H -category A(1) has a homotopy inverse if and only if the canonically
defined functor A(1) → B A(1) is a homotopy equivalence.
Theorem 1.2 Let A denote a -category such that A(n) is a complex category for every n.
Then B A(1), . . . , B k A(1), . . . is a -spectrum of categories. Furthermore, if π0 (A(1)) is a
group, then B.A = A(1), B A(1), . . . is a -spectrum of categories.
The rest of the paper is organized as follows. In section 2 we recall basic notions and
results we need from the homotopy theory (without topological spaces) of small categories.
In section 3 we introduce the concept of the (categorical) realization of a simplicial category.
This plays a central role in our work. In section 4 we recall our previous results on numerably
contractible categories and complex categories. The latter is a generalization of the notion
of CW-complex to categories. In section 5, we use the notion of realization of a category to
define the classifying category B A of a -category A. We use all of the above to prove our
main results.
2 Homotopy theory for categories
There are three different notions of homotopy in the category C at of small categories. The
notion of strong homotopy (which is the one studied for example in [5, 7, 9]) is the symmetric
transitive closure of the relation given by: f ∼ g iff there is a natural transformation between
them. The notion of weak homotopy (studied in [13, 14]) is related to the nerve (or classifying
space) of the categories: Two functors f and g are homotopic iff B f and B g are homotopic
continuous maps. Here, B: C at → T op means the classical classifying space functor which
is the composition of the nerve functor with the geometric realization. In [6], an intermediate
notion of homotopy is introduced by using path categories. In this paper we use the notion
of strong homotopy.
Throughout this section we recall the basic notions of intervals, homotopy, α-spheres,
α-suspensions and α-loops given in [9, 11].
Definition 2.1 Given α ∈ N, let Iα be the following category. The objects of Iα are the
integers 0, 1, . . . , α and the morphisms, other than the identities, are defined as follows. If r
and s are two distinct objects in Iα there is exactly one morphism from r to s if r is even and
s = r − 1 or s = r + 1 and no morphisms otherwise. The sketch of Iα is as follows (case α
odd).
Iα :
0
/1o
2
/3o
......
/α.
Definition 2.2 Two functors f, g: C → D are strong homotopic if there is an α ∈ N and
H : C × Iα → D such that H (a, 0) = f (a) and H (a, α) = g(a) for all a ∈ C. The functor
H is called a strong homotopy from f to g and it is denoted H : f g. A functor f : C → D
is a strong homotopy equivalence if there exists g: D → C such that f g 1 and g f 1.We
will denote a strong homotopy equivalence f : C → D by C −
→ D. A category C is called
strong contractible if the functor C → ∗ is a strong homotopy equivalence (C −
→ ∗).
In the rest of this paper, when we refer to homotopies and homotopy equivalences of
functors, we will mean strong homotopies and strong homotopy equivalences.
The family of cylinders Iα : C at → C at (α ∈ N) is induced by the interval categories Iα ,
taking Iα X = X × Iα . Sometimes it is necessary to replace one given cylinder by a bigger one
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of the family (see 2.20 below). The way to do this is by using some natural transformations
between them which arise from the subdivision functors between the interval categories.
Definition 2.3 Let α, β ∈ N with β ≥ α. A functor t: Iβ → Iα such that t (0) = 0 and
t (β) = α will be called a subdivision functor. A subdivision functor will also be called a
transformation. If t: Iβ → Iα is a subdivision functor and X is a category then the functor
1 × t: X × Iβ → X × Iα will also be called a subdivision functor.
Remark 2.4 Let H : f g with H : C × Iα → D. If β ≥ α there exists at least one subdivision functor t: Iβ → Iα . Thus there is a homotopy H : C × Iβ → D from f to g defined
by H = H (1 × t).
If an object such as a mapping cylinder is constructed using a particular cylinder, it will
be necessary to relate it to corresponding objects constructed using other cylinders from the
family {Iα : α ∈ N}. To this end we introduced in [9] the concept of subdivision. We explain
here this concept very briefly.
Suppose X is the colimit in C at of a finite diagram. The objects of this diagram are some
categories Oi and Iα j Oi where α j ∈ α for some finite subset α ⊂ N. Suppose also that
the maps of the diagram are defined in a way that allows us to replace the cylinders Iα j for
bigger ones Iα j in a natural way. The colimit of the new diagram where the old cylinders were
replaced by bigger ones is denoted X and it is called a subdivision of X . If we take for each
j a transformation t j : Iα j → Iα j then these transformations induce a functor T : X → X
called transformation functor. To illustrate this consider the α-mapping cylinder of a functor
f : A → B, Z αf = Iα A ∪ A B (α ∈ N). In this case we have α = {α}. If we choose now
β
β ≥ α and some t: Iβ → Iα we obtain a transformation functor T : Z f → Z αf .
If f : X → Y is any functor in C at and T : X → X is a subdivision of X we denote
f = f T : X → Y and call it a subdivision of f .
Let us finish this brief explanation recalling the following notion which is used in relative
complexes. If X is a colimit of a diagram as before, we say that the subdivision X is invariant
β
in the subobjects O j → X . For example the subdivision T : Z f → Z αf is invariant in A and
in B (but not invariant in Iα A).
Remark 2.5 The subdivision functors t: Iβ → Iα are strong homotopy equivalences relative {0} and therefore t × 1: Iβ A → Iα A are strong homotopy equivalences rel A × 0. That
β
implies that the induced transformations between mapping cylinders T : Z f → Z αf are also
strong homotopy equivalences.
Notation 2.6 Given β ≥ α ∈ N, we denote by (β ≥ α) the non-empty set of all subdivision
functors t: Iβ → Iα .
In this paper we work in the category C at∗ of pointed categories. Homotopy of base point
preserving functors is defined via reduced cylinders. The reduced cylinder Iα A of a pointed
category A is the quotient of the (non-based) cylinder A × Iα in which all the morphisms of
the form (1a0 , f ) are identified to a point.
The family of suspension functors α : C at∗ → C at∗ (α ∈ N) is also defined using the
reduced cylinders as follows.
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Definition 2.7 Given a pointed category A and α ∈ N, the α-suspension of A is defined by
the following pushout
/∗
A+ A
(i 0 ,i 1 )
Push
Iα A
/ α A
where the functor i 0 : A → Iα (resp. i 1 ) is the functor defined as i 0 (a) = (a, 0) (resp.
i 1 (a) = (a, α)).
Definition 2.8 Given α, β ∈ N such that β ≥ α, if t: Iβ → Iα is a subdivision functor, t
induces a transformation functor t: β A → α A. Given two pointed categories A and B,
the homotopy groups πnA (B) are defined as a colimit (of pointed sets), using the family of
suspensions and all transformation functors between them:
πnA (B) = colim[αn A, B]
α
where αn
α (αn−1 ). Here the bracket [X, Y ] of two pointed categories
=
the pointed set of pointed homotopy classes of functors from X to Y .
X and Y denotes
The homotopy groups are Abelian for n ≥ 2. If we take A = S 0 the discrete category
0
with two points, we denote simply πn (B) = πnS (B).
Let B: C at → T op be the classifying space functor. We recall the following result from
[9].
Proposition 2.9 Given a pointed category C, πn (C) = πn (B(C)), n ≥ 0.
Definition 2.10 Given α ∈ N, the α-sphere Sα1 is the pointed category Sα1 = α S 0 . It can
be sketched as follows (case α even).
/1o
0
/3o
2
+/
...
α−1
In analogy with topological spaces we can describe the α-suspension of a pointed category
X in terms of the smash product of X with the α-sphere introduced above.
Remark 2.11 If X is a pointed category then α X = X ∧ Sα1 .
Let C at∗ denote the category of all small categories with based object and based object
preserving functors. In C at∗ homotopy of functors is defined by reduced cylinders.
Remark 2.12 Given α ∈ N, we define the α-loop functor
α = C at∗ (Sα1 , −): C at∗ → C at∗ .
It is clear that α is right adjoint for α for every α ∈ N in the strong homotopy category
Ho(C at∗ ) = C at∗ / , i.e.
[α A, B] [A, α B].
Given a pointed category B and α ∈ N, we can describe the α-loop category α B as
follows (case α even). The objects of α B are the sequences of the form
b0
f0
/ b1 o
f1
b2
f2
/ ......
/ bα−1 o
f α−1
b0 .
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The maps of α B are commutative diagrams
b0
f0
f1
/ b1 o
b2
γ1
b0
f 0
/ b o
1
f2
/ ......
/ bα−1 o
γ2
b2
f 1
f 2
/ ......
f α−1
b0 .
γα−1
/ bα−1
o
f α−1
b0
Remark 2.13 For β ≥ α, any subdivision functor t ∈ (β ≥ α) induces a functor t ∗ : α B →
β B. If we consider all α ∈ N and all the subdivision functors, in view of 2.12 we can interpret the homotopy groups in terms of the α-loop functors
πnA (B) = colim[A, nα B].
α
Remark 2.14 Considering all subdivisions, we define a big loop functor by
(A) = colim α (A).
α
The functors α B → B induce a map
A
: πnA (B) → πn−1
(B).
This map is not in general an isomorphism.
Definition 2.15 A category A is called finite if its set of morphisms is finite. Note that if A
is a finite category then the natural map ν: colim[A, α B] → [A, B] is a bijection.
α
The following result is an immediate consequence of this.
A (B) is an isomorphism. In
Proposition 2.16 If A is finite, the map : πnA (B) → πn−1
particular for A = S 0 we obtain the isomorphisms
πn (B) = πn−1 (B) = π0 (n B)
where n B = (n−1 B) = colim nα B.
α
Remark 2.17 Note that our big loop category B is similar to the loop category introduced
by Hoff [6] which we denote by H B. In fact there is a quotient functor H B → B which
induces an isomorphism
π0 ( H B)
=
/ π0 (B).
One can use this isomorphism to give a different proof of 2.9.
Remark 2.18 If G is a groupoid then the α-loop categories α G are groupoids for all α ∈ N,
therefore G is also a groupoid.
Given a groupoid G with base point g0 , it is well-known that π1 (G) = H om G (g0 , g0 )
(cf. for example [13]) . The key point is that we use reduced cylinders to define homotopy
of loops. Bak has remarked that if ordinary cylinders are used to define homotopy, so that
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only the based object in each slice of the homotopy is preserved then the homotopy classes
of loops we get is π1 (G) made abelian.
Given a groupoid G with base point g0 , let [g0 ] denote the base point of G, i.e the class
in the colimit G = colimα α (G) of the constant loop. Recall from [11], definition 2.2,
that the morphisms in the category C at∗ (A, B) are the natural transformations γ : f → g
between base point preserving functors such that γ (a0 ) = 1b0 : f (a0 ) = b0 → g(a0 ) = b0 ,
where a0 and b0 are the base points of A and B. By definition of the categories α (G)
(see diagram above), this implies that H om G ([g0 ], [g0 ]) = 0. This verifies the well known
result that for any groupoid G, πn (G) = 0 for all n ≥ 2.
The following theorem gives conditions for a groupoid G to be delooped by a group. By
a group H we mean a groupoid with only one object. Recall that a functor f : A → B is a
weak equivalence of pointed categories if πn ( f ) is an isomorphism ∀ n ≥ 0.
Theorem 2.19 Let G be a groupoid. There exists a group H and a weak equivalence ω: G →
H if and only if π1 (G) = 0 and π0 (G) is a group.
Proof If ω exists then it is obvious that π1 (G) = 0 and π0 (G) is a group because this
is true for H .
Conversely, take H to be the group π0 (G). We consider H as a groupoid with only one
object ∗. Consider the following functor ω: G → H . If x is an object of G, let [x] denote
the class of x in π0 (G) and define
ω(x) = [ ∗
[x]
/∗]
[x]
/ ∗ ] denotes the class of the map [x]: ∗ → ∗ in H .
where [ ∗
If f : x → y is a map in G, then [x] = [y] in π0 (G) and therefore we define ω( f ) = I d.
It is clear that ω induces isomorphisms ω∗ : πn (G) → πn (H ) for all n ≥ 0.
We finish this section by recalling the notion of cofibration of categories introduced in
[9].
Definition 2.20 A functor i: A → B is a cofibration if given α ∈ N and a commutative
diagram
A
i
B
i εα
f
/ Iα A
/X
H
there exists β ≥ α and t ∈ (β ≥ α), such that the commutative diagram
β
A
i
B
iε
/ Iβ A
f
/X
Ht
β
satisfies the homotopy extension property, i.e., there exists a map G: Iβ B → X , with Gi ε =
f and G Iβ (i) = H t.
We shall frequently set H = H t.
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3 Simplicial categories and realization
In this section we introduce the concept of realization of a simplicial category, which is
analogous to the notion of realization of simplicial spaces. The realization of a simplicial
category plays a central role in the construction of spectra out of -categories.
Recall that a simplicial object in a category C is a contravariant functor from the category
to C , where is the category whose objects are the finite ordered sets n = {0, 1, . . . , n}
and the maps φ: n → m are the non-decreasing maps. Thus, a simplicial category is a
functor
A: op → C at
which takes n → An .
To define the realization of a simplicial category, one needs categorical models of the
standard n-simplices.
Definition 3.1 We define the categories n for n ≥ 0 as follows. The objects of n are the
integers 0, 1, . . . , n. Given two objects r, s of n , there exists exactly one morphism r → s
whenever r ≤ s. We can sketch these categories as follows.
n :
0 → 1 → 2 → ··· → n
Note that the nerve of n is the standard n-simplex.
Now we can define the realization of a simplicial category analogously to the realization
of a simplicial space.
For 0 ≤ i ≤ n, let δi : n−1 → n and σi : n+1 → n be the functors defined by
δi ( j) = j for j < i
σi ( j) = j for j ≤ i
δi ( j) = j + 1 for j ≥ i
σi ( j) = j − 1 for j > i.
These functors correspond to the face and degeneracy operators δi and σi in the category of
finite ordinals .
Definition 3.2 Let A: op → C at be a simplicial category. We define the standard realization of A as the category
An × n / ∼
|A| =
n≥0
Here ∼ means the equivalence relation generated by the identifications (di a, x) ∼ (a, δi x)
and (s j a, x) ∼ (a, σ j x) with di = δi∗ and s j = σ j∗ .
In order to use a generalization of Segal’s machine, we need to work with good simplicial
categories. A simplicial category is called good (in analogy with Segal’s terminology) if it
satisfies the following cofibration condition. For any n ≥ 1 and 0 ≤ i ≤ n, the inclusions
si An−1 → An are cofibrations.
Since not every simplicial category is good, one needs to replace A by a thickening τ A
before one takes the realization.
Definition
3.3 Given a simplicial category A and a subset V of {1, 2, . . . , n}, let An,V =
si An−1 and An,∅ = An . By the intersection of two subcategories of a given catei∈V
gory we mean the pullback of the inclusions. Define the categories τn A as the union of the
subcategories 1V × An,V of n1 × An .
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Remark 3.4 Note that
τ A: op → C at
defined by n → τn A, is a simplicial category and τn A is homotopy equivalent to An .
The following proposition follows immediately from [15], Proposition A.2.
Proposition 3.5 The functor A → |τ A| satisfies the following properties.
(a)
(b)
(c)
(d )
→ An for all n, then |τ A| −
→ |τ A |
If A → A is a simplicial functor such that An −
|τ (A × A )| −
→ |τ A| × |τ A |
τ A is good for any A.
→ |A|
If A is good, then |τ A| −
For the rest of this paper, the realization of a simplicial category A will mean the special
realization A → |τ A| and we will denote it simply by |A|.
Note 3.6 There is another way to remedy the problem of the cofibration condition. As it
is shown in [8], one could work with the standard realization and impose the cofibration
condition directly in the definition of a -category.
In the last section of this paper, we use simplicial pointed categories in the construction
of the machine. A simplicial pointed category is a functor
A: op → C at∗ .
Note that the realization of a simplicial pointed category is again a pointed category.
The following results will also be used in the construction of the machine.
Remark 3.7 Let ∗ be the category with one morphism. If A is a simplicial pointed category
such that A0 = ∗ then, by definition, the 1-skeleton |A|1 of the realization of A is isomorphic
to the 1-suspension 1 A1 . Moreover, if A0 is homotopy equivalent to the one point category
∗, then for any α ∈ N then reduced α-suspension α A1 is homotopy equivalent to |A|1 . The
homotopy equivalence takes the class [(a, 0)] = [(a, α)] in α A1 to the base point a0 of
|A|1 .
In particular, since |A|1 is a subcategory of |A|, we obtain a functor 1 A1 → |A| which
induces a functor
A1 → 1 |A| → |A|.
Remark 3.8 Given a simplicial pointed category A. Define the simplicial path category
P A: op → C at∗
by P A = A ◦ P. Here the functor P: → is the standard path simplicial functor
which takes the object n of to n + 1 and φ: n → m to P(φ): n + 1 → m + 1, where
P(φ)(0) = 0 and P(φ)(i) = φ(i − 1) + 1 if i > 0.
Note that, in analogy with topological spaces, the category |P A| is contractible.
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4 Numerably contractible categories
It is well known that, if X is a homotopy associative H-space such that π0 (X ) is a group
and X is numerably contractible then it admits a homotopy inverse [2]. This result is used
in [15] to develop the infinite loop machine for (topological) spaces. Recall that a space X
is numerably contractible if there is a covering of X by contractible subspaces which admits
a refinement by a locally finite partition of unity. For example, a CW-complex is numerably
contractible [2, 3]. In [12] we proved a similar result for categories.
If X is a space, any open covering {Ui } of X strongly generates X in the sense that any
continuous map f : X → Y is determined by a family of compatible maps f i : Ui → Y , i.e. a
family of maps defined in each open subspace of the covering such that f i = f j in Ui ∩ U j .
The analog of an open covering is a family of strong generators {Ai } of a category A.
A family of categories {Ai }i∈J strongly generates A if there exists a family of monomorphisms φi : Ai → A such that for any category B and any family f i : Ai → B of compatible
functors, there exists a unique f : A → B such that f φi = f i . The family {φi : Ai → A} is
called a family of strong generators of A.
A category A is contractibly generated if there exists a family of contractible categories
{Ai } which generates A. It is clear that a contractible category is contractibly generated (by
itself). The sphere-categories (see 4.5) are contractibly generated but not contractible. In
general, any complex category is contractibly generated (see 4.7).
A family of generators {Ai } is star-finite if each Ai intersects only a finite number of
(A j ) s. A family of generators {φi : Ai → A} is called numerable if it admits a star-finite
refinement. A category A is called numerably contractible if there exists a numerable family
of generators {φi : Ai → A} such that Ai are contractible for all i ∈ J .
Definition 4.1 An H -category consists of a pointed category A together with a functor
µ: A × A → A for which the constant functor c: A → A is a homotopy identity. Given an
object a ∈ A, one defines the left translation la : A → A by la (x) = µ(a, x) in objects and
la ( f ) = µ(a, f ) in maps. Similarly one can define the right translation ra : A → A.
Remark 4.2 If A is a homotopy associative H -category and π0 (A) is a group then la and
ra are homotopy equivalences for all objects a.
The main theorem of [12] asserts that numerably contractible H -categories satisfy properties in common with numerably contractible H -spaces.
Theorem 4.3 Let A be an H -category and suppose that A is numerably contractible. If for
every object a, la : A → A (resp. ra : A → A ) is a homotopy equivalence then A has a right
homotopy inverse (resp. left homotopy inverse).
As an immediate consequence of this theorem we obtain the following result which we
shall use for our infinite loop machine.
Corollary 4.4 If A is a homotopy associative H -category such that A is numerably contractible and π0 (A) is a group then A has a homotopy inverse.
Complex categories were introduced in [10] as categorical analogs of CW-complexes.
These categories satisfy nice extra properties analogous to properties of CW-complexes, for
instance they are numerably contractible as we shall show below. Also, if A is a simplicial
category such that An is a complex category for each n then the special realization |A| is also
a complex.
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One of the main differences with CW-complexes is that in C at there is a countable family
of n-spheres for every dimension n ≥ 1. That means that a (relative) complex category is
constructed by attaching cells of different dimensions and different lengths. The concept of
subdivision explained above plays also here an important role.
Definition 4.5 Let S 0 be the discrete category with two points. For n ∈ N, we define the
n-dimensional α-sphere as the pointed category Sαn = αn S 0 .
Given a pointed category A and α ∈ N, the α-cone Cα A of A is the pushout
A+ A
(i 0 ,i 1 )
(1,∗)
Push
Iα A
/A
/ Cα A
Here Iα A denotes the reduced α-cylinder of A.
Let jβ : Sαn−1 → Cβ Sαn−1 be the natural inclusion of Sαn−1 in the β-cone. Note that jβ is a
cofibration.
A category X̃ is obtained from X by attaching an n-cell of length (α, β) (or simply an
n − (α, β)-cell) if there exists a pushout
Sαn−1
jβ
Cβ Sαn−1
f
Push
/X
/ X̃
for some attaching map f : Sαn−1 → X .
A pair of categories (X, A) is a relative complex category if there exists a sequence
A = X −1 → X 0 → X 1 → · · ·
with X = colim X n such that X n is obtained from X n−1 by attaching n − (α, β)-cells.
For A = ∗ we denote the pair (X, ∗) simply by X and we call it a complex category.
We say that dim(X, A) = n if X = X n and X = X n−1 . A relative complex category
(X, A) is called finite if X is obtained by attaching a finite number of cells.
For example, Sαn is a finite complex category with the base point as 0-cell and Sαn as n-cell.
Remark 4.6 Note that, if X and Y are complex categories, the category X × Y is also a
complex category and the inclusion X ∨ Y → X × Y is a cofibration. If A is a simplicial
category such that An is a complex category for every n, then the realization |A| is a complex
category since we are considering the special realization defined in the previous section.
Theorem 4.7 If X is a complex category, then it is contractibly generated.
Proof First note that, given the following pushout in C at
Sαn−1
jβ
Cβ Sαn−1
f
Push
/X
/ X̃
if X is contractibly generated, then X̃ is contractibly generated.
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259
Since X = colim(∗ → X 0 → X 1 → · · · ), it suffices to prove the theorem for X m . Now
this follows by induction (see [12]).
Since complex categories are paracompact [10, 12], the following result is an immediate
consequence of the previous theorem.
Corollary 4.8 Every complex category is numerably contractible.
The following result follows immediately from 4.4 and 4.8.
Corollary 4.9 Suppose A is a homotopy associative H -category such that A is a complex
category. If π0 (A) is a group, then A has a homotopy inverse.
5 -categories and spectra
We start this section by defining a suitable notion of a spectrum in C at∗ . Then we redefine the
notion of -category in a slightly more general way than Segal [15] and apply a generalization
of his machine to construct spectra of categories.
Definition 5.1 A spectrum C. in C at∗ consists of a sequence of pointed categories C1 , C2 , . . .
together with functors
αi Ci → Ci+1
∀i
for some αi ∈ N. A spectrum C. is called an -spectrum if the functors Ci → Ci+1 ,
obtained by the composition of the induced functor Ci → αi Ci+1 and the natural functor
αi Ci+1 → Ci+1 , are homotopy equivalences for all i.
Example 5.2 The Eilenberg–MacLane spectrum in C at∗ . Here the individual members of
the spectrum are complex categories. See [10].
Definition 5.3 Let F be the category of pointed finite sets n = {0, 1, . . . , n} with 0 the base
point. The maps φ: n → m in F preserve the base point. Note that F is isomorphic to the
opposite of the category introduced by Segal [15]. Given n ∈ N and 1 ≤ k ≤ n, define
i k : n → 1 by i k ( j) = 0 if j = k and i k (k) = 1.
Definition 5.4 A -category is a (covariant) functor A: F → C at∗ which satisfies the following two conditions.
(a)
(b)
→∗
A(0) −
The maps pn : A(n) → A(1) × · · · × A(1) induced by the maps i k : n → 1 (1 ≤ k ≤ n)
are homotopy equivalences of categories for all n.
Note that our definition of a -category is more general than the original definition of
Segal since an equivalence of categories is clearly a homotopy equivalence ([9, 13]). If we
use the terminology of [1], we could call this a special -category.
If A: F → C at∗ is a -category then its classifying space
B(A): F → T op∗
defined in [15] is clearly a -space, since B(C × D) −
→ B(C) × B(D).
260
K-Theory (2006) 37:249–261
Remark 5.5 The smash product n ∧ m in F is defined in the obvious way. Note that n ∧ m
coincides with n.m. Given n in F , if we compose a -category A with the functor n∧−: F →
F then we obtain a -category
A(n ∧ −): F → C at∗
There is a functor φ: op → F . Given f : n → m in , φ( f ): m → n in F is defined by
φ( f )( j) = i if f (i − 1) < j ≤ f (i) and φ( f )( j) = 0 if the set {i| f (i − 1) < j ≤ f (i)} is
empty.
Via this contravariant functor, any -category A has an underlying simplicial pointed
category Aφ: op → C at∗ . If A is a -category, the realization |A| will mean the realization
of the underlying simplicial category of A.
Definition 5.6 Let A be a -category. The classifying -category
B A: F → C at∗
is defined by B A(n) = |A(n ∧ −)|. Here |A(n ∧ −)| denotes the realization of the underlying
simplicial category of the -category A(n ∧ −).
Note that B A(1) = |A|. Thus, by 3.7 we obtain a functor 1 A(1) → B A(1) which induces
a functor A(1) → B A(1). Therefore, given a -category A, we construct a spectrum of
categories
B.A:
A(1), B A(1), . . . , B k A(1), . . .
In order to say when A(1) → B A(1) is a homotopy equivalence we need to observe that
the -category structure of A makes A(1) into a homotopy associative H -category. More
precisely, the multiplication on A(1) is defined by
p2−1
m2
µ2 : A(1) × A(1) −−→ A(2) −→ A(1)
where p2−1 is any homotopy inverse of p2 : A(2) → A(1) × A(1) and m 2 is induced by the
map m: 2 → 1 in F defined by m(0) = 0 and m(1) = m(2) = 1.
The main theorem of [15] asserts that given a -space A, the space A(1) is the loop
space of B A(1) provided the H -space A(1) has a homotopy inverse. We obtain the following
analogous result.
Theorem 5.7 Let A: F → C at∗ be a -category, then the H -category A(1) has a homotopy
inverse if and only if the functor A(1) → B A(1) is a homotopy equivalence of categories.
Proof Consider the underlying simplicial category Aφ: op → C at∗ . For simplicity
of notation, we write An = Aφ(n).
Let P A be the simplicial path category of A defined in 3.8.
We prove that the diagram
A1
/ |P A|
/ |A|
A0
(1)
is homotopy cartesian if A1 has a homotopy inverse, i.e. that A1 is homotopy equivalent to
the homotopy pullback defined by A0 and |P A| over |A|. This will prove that A1 → |A|
is a homotopy equivalence, as required.
K-Theory (2006) 37:249–261
261
For each n consider the diagram
An+1
1
µ
/ A1
p
/∗
An1
where p is the projection on the last n factors and µ = µn+1 is the composition-law. If
A1 has a homotopy inverse, then this diagram is homotopy cartesian. Now, since (P A)n is
homotopy equivalent to An+1
1 , it follows that for every map ψ: n → m in , the diagram
(P A)m
Am
ψ∗
ψ∗
/ (P A)n
/ An
is homotopy cartesian. Thus, by the analog of [15, Prop.(1.6)] for simplicial categories, it
follows that the diagram (1) is homotopy cartesian.
We now consider -categories A such that A(n) is a complex category for every n. In
this case, by 4.6, the categories B k A(1) are complex categories in C at for all n. Thus, by 4.9
the H -category B k A(1) have homotopy inverses for k ≥ 1 and the H -category A(1) has a
homotopy inverse provided π0 (A(1)) is a group. Therefore, we obtain the following result.
Corollary 5.8 Let A be a -category such that A(n) is a complex category for every n. Then
B A(1), . . . , B k A(1), . . . is a -spectrum of categories. Furthermore, if π0 (A(1)) is a group,
then B.A = A(1), B A(1), . . . is a -spectrum of categories.
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