arXiv:1206.1203v2[math.CT]4Jul2012
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS
IN CAT
JOHN BOURKE
Abstract. Badzioch showed that in the category of simplicial sets each homotopy
algebra of a Lawvere theory is weakly equivalent to a strict algebra. In
seeking to extend this result to other contexts Rosick´y observed a key point to
be that each homotopy colimit in SSet admits a decomposition into a homotopy
sifted colimit of ﬁnite coproducts, and asked the author whether a similar
decomposition holds in the 2-category of categories Cat. Our purpose in the
present paper is to show that this is the case.
1. Introduction
When V is a complete and cocomplete symmetric monoidal closed category the
theory of categories enriched in V develops in much the same way as ordinary
category theory. Classical concepts, such as ﬁnite limit theories and their algebras,
have enriched analogues: if T is a small V-category with ﬁnite products one
can consider T-algebras in V, which are V-functors X : T → V preserving ﬁnite
products.
If V has a notion of weak equivalence then algebras have a natural homotopy
analogue: a homotopy algebra being given by a V-functor X : T → V which
preserves products up to weak equivalence, in the sense that the canonical map
X(A1 × . . . × An) → X(A1) × . . . × X(An) is a weak equivalence for each ﬁnite
tuple of objects of T.
Each genuine or strict algebra is a homotopy algebra and one can ask to what
extent the converse is true—with respect to the natural pointwise notion of weak
equivalence one can ask whether each homotopy algebra is weakly equivalent to a
strict one. Badzioch in [2] investigated this question in the case of simplicially enriched
categories, with theories the classical single sorted Lawvere theories viewed
as discrete simplicial categories; his main result a rigidiﬁcation theorem establishing
each homotopy algebra to be weakly equivalent to a strict algebra. This
result was extended by Bergner in [3] to cover ﬁnite product theories, again in the
simplicial setting.
In [22] Rosick´y has investigated the possibility of extending these rigidiﬁcation
results to other settings, by allowing his base of enrichment V to be a monoidal
model category other than simplicial sets, and by considering weighted limit theories
more general than ﬁnite product theories. One of his rigidiﬁcation results,
Theorem 3.3 of [22], requires that each coﬁbrant weight, or coﬁbrant object in
Date: July 5, 2012.
2000 Mathematics Subject Classiﬁcation. Primary: 18D05, 55P99.
Supported by the Grant agency of the Czech Republic under the grant P201/12/G028.
1
2 JOHN BOURKE
[J, V] with its projective model structure, admits a certain kind of colimit decomposition.
He asked the author whether such a colimit decomposition exists in
the case that V = Cat with weak equivalences the equivalences of categories, and
when the theories under consideration are just ﬁnite product theories—the special
nature of the colimit decomposition now requiring that each coﬁbrant object of
[J, Cat] can be presented as a sifted colimit of ﬁnite coproducts of representables,
in which moreover each colimit involved is homotopically well behaved in a manner
described in Section 4.2.
The aim of the present paper is to show, in Theorem 8, that this is the case.
With this result in place Rosick´y’s theorems’ 3.3 and 5.1 of [22] yield rigidiﬁcation
results for homotopy algebras of ﬁnite product theories in Cat—his Theorem 5.1
now asserts that, in Cat, each homotopy algebra of a ﬁnite product theory is weakly
equivalent to a strict algebra, a direct analogue of the results in the simplicial
setting described above.
Now the coﬁbrant objects of [J, Cat] are the ﬂexible weights of [4]. Flexible limits
and colimits have been the subject of much study in 2-category theory, and many
of the results required to give the main decomposition, in Theorem 8, are known—
our main contribution here is to put these facts together in an appropriate way.
Since the results are spread throughout the literature on 2-category theory, and
some without detailed proof, we give a thorough, and reasonably self contained,
treatment of all aspects involved in the decomposition, with the intention of making
Rosick´y’s rigidiﬁcation result in this 2-categorical setting more easily accessible.
In Section 2 we give the necessary background, beginning with a few brief remarks
on weighted limits and colimits. We recall the notion of a ﬂexible weight,
and so ﬂexible limits and colimits, describing the connection with model categories.
Examples of ﬂexible colimits are given and their properties discussed. We begin
the third section by describing those ﬂexible colimits involved in our decomposition
of a ﬂexible weight, giving a detailed treatment of reﬂexive codescent and reﬂexive
lax codescent objects. We show each of these colimits to be sifted colimits—the
case of reﬂexive codescent objects is in [15]. Combining these results a presentation
of each ﬂexible weight as a sifted ﬂexible colimit of coproducts of representables is
given—this is the main novel result of the paper. In the ﬁnal section we begin by
discussing bicolimits and their relationship with ordinary colimits. We use that
ﬁltered colimits are bicolimits in Cat [19] and reduce from arbitrary coproducts of
representables to ﬁnite coproducts, giving the ﬁnal decomposition in Theorem 8.
The author thanks Jiˇr´ı Rosick´y and Stephen Lack for useful discussions on the
content of this paper.
2. Weighted colimits and flexible colimits
2.1. Weighted colimits. When V is a complete and cocomplete symmetric
monoidal closed category one has the full theory of categories enriched in V [11].
V itself admits such an enrichment (which we will also denote by V) and for each
small V-category J we have the enriched category [J, V] whose objects, V-functors,
are called J-indexed weights. Given a diagram D : J → C its W-weighted limit is
an object {W, D} together with a V-natural transformation W → C({W, D}, D−)
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 3
called a cone, or cylinder, which induces an isomorphism
C(A, {W, D}) ∼= [J, V](W, C(A, D−))
for each A ∈ C. Dually given a diagram D : Jop → C its W-weighted colimit,
or just W-colimit to be brief, is an object W ⋆ D equipped with a cocone W →
C(D−, W ⋆ D) inducing an isomorphism
C(W ⋆ D, A) ∼= [J, V](W, C(D−, A))
for each A ∈ C.
Amongst weighted colimits we ﬁnd the familiar conical colimits such as coequalisers
and coproducts, and also tensors by objects of V, but also more complex
kinds—for instance when V = Cat we have codescent objects, Kleisli objects of
monads and many other useful colimits, some of which are described in detail in
Section 3.
2.2. Flexible colimits and coﬁbrancy. A limit {W, D} or colimit W ⋆D is said
to be ﬂexible if W is a ﬂexible weight. One can approach ﬂexible weights using
model categories without knowing anything of 2-category theory beyond general
enriched category theory, and likewise using 2-category theory without any model
categories at all. Both perspectives are important here so we recall each.
Flexible weights were ﬁrst deﬁned in 2-category theory, as a special case of the
notion of ﬂexible algebra for a 2-monad, and the results we describe now are special
cases of results of [5] concerning 2-monads. Whilst the generality of 2-monads is
mostly beyond our needs at present a little background is required.
Given a 2-category C we have the identity on objects inclusion ι : [J, C] →
Ps(J, C) with the latter 2-category having 2-functors as objects, arrows the more
general pseudonatural transformations, and modiﬁcations for 2-cells. The inclusion
is the identity on objects so that we typically omit to label its action. If C is both
complete and cocomplete the inclusion has a left adjoint Q—we refer the reader to
Section 3.2 for more detail on this. The unit and counit at W ∈ [J, C] are given by
pseudonatural and 2-natural transformations pW : W QW and qW : QW → W
respectively. The isomorphism
[J, C](QW, X) ∼= Ps(J, C)(W, X)
exhibits QW as a pseudomorphism classiﬁer, in the sense that any pseudonatural
W X factors uniquely through pW : W QW as a 2-natural transformation.
One of the triangle equations for the adjunction asserts that the pseudonatural
pW : W QW is a section of qW in Ps(J, C); in fact qW is a retract equivalence,
or surjective equivalence, in the 2-category Ps(J, C). It follows in particular that
the component of qW : QW → W at each j ∈ J is a surjective equivalence in C;
thus qW is a pointwise surjective equivalence.
Now let us interpret the above in the special case of [J, Cat]. We have seen that
the counit qW : QW → W at a weight W always admits a pseudonatural section
pW ; the weight W is said to be a ﬂexible weight [4] just when qW admits a section
in [J, Cat], in which case qW : QW → W is in fact a surjective equivalence in
[J, Cat].
On the other hand if we ignore its 2-dimensional structure then Cat is a combinatorial
model category with weak equivalences and ﬁbrations the equivalences
4 JOHN BOURKE
of categories and isoﬁbrations, and trivial ﬁbrations the surjective equivalences.
Cartesian product gives it the structure of a monoidal model category [9] so that
one can speak of model 2-categories, Cat being one of these. It was shown in [16]
that [J, Cat] obtains the projective model structure in which the weak equivalences
and ﬁbrations are pointwise as in Cat, and that a ﬂexible weight is just a coﬁbrant
object, or coﬁbrant weight, therein. This was done in the more general context of
2-dimensional monad theory, but since the proof is short we include it in the case
of weights.
Proposition 1. (Lack) When [J, Cat] is equipped with the projective model structure,
the coﬁbrant objects therein are exactly the ﬂexible weights.
Proof. In the projective model structure on [J, Cat] the trivial ﬁbrations are pointwise
as in Cat, and so the pointwise surjective equivalences. We have seen that
qW : QW → W is one of these; as such it will exhibit QW as a coﬁbrant replacement
of W if we can show QW to be coﬁbrant. Upon doing so it is clear
that W will be coﬁbrant just when qW admits a section, which is to say when W
is a ﬂexible weight. To see that QW is coﬁbrant suppose that f : X → Y is a
pointwise surjective equivalence—given an arbitrary arrow r : QW → Y we should
show it factors through f. Observe that as f is a pointwise surjective equivalence
it admits a pseudonatural section g : Y X so that fg = 1. Now consider the
following diagram
W QW Y
X
pW
///o/o/o/o
r
//
f
gr
;;;{;{;{;{;{;{
h
..
where all but h have been deﬁned. By the universal property of pW the composite
grpW : W X is uniquely of the form hpW for a 2-natural h : QW → X. Now to
check that fh = r it suﬃces, by the same universal property, to show fhpW = rpW
But we have fhpW = fgrpW = rpW as required.
Knowing that the ﬂexible weights are the coﬁbrant weights doesn’t oﬀer much
insight as to what they actually look like, or why they are interesting in 2-category
theory. Let us conclude this section by brieﬂy mentioning some examples of ﬂexible
colimits including the generating ones, and some other properties. We will describe
the examples of most importance in more detail in Section 3.
2.3. Pseudocolimits. Given a weight W and diagram D : Jop → C its W
weighted pseudocolimit W ⋆p D is deﬁned by an isomorphism C(W ⋆p D, A) ∼=
Ps(J, Cat)(W, C(D−, A)) natural in A. By the adjunction Q ⊣ ι we have a natural
isomorphism [J, Cat](QW, C(D−, A)) ∼= Ps(J, Cat)(W, C(D−, A)) so that the
pseudocolimit is nothing but the weighted colimit QW ⋆ D. That pseudocolimits
are ﬂexible is easy to see: the adjunction Q ⊣ ι generates a comonad (Q, q, ∆) on
[J, Cat] with counit the same q : Q → 1 as before; in particular each QW admits a
(co-free) coalgebra structure, and so is certainly a ﬂexible weight. Since the pseudocolimit
W ⋆p D is the genuine colimit of D weighted by a coﬁbrant replacement
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 5
of W, pseudocolimits are closely related to homotopy colimits—this relationship
was studied in [8].
2.4. Saturation, pie colimits and splittings of idempotents. Let Flex denote
the class of ﬂexible weights so that Flex(J) ⊂ [J, Cat] consists of the J-indexed
ﬂexible weights. It was shown in [4] that the ﬂexible weights form a saturated
class1
in the sense of [14]. This means that, for each J, Flex(J) contains the representables
and is closed in [J, Cat] under ﬂexible colimits (not just J-indexed ones).
Moreover four kinds of colimit suﬃce to construct all ﬂexible weights—namely
co(p)roducts, co(i)nserters, co(e)quiﬁers and (s)plittings of idempotents, by which
we mean that these four colimits are ﬂexible and Flex(J) = PIES∗(J), the closure
of the representables in [J, Cat] under these four kinds of colimit.
It follows that if one can construct W-colimits out of these four kinds in a general
2-category then W is ﬂexible: one uses that W = W ⋆Y for the Yoneda embedding
Y and carries out the corresponding construction in [J, Cat]. This fact provides a
convenient way to test whether a particular weight is ﬂexible—examples which are
easily seen to be ﬂexible in this manner are coinverters, Kleisli objects of monads,
the codescent objects of the following section along with numerous others—many
such cases were described in [12] and [4].
Though not relevant in what follows it is perhaps worth mentioning that if we
drop splittings of idempotents from the above and take the closure PIE∗(J) ⊂
[J, Cat] we get what are called the pie weights [20]. Apart from splittings of
idempotents essentially all ﬂexible colimits, such as those just mentioned, are pie.
The pie weights can be recognised as precisely those admitting coalgebra structure
for the comonad Q on [J, Cat]—see [18] or [6]—this can be interpreted as saying
that they are the algebraically coﬁbrant objects in [J, Cat] in the sense of [21], a
perspective which was further explored in [6].
2.5. The importance of ﬂexible limits and colimits in 2-category theory.
Let us brieﬂy indicate some reasons for the interest in ﬂexibility. Primary objects of
study in 2-dimensional universal algebra are 2-categories, such as the 2-category of
monoidal categories and strong monoidal functors, whose morphisms only preserve
structure up to isomorphism. Such 2-categories generally admit pie limits [5],
and if the structure involved is itself “ﬂexible”, such as monoidal structure, they
also admit ﬂexible limits [4]; note that in the full sub 2-category containing the
more “rigid” strict monoidal categories idempotents need not split. The precise
distinction is that monoidal categories are the algebras for a ﬂexible 2-monad [10, 5]
whereas strict monoidal categories are not.
3. Sifted flexible colimits and a first decomposition
A weight W ∈ [J, Cat] is said to be sifted if ﬁnite products commute with Wcolimits
in Cat. This is to say that the 2-functor W ⋆− : [Jop, Cat] → Cat preserves
ﬁnite products. In the present section we describe a number of weights which are
both sifted and ﬂexible and give our ﬁrst decomposition result.
The two key kinds of colimits are reﬂexive codescent and reﬂexive lax codescent
objects—that the former are sifted is a result of Lack, Proposition 4.3 of [15],
1Saturated classes were originally called closed in [1].
6 JOHN BOURKE
though the proof only outlined. We ﬁll in the details here and follow a suggestion of
Lack to extend this result to the lax setting. In both cases we follow the argument
outlined in [15]—to apply the following lemma2
of the same paper, which reduces
the colimits to be computed to manageable special cases.
Lemma 2. (Lack) A weight W : J → Cat is sifted if W ⋆ − : [Jop, Cat] → Cat
preserves ﬁnite products of representables. If J has a terminal object which is
preserved by W then W is sifted so long as W ⋆ − preserves binary products of
representables.
Let us remark upon our terminology concerning codescent objects—this is based
upon [15] and ﬁts well with the appearance of codescent objects in 2-dimensional
monad theory. What we call lax codescent and codescent objects have also been
called codescent and isocodescent objects respectively—see [23] for instance. We
will only consider the notion of a reﬂexive lax codescent or reﬂexive codescent
object here which relate to the irreﬂexive kind [15] as reﬂexive coequalisers do
to coequalisers—a notable distinction is that only the reﬂexive variants commute
with ﬁnite products in Cat and Set respectively.
3.1. Reﬂexive lax codescent objects. Truncating the simplicial category ∆
at the ordered set with three elements gives a full subcategory ∆2 ⊂ ∆; now
restricting the usual embedding ∆ → Cat along the inclusion yields the weight
Wl : ∆2 → Cat for reﬂexive lax codescent objects. A diagram ∆op
2 → C in a
2-category C consists of a truncated simplicial object as on the left below
(1) A2 A1 A0
d //
ioo
c
//
p
//
m //
q
//roo
loo
A1
A0
A
A0
d <<②②②
c ""❊❊❊
f
""❊❊❊
f
<<②②②
η
In elementary terms its reﬂexive lax codescent object A is speciﬁed by a triple (A,
f : A0 → A, η : fd ⇒ fc) as above satisfying the two equations for a lax codescent
cocone. The ﬁrst of these asserts the equality
A2 A1
A0
A
A0
A1
A1
p
BB☎☎☎☎☎
q ✿✿✿✿✿
d //
c
//
d
BB☎☎☎☎☎
c ✿✿✿✿✿
f
✿✿✿✿✿✿
f
BB☎☎☎☎☎☎
m // η
= A2 A0
A0
A
A0
A1
A1
p
BB☎☎☎☎☎
q ✿✿✿✿✿
d //
c
//
c ✿✿✿✿✿
d
BB☎☎☎☎☎
f //
f
✿✿✿✿✿✿
f
BB☎☎☎☎☎☎
η
η
whilst the second equation asserts that ηi : f = fdi ⇒ fci = f is an identity
2-cell. As with all 2-categorical colimits it has both a 1 and 2-dimensional aspect
to its universal property; the 1-dimensional aspect asserts that given any other
such cocone (B, g, θ) there exists a unique arrow g′ : A → B such that g′f = g and
2 In [15] the hypothesis on the terminal object does not appear in the statement of the lemma
but is discussed in the proof.
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 7
g′η = θ; the 2-dimensional aspect asserts that given a second such triple (B, h, φ)
together with a 2-cell ρ : g ⇒ h rendering the square
gd gc
hd hc
θ +3
ρd
ρc
φ
+3
commutative, then there exists a unique 2-cell ρ′ : g′ ⇒ h′ between the induced
factorisations such that ρ′ ◦ f = ρ.
The most important example of such colimits for our concerns is the following
example from [23]. A small category A can be presented as an internal category in
Set by taking its truncated nerve. The reader can interpret diagram (1) above in
this manner, so that A0, A1 and A2 are the sets of objects, arrows and composable
pairs of A, with the maps d and c the domain and codomain projections, and so
on. Viewing each of these sets as discrete categories one can view this internal
category in Set as an internal category in Cat whose reﬂexive lax codescent object
is exactly A. The universal cocone (A, f, η) has f : A0 → A the identity on objects
inclusion, and η : fd ⇒ fc the natural transformation whose component ηα at an
object α : x → y ∈ A1 is simply α : x = fdα → fcα = y itself, now viewed as an
arrow of A. We leave to the reader the worthwhile exercise of checking that this
is indeed the claimed colimit.
In order to see that reﬂexive lax codescent objects are sifted colimits in Cat it
will be worth being precise about the manner in which we passed from the category
A to the corresponding internal category in Cat; this was achieved by taking the
singular functor Cat → [∆op
2 , Set] associated to the embedding ∆2 → Cat and
postcomposing by D∗ : [∆op
2 , Set] → [∆op
2 , Cat] where D : Set → Cat is the functor
viewing each set as a discrete category; let us write Nd : Cat → [∆op
2 , Cat] for
the composite functor (note that this is not a 2-functor). As a composite of limit
preserving functors we observe that Nd preserves limits.
Consider the 2-functor Wl ⋆ − : [∆op
2 , Cat] → Cat which takes reﬂexive lax
codescent objects and write (Wl ⋆−)0 : [∆op
2 , Cat] → Cat for its underlying functor.
The content of our example is that we have an isomorphism Wl ⋆ Nd(A) ∼= A
for each category A; moreover this is easily seen to be a natural isomorphism
(Wl ⋆ −)0 ◦ Nd
∼= 1.
A diagram X ∈ [∆op
2 , Cat] lies in the essential image of Nd just when it is an
internal category in Cat with each X(j) discrete; let us call such diagrams in Cat
pointwise discrete categories. The relevance of this notion is that the representables
∆2(−, i) : ∆op
2 → Cat, corresponding to our weight Wl : ∆2 → Cat, all share this
form. To see this let i ∈ {0, 1, 2} and consider the following composite
∆op
2 ∆op Set Cat
j
// ∆(−,i)
// D //
In the middle the representable ∆(−, i) is well known to be the nerve of a category,
so that its restriction ∆(−, i) ◦ j is one too. As ∆2 is a full subcategory of ∆ so
the restriction ∆(−, i) ◦ j is just ∆2(−, i) which is hence an internal category.
But we are supposed to be considering ∆2 as a 2-category and its Cat-valued
representables; however since ∆2 is locally discrete the corresponding Cat-valued
8 JOHN BOURKE
representable is just the composite D ◦ ∆2(−, i), which, as D preserves pullbacks,
is a pointwise discrete category in Cat. We can now prove:
Proposition 3. Reﬂexive lax codescent objects are sifted ﬂexible colimits.
Proof. Reﬂexive lax codescent objects can be constructed by forming a coinserter
followed by two coequiﬁers and are consequently ﬂexible colimits (see 2.4)—this
construction is described in Proposition 2.1 of [15] for a more general kind of lax
codescent object.
With regards siftedness observe that ∆2 has a terminal object preserved by its
inclusion to Cat so that it suﬃces, by Lemma 2, to show that Wl ⋆− : [∆op
2 , Cat] →
Cat preserves binary products of representables. Each representable is a pointwise
discrete category—we will show Wl ⋆− preserves binary products of these instead.
To show Wl ⋆− preserves the product X ×Y of such a pair it suﬃces to show that
its underlying functor (Wl ⋆ −)0 does so. But now X and Y lie in the essential
image of Nd so that, as Cat has products and Nd preserves them, the product also
lies in the essential image. Consequently we need only show that the composite
(Wl⋆−)0◦Nd preserves binary products. Being naturally isomorphic to the identity
functor this is the case.
3.2. Reﬂexive codescent objects. The weight Wi : ∆2 → Cat for reﬂexive
codescent objects is obtained from the weight Wl : ∆2 → Cat for reﬂexive lax
codescent objects by postcomposing Wl by the reﬂection Cat → Gpd to groupoids,
and then passing back via the inclusion Gpd → Cat. In elementary terms, given
a diagram as in (1), its reﬂexive codescent object is speciﬁed by a triple (A, f, η)
satisfying the same equations as in the lax case, with the exception that η is now
required to be an invertible 2-cell; moreover the 1-dimensional universal property
of A only quantiﬁes over triples (B, g, θ) in which θ is invertible, whilst the 2dimensional
universal property is the same as before.
The relevant example concerns the construction of the pseudomorphism classiﬁer
QX of a diagram X : J → C in a complete and cocomplete 2-category C as arises
from the adjunction (ι : [J, C] ⇆ Ps(J, C) : Q) discussed in 2.2. To explain how
this goes observe that restriction U : [J, C] → [obJ, C] along the inclusion of the
discrete 2-category with the same objects as C has a left 2-adjoint F and that U
is moreover monadic. The adjunction (ǫ, F ⊣ U, η) gives FU the structure of a
comonad on [J, C] and so, in the usual way, yields for each X ∈ [J, C] a (truncated)
simplicial object
(FU)3X (FU)2X FUX
ǫF UX
//
F ηUX
oo
F UǫX
//
//
//
//
oo
oo
where we have omitted to label the higher face and degeneracy maps. The
reﬂexive codescent object of this diagram in [J, C] is exactly QX.
That this is the case is best understood in terms of two dimensional monad
theory: the monadic adjunction F ⊣ U induces a 2-monad T = UF whose 2category
of strict algebras and strict morphisms T-Algs is [J, C] whilst the Tpseudomorphisms,
as belonging to the 2-category T-Alg, are precisely the pseudonatural
transformations of Ps(J, C)—this is shown in Section 6.6 of [5]. It follows
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 9
that the inclusion [J, C] → Ps(J, C) coincides with the inclusion T-Algs → T-Alg
that views each strict T-algebra morphism as a pseudomorphism. The formula for
QD above is then a special case of the formula for the pseudomorphism classiﬁer
QA of a T-algebra A as a reﬂexive codescent object of free algebras. This formula
was ﬁrst described in [15] in a more general setting relevant to pseudoalgebras—a
description better suited to the present level of generality is given in Section 4.2 of
[17]. The most important case for us is when C = Cat—at a weight W the above
presentation exhibits QW as a reﬂexive codescent object of free weights.
In understanding that reﬂexive codescent objects are sifted colimits in Cat it
will be useful to break down their construction into two steps: as a reﬂexive lax
codescent object followed by a coinverter. It is not worth the eﬀort to describe the
weight for coinverters here—see [12]—it suﬃces to say that the domain 2-category
consists of a single 2-cell, so that, correspondingly, one forms the coinverter of a
2-cell α ∈ C(A, B)(f, g) in a 2-category C; this consisting of a pair (C, h) as on the
left below
A B C
f
g
AAα
h // A1
A0
A
A0
B
d <<②②②
c ""❊❊❊
f
""❊❊❊
f
<<②②②
η
f
<<②②②
g
//
in which hα is invertible; the 1-dimensional universal property is that given any
k : B → D with kα invertible there exists a unique k′ : C → D such that k′h = k;
its 2-dimensional universal property asserts that, for each object D, the induced
functor C(h, D) : C(C, D) → C(B, D) is fully faithful.
Now it is easily seen that given a diagram as in (1) we can form its reﬂexive
codescent object in two steps—ﬁrstly forming the reﬂexive lax codescent object
(A, f, η) and then the coinverter (B, g) of the resulting 2-cell η, as drawn on the
right above. In the following proposition we shall use this construction to show
that reﬂexive codescent objects are sifted colimits in Cat; however it is not the
case that coinverters are themselves sifted, but only reﬂexive coinverters—this was
shown using a 3×3 argument in [13]. The distinction between a reﬂexive coinverter
and a coinverter is in the input; a coinverter is said to be reﬂexive when the input
2-cell α, as above, admits a splitting, in the sense of an arrow k : B → A such that
fk = 1, gk = 1 and αk = 1.
We need one further auxiliary concept—that of a liberal arrow. An arrow f :
A → B of a 2-category C is said to be liberal if it is conservative in Cop—this means
that a 2-cell α : g ⇒ h ∈ C(B, C) out of B is invertible whenever the composite
αf is. We will have use for the fact that each bijective on objects functor is liberal
in Cat. Finally observe that given a diagram
D A B C
f
g
AAα
h //e //
with e liberal then h exhibits C as the coinverter of α if and only if it exhibits
C as the coinverter of αe; this follows from the fact that for any r : B → E the
composite rα is invertible just when rαe is. With this in place we can prove:
10 JOHN BOURKE
Proposition 4. (Lack) Reﬂexive codescent objects are sifted ﬂexible colimits.
Proof. That reﬂexive codescent objects are ﬂexible follows from their construction
via reﬂexive lax codescent objects and coinverters, both of which are ﬂexible colimits.
That coinverters are ﬂexible, constructible from coinserters and coequiﬁers,
can be found in Proposition 4.2 of [12].
With regards siftedness observe that the weights Wi and Wl have the same
domain so that the associated representables coincide—these are pointwise discrete
categories as in 3.1. Again Wi preserves the terminal object so that, as in the proof
of Propostion 3, Wi will be sifted if we can show that the composite (Wi ⋆−)0 ◦Nd :
Cat → [∆op
2 , Cat] → Cat preserves binary products. We will show this to be true by
breaking this functor down into several components. Given A ∈ Cat the reﬂexive
codescent object of NdA is obtained by forming the reﬂexive lax codescent object
of NdA—this is just (A, f, η) as described in 3.1—followed by the coinverter (B, g)
of η as on the left below
A1
A0
A
A0
B
d <<②②②
c ""❊❊❊
f
""❊❊❊
f
<<②②②
η
f
<<②②②
g
// A1 A2 A B
t // λ
p
q
??
g
//
Let us form the arrow category A2 of A which comes equipped with an evident
pair of projections and natural transformation λ : p ⇒ q ∈ Cat(A2, A) with the
universal property that any natural transformation into A factors uniquely through
it; we factorise η as λt accordingly as indicated on the right above. Explicitly t is
given by the map which assigns to an object of A1, an arrow of A, the corresponding
object of the arrow category A2; thus t is bijective on objects. As such it is
liberal so that the coinverter B of η is equally the coinverter of λ. Therefore
(Wi ⋆ −)0 ◦ Nd : Cat → [∆op
2 , Cat] → Cat is equally just the functor which ﬁrst
assigns to a category A the 2-cell (A2, λ : p ⇒ q, A) and then its coinverter;
certainly the ﬁrst assignment preserves products since arrow categories are limits
(cotensors with the free arrow) in Cat. Furthermore the 2-cell λ is reﬂexive, split
by the functor i : A → A2 that assigns to an object of A the identity arrow upon it;
since reﬂexive coinverters commute with ﬁnite products we deduce the claim.
3.3. Decomposition of a ﬂexible weight as a sifted ﬂexible colimit of
coproducts of representables. As well as using the above kinds of codescent
object in the following decomposition, we will also split an idempotent. Splittings
of idempotents are of course absolute colimits and so sifted; and ﬂexible as
discussed in 2.4. With these results in place we give our ﬁrst decomposition.
Theorem 5. Each ﬂexible weight lies in the closure of the coproducts of representables
under sifted ﬂexible colimits.
Proof. If W is a ﬂexible weight then qW : QW → W has a section w : W → QW
so that W is the splitting of the idempotent w ◦ qW on QW, a sifted ﬂexible
colimit. Now recall from 3.2 the adjunction (U : [J, Cat] ⇆ [obJ, Cat] : F) given
by restriction and left Kan extension along the inclusion obJ → J. As described in
3.2 the weight QW is a reﬂexive codescent object of free weights, those of the form
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 11
FX for a family X : obJ → Cat, and so a sifted ﬂexible colimit of free weights by
Proposition 4. For j ∈ J each category Xj is, by 3.1, the reﬂexive lax codescent
object of its truncated nerve Nd(Xj) as (unlabelled) on the left below
X(j)2 X(j)1 X(j)0
//
oo
//
//
//
//
oo
oo
X2 X1 X0
//
oo
//
//
//
//
oo
oo
so that X is the reﬂexive lax codescent object in [obJ, Cat] of the diagram on
the right which pointwise evaluates to that on the left. Applying the left adjoint
F : [obJ, Cat] → [J, Cat] we deduce that FX is a reﬂexive lax codescent object of
the FXi, in particular a sifted ﬂexible colimit of the FXi by Proposition 3.
For each i ∈ {0, 1, 2} the family Xi : obJ → Cat takes its values amongst
the discrete categories. Any Y : obJ → Cat with this property is the coproduct
Y = ΣjY (j).obJ(j, −) where, since obJ is discrete, the representable obJ(j, −) is
just the characteristic function at j. By the left Kan extension formula for F it
is easy to see that F(obJ(j, −)) = J(j−), whence FY = F(ΣjY (j).obJ(j, −)) =
ΣjY (j).J(j, −) is a coproduct of representables; in particular for each i ∈ {0, 1, 2}
the weight F(Xi) is a coproduct of representables. Thus W can be formed in three
steps by taking sifted ﬂexible colimits of coproducts of representables.
4. Filtered colimits and the reduction to finite coproducts
In this ﬁnal section we extend the decomposition of Theorem 5, using ﬁltered
colimits to reduce from arbitrary coproducts of representables to ﬁnite coproducts.
Whilst ﬁltered colimits are not ﬂexible in general they do exhibit some good homotopical
behaviour in Cat which is crucial for our decomposition—namely, they
are bicolimits in Cat.
4.1. Bicolimits. Given a weight W : J → Cat and diagram D : Jop → C the
W-bicolimit W ⋆b D [12] is speciﬁed by a pseudococone W C(D−, W ⋆b D) ∈
Ps(J, Cat) such that the induced functor
C(W ⋆b D, A) → Ps(J, Cat)(W, C(D−, A))
is an equivalence for each A ∈ C. The bicolimit is only determined up to equivalence
by its deﬁning property; the pseudocolimit, if it exists, provides a canonical
instance.
We call a genuine colimit W ⋆ D a bicolimit if its colimiting cocone W →
C(D−, W ⋆ D) in fact exhibits W ⋆ D as the W-bicolimit. This amounts to saying
that the composite
C(W ⋆ D, A) ∼= [J, Cat](W, C(D−, A)) → Ps(J, Cat)(W, C(D−, A))
is an equivalence for each A ∈ C. Since the ﬁrst component is an isomorphism this
is equally to say that the inclusion
[J, Cat](W, C(D−, A)) → Ps(J, Cat)(W, C(D−, A))
is an equivalence for each A ∈ C.
If W is ﬂexible then qW : QW → W is an equivalence in [J, Cat] so that
q∗
W : [J, Cat](W, C(D−, A)) → [J, Cat](QW, C(D−, A)) is an equivalence for each
12 JOHN BOURKE
A. Composing this map with the canonical isomorphism [J, Cat](QW, C(D−, A)) ∼=
Ps(J, Cat)(W, C(D−, A)) yields the above inclusion, so that any ﬂexible colimit is
a bicolimit.
It is not true that ﬁltered colimits are ﬂexible nor that they are bicolimits in each
2-category. They are, however, bicolimits in Cat—this was shown in Lemma 5.4.9
of [19] by directly calculating ﬁltered colimits in Cat. We give a diﬀerent proof
below, which follows easily from the equivalence of (1) and (3) in the following.
Proposition 6. Let C be a complete and cocomplete 2-category consider a weight
W ∈ [J, Cat]. The following are equivalent.
(1) W-colimits are bicolimits in C.
(2) For each diagram D the map qW ⋆ D : QW ⋆ D → W ⋆ D is an equivalence in
C.
(3) For each pointwise equivalence f : D → E of diagrams the induced W ⋆ D →
W ⋆ E is an equivalence in C.
Proof. To say that qW ⋆ D : QW ⋆ D → W ⋆ D is an equivalence is equally to say
that the functor C(qW ⋆ D, A) : C(W ⋆ D, A) → C(QW ⋆ D, A) is an equivalence
for each A ∈ C. The canonical isomorphisms render this isomorphic to the inclusion
[J, Cat](W, C(D−, A)) → Ps(J, Cat)(W, C(D−, A)) so that (1) and (2) are
equivalent.
That (1) implies (3) is also straightforward. Assuming (1) consider a pointwise
equivalence f : D → E of diagrams in [Jop, C]. This induces a pointwise equivalence
f∗ : C(E−, A) → C(D−, A) ∈ [J, Cat]. Whiskering by f∗ induces a commutative
square
[J, Cat](W, C(E−, A)) Ps(J, Cat)(W, C(E−, A))
[J, Cat](W, C(D−, A)) Ps(J, Cat)(W, C(D−, A))
//
//
in which the horizontal arrows are the inclusions, both of which are equivalences
since W-colimits are bicolimits in C. The pointwise equivalence f∗ is a genuine
equivalence in Ps(J, Cat) so that the right vertical arrow is an equivalence whence,
by 2 out of 3, the left vertical arrow is an equivalence too. Therefore the isomorphic
C(W ⋆ f, A) : C(W ⋆ E, A) → C(W ⋆ D, A) is an equivalence for each A so that
W ⋆ f is itself an equivalence.
It remains to show that (3) implies (2). Recall the adjunction (ι : [J, C] ⇆
Ps(J, C) : Qc) from 2.2 with counit qc : Qc → 1. The key point here is that we
have an isomorphism λ : QW ⋆ D ∼= W ⋆ QcD compatible with the counits in the
sense that the triangle
QW ⋆ D
W ⋆ QcD
W ⋆ Dλ
W ⋆qc
D
33❢❢❢❢❢❢❢❢❢
qW ⋆D
++❳❳❳❳❳❳❳❳❳
commutes. This is just the colimit variant of a result on limits of [8]. To see where
the isomorphism comes from recall that the cotensor AX (or power) of an object
A ∈ C by a category X is the limit deﬁned by a natural isomorphism
C(B, AX
) ∼= Cat(X, C(B, A))
A COLIMIT DECOMPOSITION FOR HOMOTOPY ALGEBRAS IN CAT 13
Given a weight W : J → Cat and diagram D : Jop → C this isomorphism extends
in a pointwise manner to a natural isomorphism as on the left below
[Jop
, C](D, AW −
) ∼= [J, Cat](W, C(D−, A))
which, moreover, is compatible with pseudonatural transformations in the sense
that Ps(Jop, C)(D, AW −) ∼= Ps(J, Cat)(W, C(D−, A)). Applying these isomorphisms
back and forth, together with the universal properties of QW and QcD,
yields an isomorphism [J, Cat](QW, C(D−, A)) ∼= [J, Cat](W, C(QcD−, A)) and so
the claimed λ : QW ⋆ D ∼= W ⋆ QcD. It is straightforward to check compatibility
with the counit.
As discussed in 2.2 the map qc
D is always a pointwise equivalence so that, by
assumption, W ⋆ qc
D is an equivalence. As λ is an isomorphism it follows that
qW ⋆ D is an equivalence for each D proving (2).
Corollary 7. (Makkai-Par´e) Filtered colimits are bicolimits in Cat.
Proof. As Cat is a (ﬁnitely) combinatorial model category a ﬁltered colimit of
equivalences is again an equivalence [7]. The equivalence of (1) and (3) in Proposition
6 gives the result.
4.2. The ﬁnal decomposition. Our terminology below diﬀers slightly from that
used by Rosick´y in 3.2 and 3.3 of [22]. Given a suitable monoidal model category
V, such as our Cat or SSet, and a weight W ∈ [J, V] with coﬁbrant replacement
qc : Wc → W in the projective model structure, he calls W homotopy invariant if
for each objectwise coﬁbrant diagram D : Jop → V the induced morphism qc ⋆ D :
Wc ⋆ D → W ⋆ D is a weak equivalence in V. In Cat the coﬁbrant replacement of
a weight W is the map qW : QW → W considered throughout whilst every object
in Cat is coﬁbrant; it now follows from Proposition 6 that a weight W : J → Cat is
homotopy invariant in Rosick´y’s sense just when W-colimits are bicolimits in Cat.
For the ﬁnal decomposition we need a notion of closure which quantiﬁes over
diagrams as well as weights. Consider a full subcategory : A → B and a class Φ
of pairs of weights and diagrams Φ = {(Wi : Ji → Cat, Di : Jop
i → B); i ∈ I} with
each colimit Wi ⋆ Di existing in B. By the 1-step closure of A in B under colimits
of type Φ we mean the full subcategory of B consisting of A together with each
colimit of the form Wi ⋆ Di for (Wi, Di) ∈ Φ where Di is a diagram taking values
in A. In a similar manner the 2-step closure is obtained by adding Φ-colimits with
diagrams taking values in the 1-step closure, and so on.3
Theorem 8. Each ﬂexible weight of [J, Cat] lies in the (4-step) closure of the
ﬁnite coproducts of representables in [J, Cat] under colimits of type Φ, where a
pair (W : K → Cat, D : Kop → [J, Cat]) belongs to Φ just when
(1) W is a sifted weight.
(2) W-colimits are bicolimits in Cat.
(3) Each diagram D takes values amongst ﬂexible weights and each colimit W ⋆D
is ﬂexible.
Proof. Arbitrary coproducts of representables are ﬂexible by 2.4. In any category,
or 2-category, an arbitrary coproduct can be constructed using ﬁltered colimits
3The limit of this process is the closure discussed in 3.5 of [11].
14 JOHN BOURKE
of ﬁnite coproducts—in particular each coproduct of representables can be constructed
as a ﬁltered colimit of ﬁnite coproducts of representables in [J, Cat]. Filtered
colimits certainly commute with ﬁnite products in Cat, and so are sifted, and
are bicolimits by Corollary 7; thus arbitary coproducts lie in the 1-step closure.
We saw in Theorem 5 that any ﬂexible weight can be constructed from coproducts
of representables in three stages by taking sifted ﬂexible colimits, always of
ﬂexible weights. As discussed in 4.1 ﬂexible colimits are bicolimits in Cat—thus
each ﬂexible weight lies in the 4-step closure.
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Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz