nLab
homotopy limit

Context

Homotopy theory

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Limits and colimits

Contents

Idea

In great generality, a homotopy limit is a way of constructing appropriate sorts of limits in a (weak) higher category and in general and in (∞,1)-category theory in particular, using some presentation of that higher category by a 1-categorical structure. The general study of such presentations is homotopy theory.

In classical homotopy theory, the presentation is given by a category with weak equivalences, possibly satisfying extra axioms such as those of a homotopical category, a category of fibrant objects, or a model category. Such structures are considered to present an (∞,1)-category, and homotopy limits give a way of constructing the appropriate sort of (∞,1)-categorical limits.

In enriched homotopy theory, the presentation is given by an enriched model category or an enriched homotopical category, and it may presents an “enriched (,1)(\infty,1)-category” or be a more powerful presentation of a bare (,1)(\infty,1)-category (notably if the enrichment is over sSet). In the enriched category theory context the appropriate notion is a weighted homotopy limit, which may construct “weighted (,1)(\infty,1)-limits” in the presented “enriched (,1)(\infty,1)-category” or may be a more powerful tool for constructing plain (,1)(\infty,1)-categorical limits (in particular if the enrichment is over sSet). Note that as yet, no fully general notion of “enriched (,1)(\infty,1)-category” exists; see homotopical enrichment.

Maybe the most commonly encountered setup for homotopy limits is that where the (∞,1)-category in question is presented by a simplicial model category. See also homotopy Kan extension, of which (globally defined) homotopy (co)limits are a special case.

Definitions

As for ordinary limits, there are two ways to define homotopy limits:

  • with explicit constructions that satisfy a local universal property: the homotopy limit object “represents homotopy-coherent cones up to homotopy.”

  • as derived functors of a homotopy Kan extension that satisfy a global universal property: the homotopy limit functor is “universal among homotopical approximations to the strict limit functor.”

One of the central theorems of the subject is that in good cases, the two give equivalent results; see below.

Global definition

Let CC be a category with weak equivalences and let DD be a (small) diagram category. Make the functor category [D,C][D,C] into a category with weak equivalences by taking the weak equivalences to be those natural transformations which are objectwise weak equivalences in CC.

The ordinary limit and colimit operations over DD-diagrams are (as described there) the right and left adjoints of the functor const:C[D,C]const : C \to [D,C], or equivalently left and right Kan extension along the unique functor !:C*!\colon C\to * to the terminal category.

[D,C]lim=Ran !constcolim=Lan !C. [D,C] \stackrel{\stackrel{\overset{colim = Lan_!}{\to}}{\overset{const}{\leftarrow}}}{\underset{lim = Ran_!}{\to}} C \,.

The (globally defined) homotopy limit and colimit are accordingly the right and left homotopy Kan extension along !:C*!\colon C\to *:

  • The homotopy limit of a functor F:DCF : D \to C is, if it exists, the image of FF under the right derived functor of the limit functor lim D:[D,C]Clim_D : [D,C] \to C with respect to the given weak equivalences on CC and the objectwise weak equivalences on [D,C][D,C]:

    holim DF:=(lim D)F. holim_D F := (\mathbb{R} lim_D)F \,.
  • The homotopy colimit of a functor F:DCF : D \to C is, if it exists, the image of FF under the left derived functor of the colimit functor colim D:[D,C]Ccolim_D : [D,C] \to C with respect to the given weak equivalences on CC and the objectwise weak equivalences on [D,C][D,C]:

hocolim DF:=(𝕃colim D)F. hocolim_D F := (\mathbb{L} colim_D)F \,.

Alternative definitions can be formulated at the level of the homotopy category W 1CW^{-1} C one defines a localized version Δ¯ I:W 1CW I 1C I\bar{\Delta}^I : W^{-1} C\to W_I^{-1} C^I of the diagonal functor Δ I:CC I\Delta^I : C\to C^I and define the homotopy limits and colimits as the adjoints of Δ¯ I\bar{\Delta}^I (at least at the points where the adjoints are defined). Here W IMor(C I)W_I\subset Mor(C^I) are the morphisms of diagrams whose all components are in WMor(C)W\subset Mor(C). The above definitions via derived functors (Kan extensions) follow once one applies the general theorem that the derived functors of a pair of adjoint functors are also adjoint and noticing that (Δ I,Δ¯ I)(\Delta^I,\bar{\Delta}^I) is a morphism of localizers (and in particular that Δ¯ I\bar{\Delta}^I with the identity 2-cell is a Kan extension (simultaneously left and right)).

In the enriched case, this must be suitably modified to deal with weighted limits as well as enrichment of both CC and DD.

In particular, if CC is equipped with the extra structure of a simplicial model category and KK is an (small) sSet-enriched category we may also hope to equip the enriched functor category [D,C][D,C] with the structure of a simplicial model category. There are two different “canonical” such structures:

(both of which have the same objectwise weak equivalences and are in fact Quillen equivalent). When these model structures exist (as they do when CC is combinatorial), limit and colimit constitute then two sSetsSet-enriched Quillen adjunctions

[D,C] projconstcolimC [D,C]_{proj} \stackrel{\overset{colim}{\to}}{\underset{const}{\leftarrow}} C

and

[D,C] injlimconstC. [D,C]_{inj} \stackrel{\overset{const}{\leftarrow}}{\underset{lim}{\to}} C \,.

(All proofs and other technical details on this are at homotopy Kan extension.)

These present directly the corresponding adjoint (∞,1)-functors (as described there)

(,1)Func(D,C )C (\infty,1)Func(D,C^\circ) \stackrel{\to}{\stackrel{\leftarrow}{\to}} C^\circ

by precomposition with a cofibrant replacement functor (for the colimit) and a fibrant replacement functor (for the limit).

Local definition

The local definition requires making precise the notion of a homotopy commutative cone over a diagram.

For the case of SimpSet-enrichment one elegant way to do so is in terms of suitable weighted limits as described in the example section at weighted limit: a homotopy commutative cone with tip cCc \in C over a diagram F:KCF : K \to C in an SimpSet\Simp\Set-enriched category CC is a natural transformation WC(c,F()):KSimpSetW \Rightarrow C(c,F(-)) : K \to \Simp\Set where the weight functor WW is not constant on the point, as for ordinary limits, but is given by W:kN(K/k)W : k \mapsto N(K/k).

The same idea works if we are enriched over a category VV that is not SimpSet\SimpSet but is itself enriched over SimpSet\Simp\Set, such as topological spaces or spectra, since then any VV-category becomes a SimpSet\Simp\Set-category as well in a natural way. Finally, although a general model category need not be enriched over anything, it is always “almost” enriched over SimpSet\Simp\Set, and so one can still make sense of this using the techniques of framings and resolutions; see the books of Hirschhorn and Hovey.

Following the reasoning described in Example 1 of representable functor one then defines the homotopy limit LL of a functor F:KCF: K \to C to be a representing object for such homotopy cones, in the sense that we have a (weak) equivalence

Map(X,L)HoCones(X,F) Map(X,L) \simeq HoCones(X,F)

of hom-objects (spaces or simplicial sets in the classical context; enriched hom-objects in the enriched context).

Global versus local

The global definition is formulated in terms of weak equivalences only, while the local definition is formulated in terms of homotopies only. However, in practical cases, derived functors exist because their input objects (in this case, the diagram FF) can be replaced by “good” (fibrant and/or cofibrant) objects in such a way that weak equivalences become homotopy equivalences. The derived functor of limlim at the input object FF is then computed by applying the ordinary functor limlim to a good replacement RFR F of FF.

It then turns out that the “good” (precisely, “fibrant”) replacement RFR F “builds in” precisely the right homotopies so that an ordinary cone over RFR F is the same as a homotopy-commutative cone over FF. Therefore, lim(RF)lim (R F), which is the global homotopy-limit of FF, is a representing object for homotopy-commutative cones over FF, and thus is also a local homotopy-limit of FF. There is a dual argument for colimits using cofibrant replacements.

Formal versions of this argument can be found in many places. Perhaps the original statement can be found in XI.8.1 of:

  • A. K. Bousfield and D. M. Kan, Homotopy limits, completions, and localizations.

(As was often the case with Kan’s papers at that time, there are some details omitted from that treatment, but most are, as he claimed, quite easy to complete.) For another approach in an algebraic context, there is a description in Illusie’s thesis.

An abstract version in modern language, with proof, can be found in

Strictness

Another notable difference between the local and global definitions is that the global definition can only ever define the homotopy limit up to weak equivalence (isomorphism in the homotopy category), while in the local definition we could require, if we wanted to, an actual isomorphism

Map(X,L)HoCones(X,F) Map(X,L) \cong HoCones(X,F)

of hom-objects, rather than merely a weak equivalence. By analogy with strict 2-limits, we may call such an object a strict homotopy limit.

Frequently a strict homotopy limit does in fact exist, and can be constructed as a weighted limit in the ordinary (enriched) category in question. In such cases, the strict homotopy limit may be easier to compute with than an arbitrary homotopy limit merely known to have the up-to-weak-equivalence universal property. Thus, sometimes when people say the homotopy limit they refer mean a strict homotopy limit.

When a strict homotopy limit exists, an arbitrary homotopy limit may be defined as an object which is (weakly) equivalent to the strict homotopy limit.

Derivators

It is noteworthy that the homotopy limit and colimit in a category with weak equivalences are drastically different from the ordinary limit and colimit in the corresponding homotopy category: the universal property of the full (,1)(\infty,1)-categorical limits and colimits crucially does take into account the explicit higher cells and does not work just up to any higher cells.

This (obvious) observation is a very old one and can be seen to be one of the driving forces behind the insight that just working with homotopy categories misses crucial information, something which today is understood as the statement that a homotopy category is just the decategorification of an (∞,1)-category.

While the full theory of (∞,1)-categories is one way to impose the correct notion of higher categorical limits, there are other ways to deal with issue. Due to Alexander Grothendieck is the technique of using derivators for keeping track of homotopy limits.

Roughly, the idea of a derivator is that while the single homotopy category Ho(C)Ho(C) of a category CC with weak equivalences is not sufficient information, the homotopy limit of a diagram in [D,C][D,C] is encoded in the homotopy category Ho([D,C])Ho([D,C]) of that functor category (this is after all the domain of the plain 1-categorical derived functor of the limit functor). Accordingly, what is called the derivator of a category with weak equivalences CC is a collection of all the homotopy categories Ho([D,C])Ho([D,C]) of all the possible diagram categories [D,C][D,C], as DD runs over all small categories. See there for more details.

Computational methods

Above we defined homotopy (co)limits in general. There are various more specific formulas and algorithms for computing homotopy (co)limits. Here we discuss some of these

Ordinary (co)limits over resolved diagrams

The direct prescription for computing the value of a right or left derived functor between model categories is by evaluating the original functor on a fibrant or cofibrant resolution of the given object.

For the derived functors of limit and colimit

[D,C]lim constlim C [D,C] \stackrel{\overset{\lim_\leftarrow}{\to}}{\stackrel{\overset{const}{\leftarrow}}{\underset{\lim_\to}{\to}}} C

let for instance

Q proj:[D,C][D,C]Q_{proj} : [D,C] \to [D,C] be a cofibrant replacement for the projective model structure on functors, so that for any diagram FF the diagram Q projFQ_{proj} F is a projectively cofibrant diagram (see there for more details). Then the homotopy colimit is presented by the ordinary colimit over Q projFQ_{proj} F:

(𝕃lim)FlimtQ proj(F). (\mathbb{L}\underset{\to}{\lim}) F \simeq \underset{\to}{\limt} Q_{proj}(F) \,.

This is sometimes called the Quillen formula for computing homotopy colimits. Analogously with P injP_{inj} a fibrant replacement functor for the injective model structure, we have

holimFlimP inj(F). holim F \simeq lim P_{inj}(F) \,.

Often, however, it is inconvenient to produce a resolution of a diagram. Because often all the work is in finding the resolution, while it is easy to evaluate the original functor on it. Therefore one wants ways to slightly change the setup of the problem such that the computation of the resolutions becomes more systematic. One such way is the use of derived (co)ends, discussed below.

Resolved (co)ends

Let Q projQ'_{proj} a cofibrant replacement functor for [D op,sSet Quillen] proj[D^{op}, sSet_{Quillen}]_{proj} (notice the opposite category) and Q injQ_{inj} one for [D,C] inj[D,C]_{inj}. Let *[D op,sSet]* \in [D^op,sSet] simplicial presheaf constant on the terminal simplicial set and Q proj()Q'_{proj}(') its projective cofibrant replacement.

Then we also have the coend expression

hocolimF DQ proj(*)Q inj(F), hocolim F \simeq \int^{D} Q'_{proj}(*) \cdot Q_{inj}(F) \,,

where in the integrand we have the tensoring of the simplicial model category CC over sSet.

If DD happens to be a Reedy model category we can equivalently use in this expression also the Reedy model structures [D,C] Reedy[D,C]_{Reedy} and [D op,C] Reedy[D^{op}, C]_{Reedy} and obtain the homotopy colimit as

hocolimF DQ Reedy(*)Q Reedy(F), hocolim F \simeq \int^{D} Q'_{Reedy}(*) \cdot Q_{Reedy}(F) \,,

This formula is sometimes called the Bousfield-Kan formula (see also Bousfield-Kan map). The coend is a weighted limit and this formula for the plain homotopy colimit can be understood the left derived weighted colimit which trivial weight (the underived weight is trivial, but becomes non-trivial after derivation – this extra complexity helps reduce the complexity for the replacement for the functor FF itself).

In detail, let VV be a monoidal model category and CC a VV-enriched model category (for instance V=sSet QuillenV = sSet_{Quillen} as in the above discussion). Let DD be a small VV-enriched category. Then for a any given weight

W:D opV W : D^{op} \to V

we have a VV-adjunction

[D,C]const() WW D()D, [D,C] \stackrel{\overset{W \otimes_D (-)}{\to}}{\underset{const(-)^{W}}{\leftarrow}} D \,,

where the left adjoint is the weighted colimit given by the coend

W DF= dDW(d)F(d) W \otimes_D F = \int^{d \in D} W(d) \otimes F(d)

(with the tensoring over VV in the integran) and where the right adjoint is formed using the cotensor over VV.

The crux now is that as WW varies, the left adjoint here is a left Quillen bifunctor

()():[D op,V]×[D,C]C. \int (-)\cdot (-) : [D^{op}, V] \times [D,C] \to C \,.

For details on this see Quillen bifunctor or around page 9 of (Gambino).

From the fact that this is a Quillen bifunctor and using the observation that for the trivial weight W=const1W = const 1 the wighted colimit reduces to an ordinary colimit, follows the above Bousfield-Kan-type formula for the homotopy colimit.

Bar constructions

A general way of obtaining resoltions that compute homotopy (co)limits involves bar constructions. (…)

Homotopy limits versus higher categorical limits

As a special case of enriched homotopy theory, we may consider model categories or homotopical categories that are enriched over a notion of nn-category as presentations for (n+1)(n+1)-categories. (Here we allow nn to also be of the form (n,r), with the obvious convention that (n,r)+1=(n+1,r+1)(n,r)+1 = (n+1,r+1) and +1=\infty+1=\infty.) For example:

  • simplicial sets are models for \infty-groupoids ((,0)(\infty,0)-categories), so simplicial model categories are presentations for (,1)(\infty,1)-categories. Of course, arbitrary model categories are also presentations for (,1)(\infty,1)-categories, but simplicial model categories are easier to work with, and in particular easier to construct homotopy limits in.

  • A Cat-enriched category is just a strict 2-category, so a model 2-category or homotopical 2-category? is a presentation of a weak 2-category (or bicategory).

  • A (2Cat, Gray(2Cat,\otimes_{Gray})-enriched category is a Gray-category, and a model Gray-category? or homotopical Gray-category? is a presentation of a weak 3-category (or tricategory).

If CC is a category enriched over (n1)(n-1)-categories and we are considering it to be an nn-category (which happens to be strict at the bottom level), it is natural to define a “weak equivalence” in the underlying ordinary category of CC to be a morphism that is an nn-category-theoretic equivalence. We call this the natural or trivial homotopical structure on CC. In certain cases (such as n=2)n=2) it can be made into a model structure, also called natural or trivial.

Since higher categorical limits are generally defined as representing objects for cones that commute up to equivalence, it is unsurprising that if CC has a natural homotopical structure, locally-defined homotopy limits and nn-limits coincide. For n=1n=1 this is trivial. For n=2n=2 it is proven in (Gambino) (particularly section 6). For n=(,1)n=(\infty,1) it is proven in (among other places) Lurie’s book, section 4.2.4. The case n=3n=3 ought to be approachable in theory, but doesn’t seem to have been done (probably partly because the general theory of 3-limits is fairly nonexistent).

On the other hand, we may also consider a category CC enriched over nn-categories with a larger class of weak equivalences than just the nn-categorical equivalences. Then CC presents an nn-category (its “homotopy nn-category”) obtained by formally turing these weak equivalences into nn-categorical equivalences. Homotopy limits in CC with this homotopical structure should then present nn-limits in its homotopy nn-category. In the case n=(,1)n=(\infty,1) this is also essentially in Lurie’s book; for other values of nn it may not be in the literature.

Homotopy limits versus limits in the homotopy category

It is important to note that homotopy limits and limits in the homotopy category are, in general, incomparable. A homotopy limit need not be a limit in the homotopy category, while a limit in the homotopy category need not be a homotopy limit.

It is generally true that homotopy products (and coproducts) are also products and coproducts in the homotopy category. Some other homotopy limits induce the corresponding notion of weak limit in the homotopy category; for instance, homotopy pullbacks become weak pullbacks in the homotopy category. However, even this is not true for all types of homotopy limit.

On the other hand, homotopy categories do not usually have many limits and colimits at all (aside from products and coproducts). An explicit proof that Ho(Cat)Ho(Cat) does not have pullbacks can be found here. But even if a homotopy category does happen to have limits and colimits, these need not be the same as homotopy limits.

For instance, every chain complex over a field kk is quasi-isomorphic to its homology, regarded as a chain complex with zero differentials; and between chain complexes of the latter form, quasi-isomorphisms are just isomorphisms. Thus, the homotopy category of chain complexes over kk is equivalent to the category of graded kk-vector-spaces. This is complete and cocomplete as a category, but its limits and colimits are not the same as the homotopy limits and colimits arising from its presentation as the homotopy category of chain complexes. In particular, chain complexes are a stable (infinity,1)-category, so every homotopy pullback square is also a homotopy pushout square and vice versa; but nothing of the sort is true in graded vector spaces as a 1-category.

Examples

General weighted colimit formula for homotopy colimits

Let

There is a general formula for the homotopy colimit over FF in terms of a coend or weighted colimit in CC, using the following ingredients:

Ingredients

For CC a combinatorial simplicial model category as above and for DD any simplicially enriched category there is the projective and the injective global model structure on functors on the enriched functor category [D,C][D,C].

  • In the projective model structure [D,C] proj[D,C]_{proj} the fibrations and the local equivalences are objectwise those of CC

  • In the injective model structure [D,C] inj[D,C]_{inj} the cofibrations and the local equivalences are objectwise those of CC.

Each of these is itself a combinatorial simplicial model category, so in particular the small object argument applies in these using which one obtains cofibrant replacement functors

Q proj:[D,C][D,C] Q_{proj} : [D, C] \to [D, C]

and

Q inj:[D,C][D,C]. Q_{inj} : [D, C] \to [D, C] \,.

That CC is a simplicial model category implies in particular that it is tensored over SSet and that the tensoring functor

:C×SSetC \otimes : C \times SSet \to C

is a left Quillen bifunctor. By the properties of Quillen bifunctors discussed there, it follows that the coends over the tensor in the form

D()():[D op,SSet] proj×[D,C] injC \int^D (-) \otimes (-) : [D^{op},SSet]_{proj} \times [D,C]_{inj} \to C

and in the form

D()():[D op,SSet] inj×[D,C] projC \int^D (-) \otimes (-) : [D^{op},SSet]_{inj} \times [D,C]_{proj} \to C

both themselves left Quillen bifunctors.

Write

*:D opSSet {*} : D^{op} \to SSet

for the functor that sends everything to the identity on the singleton set. This is the tensor unit in the monoidal category [D op,SSet][D^{op},SSet].

General formula

Theorem

With the above assumptions and ingredients, the homotopy colimit over F:DCF : D \to C is given either by

hocolimF= DQ proj(*)Q inj(F) hocolim F = \int^D Q_{proj}({*}) \otimes Q_{inj}(F)

or by

hocolimF= DQ inj(*)Q proj(F). hocolim F = \int^D Q_{inj}({*}) \otimes Q_{proj}(F) \,.
Proof

By the fact that the coend over the tensor appearing here is a Quillen bifunctor.

This is disucssed for instance in section 4 of (Gambino).

Examples

Homotopy colimits over simplicial diagrams

Let D=Δ opD = \Delta^{op} be the opposite category of the simplex category.

Proposition

A cofibrant replacement of the terminal object *{*} in the projective global model structure on functors [Δ,SSet][\Delta, SSet] is the the fat simplex-functor that assigns to [n][n] the nerve of opposite category of the undercategory of Δ op\Delta^{op} under [n][n]

N(/Δ op) op:ΔSSet. N(-/\Delta^{op})^{op} : \Delta \to SSet \,.
Proof

For instance prop 14.8.8 in

  • Hirschhorn, Model categories and their localization

Notice that if F:Δ opCF : \Delta^{op} \to C takes values in cofibrant objects of CC, then it is itself cofibrant as an object of [Δ op,C] inj[\Delta^{op},C]_{inj}. In that case no further cofibrant replacement of FF is necessary and it therefore follows with the general formula and the above proposition that the homotopy colimit over FF is given by the formulas

hocolimF= [n]ΔN([n]/Δ) opF(n). hocolim F = \int^{[n] \in \Delta} N([n]/\Delta)^{op} \otimes F(n) \,.

This is famously the formula introduced and used by Bousfield and Kan (but there originally missing the necessary condition that FF be objectwise cofibrant). See Bousfield-Kan map.

Homotopy pushouts

Let in the above general formula D={acb}D = \{a \leftarrow c \to b\} be the walking span. Ordinary colimits parameterized by such DD are pushouts. Homotopy colimits over such DD are homotopy pushouts.

In this simple case, we have the following simple observation:

Observation

For DD as above, the terminal functor *:DSSet{*} : D \to SSet is already cofibrant in [D,SSet] inj[D,SSet]_{inj}.

Moreover

Observation

For DD as above, a functor F:DCF : D \to C is cofibrant in [D,C] proj[D,C]_{proj} if

  • it sends both morphisms cac \to a and cbc \to b to cofibrations

  • it sends cc (and hence also aa and bb) to cofibrant objects in CC.

Since a coend *F\int {* } \otimes F over a tensor product where the first factor in the integrand in the tensor unit is just an ordinary colimit over the remaining FF, it follows that if FF is of the form of the above observation, then the ordinary colimit over FF already computes the homotopy pushout:

hocolimF=lim F. hocolim F = \lim_\to F \,.

The dual version of this statement (for homotopy limits and homotopy pullbacks) is discussed in more detail in the examples below.

Homotopy pullbacks

Here we consider special cases of homotopy pullback in more detail.

Let D={102}D = \{ 1\to 0 \leftarrow 2\} be the pullback diagram, so that limits over it compute pullbacks, and assume that F:DCF : D \to C is such that

F(1)F(0)F(2) F(1) \to F(0) \leftarrow F(2)

satisfies

  • F(i)F(i) is fibrant for all ii;
  • and either F(1)F(0)F(1) \to F(0) or F(2)F(0)F(2) \to F(0) is a fibration;

then

  • holim DFholim_D F exists;
  • and is weakly equivalent to the ordinary limit holim DFlim DFholim_D F \stackrel{\simeq}{\to} lim_D F.

Conversely this means that on an arbitrary pullback diagram holim DFholim_D F can be computed by finding a natural transformation FRFF \Rightarrow R F whose component morphisms are weak equivalences and such that RFR F satisfies the above conditions.

Based loop objects

For BB any pointed object with point ptpt BBpt \stackrel{pt_B}{\to} B the homotopy pullback of the point along itself is the loop space object of BB

Ω pt * * B, \array{ \Omega_{pt} &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& B } \,,

i.e.

holim(ptpt BBpt Bpt)Ω ptB. holim( pt \stackrel{pt_B}{\to} B \stackrel{pt_B}{\leftarrow} pt) \;\stackrel{\simeq}{\to}\; \Omega_{pt} B \,.

One way to compute this using the above prescription by noticing that the generalized universal bundle E ptB\mathbf{E}_{pt} B provides a fibrant replacement of the pullback diagram in that we have

pt B pt Id Id E ptB B pt \array{ pt &\to& B &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{pt}B &\to& B &\leftarrow& pt }

with all vertical morphisms weak equivalences and with the left bottom horizontal morphism a fibration.

More on that in the further examples below.

Fibration sequences

If CC is a pointed object, with point *C{*} \to C, then for a homotopy pullback of the form

A * B C \array{ A &\to& {*} \\ \downarrow && \downarrow \\ B &\to& C }

the sequence ABCA \to B \to C is called a fibration sequence. The object AA is the homotopy kernel or homotopy fiber of BCB \to C. Since homotopy pullback squares compose to homotopy pullback squares, the homotopy kernel of a homotopy kernel is not trivial, but is a loop space object

ΩC A * * B C. \array{ \Omega C &\to& A &\to& {*} \\ \downarrow && \downarrow && \downarrow \\ {*} &\to& B &\to& C } \,.

Homotopy pullback of a point over a group / universal bundles

As a special case of the above general example we get the following.

Let C=C = Grpd equipped with the canonical model structure. Write GG for a group regarded as a discrete monoidal groupoid (elements of GG are the objects of the groupoids and all morphisms are identities) write and BG\mathbf{B}G for the corresponding one-object groupoid (single object, one morphism per element of GG). Write ptpt for the terminal groupoid (one object, no nontrivial morphism). Notice that there is a unique functor ptBGpt \to \mathbf{B}G. Then we have

holim( pt pt BG)G. holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} G \,.

To see this, we compute using the above prescription by finding a weakly equivalent pullback diagram such that one of its morphisms is a fibration. This is achieved in particular by the generalized universal bundle ptEG>BGpt \stackrel{\simeq}{\to} \mathbf{E}G \to\gt \mathbf{B}G, where EG\mathbf{E}G is the action groupoid G//GG//G of GG acting on itself by multiplication from one side. So we have a weak equivalence of pullback diagrams

pt BG pt = = EG BG pt \array{ pt &\to& \mathbf{B}G &\leftarrow& pt \\ \downarrow^{\simeq} && \downarrow^= && \downarrow^= \\ \mathbf{E}G &\to& \mathbf{B}G &\leftarrow& pt }

and the homotopy limit in question is weakly equivalent to the ordinary limit over the lower diagram. That is directly seen to be Disc(Obj(EG))=Disc(Obj(G//G))=Disc(G)Disc(Obj(\mathbf{E}G)) = Disc(Obj(G//G)) = Disc(G) which we just write as GG:

holim( pt pt BG)lim( pt EG BG)=G. holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} lim \left( \array{ && pt \\ && \downarrow \\ \mathbf{E}G &\to& \mathbf{B}G } \right) = G \,.

This example is important in the context of groupoidification and geometric function theory, as described there. A closely related example is the following: a functor ρ:BGTop\rho:\mathbf{B}G\to {Top} is the datum of a toplogical space XX equipped with an action of GG. Then, colim(ρ)=X/Gcolim(\rho)=X/G whereas hocolim(ρ)=EG× GXhocolim(\rho)=\mathbf{E}G\times_G X, see equivariant cohomology.

Homotopy pullback of a subgroup over a group

The above example generalizes straightforwardly to the case where the trivial inclusion ptBGpt \to \mathbf{B}G is replaced by any inclusion BHBG\mathbf{B}H \hookrightarrow \mathbf{B}G of any subgroup HH of GG pretty much literally by replacing ptpt by BH\mathbf{B}H throughout.

One finds

holim( BH BH BG)H\\G//H holim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} H \backslash\backslash G//H

where on the right we have the action groupoid of H×HH \times H acting on GG by multiplication from the left (first factor) and the right (second factor). (See for instance at Hecke category for an application.)

To see this, we again build a fibrant replacement of the pullback diagram. Following the constructions at generalized universal bundle consider first the groupoid E BHG\mathbf{E}_{\mathbf{B}H}G given by the pullback diagram

E BHG BH [I,BG] d 0 BG d 1 BG. \array{ \mathbf{E}_{\mathbf{B}H}G &\to& \mathbf{B}H \\ \downarrow && \downarrow \\ [I, \mathbf{B}G] &\stackrel{d_0}{\to}& \mathbf{B}G \\ \downarrow^{d_1} \\ \mathbf{B}G } \,.

As at generalized universal bundle one proves that the left vertical morphism E BHGBG\mathbf{E}_{\mathbf{B}H}G \to \mathbf{B}G is a fibration.

Now, notice (which was implicit in the above example) that since [I,BG][I,\mathbf{B}G] is a path object in a category of fibrant objects we have a section BG σ[I,BG] \mathbf{B}G \stackrel{\simeq}{\to}^\sigma [I, \mathbf{B}G] of [I,BG]d 0BG[I,\mathbf{B}G] \stackrel{d_0}{\to} \mathbf{B}G. In the above pullback diagram this induces a morphism BHσE BHG \mathbf{B}H \stackrel{\sigma}{\to} \mathbf{E}_{\mathbf{B}H}G making the obvious diagram commute. Now, the latter morphism, being the pullback of an acyclic fibration is an acyclic fibration, so its right inverse σ\sigma is a weak equivalence. This way we obtain the morphism of pullback diagrams

BH BG BH σ Id Id E BHG > BG BH \array{ \mathbf{B}H &\to& \mathbf{B}G &\leftarrow& \mathbf{B}H \\ {}^\simeq\downarrow^{\sigma} && \downarrow^{Id} && \downarrow^{Id} \\ \mathbf{E}_{\mathbf{B}H}G &\to \gt& \mathbf{B}G &\leftarrow& \mathbf{B}H }

which is objectwise a weak equivalence and such that the horizontal morphism on the bottom left is a fibration. By the above statement the ordinary limit of the lower horizontal diagram is weakly equivalent to the homotopy limit we are looking for. But this is manifestly the desired action groupoid:

holim( BH BH BG)lim( BH E BHG BG)lim( [I,BG] d 0×d 1 BH×BH BG×BG)=H\\G//H. holim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \stackrel{\simeq}{\to} \lim\left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{E}_{\mathbf{B}H} G &\to& \mathbf{B}G } \right) \simeq \lim\left( \array{ && [I,\mathbf{B}G] \\ && \downarrow^{d_0 \times d_1} \\ \mathbf{B}H \times \mathbf{B}H &\to& \mathbf{B}G \times \mathbf{B}G } \right) = H \backslash\backslash G//H \,.

This example, too, is important at geometric function theory.

Homotopy span traces

  • see the homotopy span traces discussed at span trace for more examples of homotopy pullbacks

Homotopy colimits over simplicial diagrams

Proposition

Every simplicial set is the homotopy colimit over its cells.

Precisely: for XX \in sSet a simplicial set, let

X˜:Δ opSetsSet \tilde X : \Delta^{op} \to Set \hookrightarrow sSet

be the corresponding bisimplicial set which in degree kk is the the constant simplicial set on the set X kX_k of kk-simplices.

For the standard homotopical structure on sSet Δ opsSet^{\Delta^{op}}, the homotopy colimit over X˜\tilde X is equivalent to the origianal XX:

hocolimX˜X hocolim \tilde X \stackrel{\simeq}{\to} X

in the standard model structure on simplicial sets.

Proof

Use the Reedy model structure [Δ op,sSet Quillen] Reedy[\Delta^{op}, sSet_{Quillen}]_{Reedy}. With the coend recipe for the hocolim discussed above, it follows that the hocolim is the coend

[k]ΔQ Reedy(*)×Q Reedy(X˜), \int^{[k] \in \Delta} Q'_{Reedy}(*) \times Q_{Reedy}(\tilde X) \,,

where Q Reedy()Q'_{Reedy}(\cdots) is a cofibrant resolution in the Reedy model structure [Δ,sSet Quillen] Reedy[\Delta,sSet_{Quillen}]_{Reedy} and Q Reedy(...)Q_{Reedy}(...) in [Δ op,sSet Quillen] Reedy[\Delta^{op}, sSet_{Quillen}]_{Reedy}. But by the discussion at Reedy model structure – simplex category we have that

  • X˜\tilde X is always cofibrant in [Δ op,sSet Quillen] Reedy[\Delta^{op},sSet_{Quillen}]_{Reedy};

  • a cofibrant resolution of the point in [Δ,sSet Quillen] Reedy[\Delta, sSet_{Quillen}]_{Reedy} is given by Δ[]:ΔsSet\Delta[-] : \Delta \to sSet.

It follows that the hocolim is given by

[k]ΔΔ[k]×X k. \int^{[k] \in \Delta} \Delta[k] \times X_k \,.

By the co-Yoneda lemma this is isomorphic to XX.

Remark

More generally with this kind of argument it follows that generally the homotopy colimit over a simplicial diagram of simplicial sets is represented by the diagonal simplicial set of the corresponding bisimplicial set.

Remark

This kind of argument has many immediate generalizations. For instance for C=[K op,sSet Quillen] injC = [K^{op}, sSet_{Quillen}]_{inj} the injective model structure on simplicial presheaves over any small category KK, or any of its left Bousfield localizations, we have that the cofibrations are objectwise those of simplicial sets, hence objectwise monomorphisms, hence it follows that every simplicial presheaf XX is the hocolim over its simplicial diagram of component presheaves.

For the following write Δ:ΔsSet\mathbf{\Delta} : \Delta \to sSet for the fat simplex.

Observation

The fat simplex is Reedy cofibrant.

Proof

By the discussion at homotopy colimit, the fat simplex is cofibrant in the projective model structure on functors [Δ,sSet Quillen] proj[\Delta, sSet_{Quillen}]_{proj}. By the general properties of Reedy model structures, the identity functor [Δ,sSet Quillen] proj[Δ,sSet Quillen] Reedy[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy} is a left Quillen functor, hence preserves cofibrant objects.

Corollary

For X[Δ op,C]X \in [\Delta^{op}, C] a Reedy cofibrant object, the Bousfield-Kan map

[n]Δ[n]X n [n]Δ[n]X n \int^{[n]} \mathbf{\Delta}[n] \cdot X_n \to \int^{[n]} \Delta[n] \cdot X_n

is a weak equivalence in CC.

Proof

The coend over the tensor is a left Quillen bifunctor

()():[Δ,sSet Quillen] Reedy×[Δ op,C] Reedy \int (-)\cdot (-) : [\Delta,sSet_{Quillen}]_{Reedy} \times [\Delta^{op}, C]_{Reedy}

(as discussed there). Therefore with its second argument fixed and cofibrant it is a left Quillen functor in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the factorization lemma). By the above discussion, both Δ[n]\mathbf{\Delta}[n] and Δ[]\Delta[-] are indeed cofibrant in [Δ,sSet Quillen] Reedy[\Delta,sSet_{Quillen}]_{Reedy}. Clearly the functor Δ[]Δ[]\mathbf{\Delta}[-] \to \Delta[-] is objectwise a weak equivalence in sSet QuillensSet_{Quillen}, hence is a weak equivalence.

Proposition

Let i:Δ fΔ i : \Delta_f \hookrightarrow \Delta be the inclusion into the simplex category of all the monomorphisms (all the face maps).

This inclusion is a homotopy-initial functor. As a consequence, homotopy colimits of shape Δ\Delta can equivalently be computed after their restriction to Δ f\Delta_f

hocolim(Δ opFC)hocolim(Δ f opi opΔ opFC). hocolim( \Delta^{op} \stackrel{F}{\to} C) \simeq hocolim( \Delta_f^{op} \stackrel{i^{op}}{\to} \Delta^{op} \stackrel{F}{\to} C) \,.

See (Dugger, example 18.2).

Homotopy colimits over diagrams of spaces

By geometric realization

The following is sometimes in the literature taken as the definition of homotopy colimits of diagrams of spaces. It is one of the earliest formulas for there.

Definition

Let DD be a category and F:DF : D \to Top a [{functor].

Define a simplicial topological space sFsF by setting

sF:[n] d 0d 1d nF(d n) sF : [n] \mapsto \coprod_{d_0 \leftarrow d_1 \leftarrow \cdots \leftarrow d_n} F(d_n)

and using the obvious face and degeneracy maps: face maps act by mapping components of the coproducts over one sequence of morphisms to one obtained by deleting outer arrows or composing inner arrows. If the rightmost arrow is deleted, then the component map is not the identity but is F(d n)D(d n1)F(d_n) \to D(d_{n-1}). The degeneracy maps similarly introduce identity morphisms.

Theorem

Let DD be a category and F:DF : D \to Top a functor. Then the homotopy colimit of FF is equivalent to the geometric realization of simplicial topological spaces of sFs F:

hocolimFsF. hocolim F \simeq \vert sF \vert \,.

This is an application of the bar-construction method.

See for instance (Dugger, part 1) for an exposition.

Higher homotopy van Kampen theorem

Theorem

Let XX be a topological space, write Op(X)Op(X) for its category of open subsets and let

χ:COp(C) \chi : C \to Op(C)

a functor out of a small category CC such that

Then:

the canonical morphism in sSet out of the colimit

lim SingχSing(X) {\lim_\to} Sing \circ \chi \to Sing(X)

into the singular simplicial complex of XX exhibits Sing(X)Sing(X) as the homotopy colimit hocolimSingχhocolim Sing \circ \chi.

See higher homotopy van Kampen theorem for details.

Sequential homotopy (co)limits

For

D={012} D = \{0 \to 1 \to 2 \to \cdots\}

the cotower category, a colimit of shape DD is called a sequential colimit. For CC a combinatorial model category it is easy to characterize the cofibrant objects in the projective model structure on functors [D,C] proj[D, C]_{proj}: these are those cotower diagrams all whose morphisms are cofibrations and whose 0th object (and hence all objects) are cofibrant.

So given a cotower with such a property, its homotopy colimit is just the ordinary sequential colimit in CC.

Dually for sequential limits over a tower diagram.

A standard application for this is for instance the construction of the classifying space BU=lim nBU(n)B U = \underset{\to_n}{\lim} B U(n) for reduced topological K-theory. See there for more.

Descent objects

Descent objects as they appear in descent and codescent are naturally conceived as homotopy limits. See also infinity-stack.

Homotopy (co)limits of simplicial pre(sheaves)

The local model structure on simplicial presheaves SPSh(C) proj/inj locSPSh(C)_{proj/inj}^{loc} over a site CC serve as models for ∞-stack (∞,1)-toposes.

Here we discuss some properties of homotopy limits and colimits in such model categories of simplicial presheaves.

Preservation of homotopy pullback by inverse images

For C,CC, C' two sites, a geometric morphism p:Sh(C)Sh(C)p : Sh(C) \stackrel{\leftarrow}{\to} Sh(C') of sheaf toposes induces correspondingly an adjunction

p:SSh(C)SSh(C):p * p : SSh(C) \stackrel{\leftarrow}{\to} SSh(C') : p^*

of simplicial (pre)sheaves. One would like this to extend to a Quillen adjunction that recalls the fact that it came from a geometric morphism by the fact that the left adjoint inverse image functor SSh(C)SSh(C)SSh(C') \to SSh(C) preserves finite homotopy limits.

In particular, if CC and CC' have the same underlying category but CC' the trivial coverage, then the geometric morphism in question is the inclusion of a reflective subcategory which typically induces a Bousfield localization of model categories that models the injection of a reflective (∞,1)-subcategory of ∞-stacks into \infty-presheaves. Here the morphism SPSh(C)SPSh(C)SPSh(C') \to SPSh(C) is \infty-stackification and should preserve finite homotopy limits.

The following result says that a strong version of this statement is true, at least for the preservation of homotopy pullbacks.

Theorem

Let p:Sh(C)Sh(C)p : Sh(C) \to Sh(C') be a geometric morphism of Grothendieck toposes. Let p *:Sh(C)Sh(C)p^* : Sh(C') \to Sh(C) be the corresponding inverse image functor and let sp *:SSh(C)SSh(C)s p^* : SSh(C') \to SSh(C) be its degreewise extension to functor of simplicial sheaf categories.

Regarded as a functor between the corresponding local injective model structures on simplicial sheaves on both sides

sp *:SSh(C) inj locSSh(C) inj loc s p^* : SSh(C')_{inj}^{loc} \to SSh(C)_{inj}^{loc}

this functor preserves homotopy pullbacks.

Proof

This appears as theorem 1.5 in

References

General

The classical references are

  • A. K. Bousfield and Dan Kan, Homotopy limits, completions, and localizations, Springer LNM 304, (see in particular chapter XI)

and

  • R. M. Vogt, Homotopy limits and colimits, Math. Z., 134 (1973) 11-52.

More recently one has:

An introduction is

A general overview via universal properties is in the

In

  • Wojciech Chacholski and Jerome Scherer, Homotopy theory of diagrams (arXiv)

is given a global definition of homotopy (co)limit as 4.1, p. 14, and it is discussed how to compute homotopy (co)limits concretely using local constructions. For instance the above statement on the computation of homotopy pullbacks is proposition 2.5, p. 15

A nice discussion of the expression of homotopy colimits in terms of coends is in

A collection of examples and exercises is in

See also

Homotopy limits for triangulated categories are studied in

Other references are

  • Philip Hirschhorn, Model categories and their localizations. Defines and studies (local) homotopy limits in model categories.

  • Dwyer, Hirschhorn, Kan, Smith, Homotopy limit functors in model categories and homotopical categories. Defines global homotopy limits in homotopical categories and computes them using local constructions.

  • Michael Shulman, Homotopy limits and colimits and enriched homotopy theory, math.CT/0610194. Constructs and compares local and global weighted homotopy limits in enriched homotopical categories. (a query on this paper is at nnForum here)

  • Nicola Gambino, Homotopy limits for 2-categories (pdf), published as: Mathematical Proceedings of the Cambridge Philosophical Society 145 (2008) 43-63.) Proves that homotopy limits in a 2-category with its natural model structure coincide with 2-categorical pseudo-limits, and hence give 2-limits.

  • Jacob Lurie, Higher Topos Theory. Lots of stuff about (,1)(\infty,1)-categories, including the computation of homotopy limits (section 4.2.4).

  • Andre Hirschowitz, Carlos Simpson, Descent pour les n-champs. Probably there is some good stuff in here about homotopy limits and limits in (,n)(\infty,n)-categories.

  • Beatriz Rodriguez Gonzalez, Realizable homotopy colimits (arXiv:1104.0646)

  • MathOverflow question: universal-problem-that-motivates-the-definition-of-homotopy-limits

Discussion in the context of the (infinity,1)-Grothendieck construction is in

In homotopy type theory

A formalization of some aspects of homotopy limits in terms of homotopy type theory is Coq-coded in

Revised on March 4, 2014 20:07:29 by Urs Schreiber (89.204.153.222)