related by the Dold-Kan correspondence
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 -category” or be a more powerful presentation of a bare -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 -limits” in the presented “enriched -category” or may be a more powerful tool for constructing plain -categorical limits (in particular if the enrichment is over sSet). Note that as yet, no fully general notion of “enriched -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.
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.”
One of the central theorems of the subject is that in good cases, the two give equivalent results; see below.
Let be a category with weak equivalences and let be a (small) diagram category. Make the functor category into a category with weak equivalences by taking the weak equivalences to be those natural transformations which are objectwise weak equivalences in .
The ordinary limit and colimit operations on -diagrams are (as described there) the right and left adjoints of the functor , or equivalently left and right Kan extension along the unique functor to the terminal category.
The (globally defined) homotopy limit and colimit are accordingly the right and left homotopy Kan extension along :
The homotopy limit of a functor is, if it exists, the image of under the right derived functor of the limit functor with respect to the given weak equivalences on and the objectwise weak equivalences on :
The homotopy colimit of a functor is, if it exists, the image of under the left derived functor of the colimit functor with respect to the given weak equivalences on and the objectwise weak equivalences on :
Alternative definitions can be formulated at the level of the homotopy category one defines a localized version of the diagonal functor and define the homotopy limits and colimits as the adjoints of (at least at the points where the adjoints are defined). Here are the morphisms of diagrams whose all components are in . 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 is a morphism of localizers (and in particular that 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 and .
In particular, if is equipped with the extra structure of a simplicial model category and is an (small) sSet-enriched category we may also hope to equip the enriched functor category 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 is combinatorial), limit and colimit constitute then two -enriched Quillen adjunctions
(All proofs and other technical details on this are at homotopy Kan extension.)
These present directly the corresponding adjoint (∞,1)-functors (as described there)
by precomposition with a cofibrant replacement functor (for the colimit) and a fibrant replacement functor (for the limit).
The local definition requires making precise the notion of a homotopy commutative cone on 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 on a diagram in an -enriched category is a natural transformation where the weight functor is not constant on the point, as for ordinary limits, but is given by .
The same idea works if we are enriched over a category that is not but is itself enriched over , such as topological spaces or spectra, since then any -category becomes a -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 , 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 of a functor to be a representing object for such homotopy cones, in the sense that we have a (weak) equivalence
of hom-objects (spaces or simplicial sets in the classical context; enriched hom-objects in the enriched context).
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 ) can be replaced by “good” (fibrant and/or cofibrant) objects in such a way that weak equivalences become homotopy equivalences. The derived functor of at the input object is then computed by applying the ordinary functor to a good replacement of .
It then turns out that the “good” (precisely, “fibrant”) replacement “builds in” precisely the right homotopies so that an ordinary cone on is the same as a homotopy-commutative cone on . Therefore, , which is the global homotopy-limit of , is a representing object for homotopy-commutative cones on , and thus is also a local homotopy-limit of . 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:
(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
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
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.
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 -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 of a category with weak equivalences is not sufficient information, the homotopy limit of a diagram in is encoded in the homotopy category 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 is a collection of all the homotopy categories of all the possible diagram categories , as runs over all small categories. See there for more details.
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
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
let for instance
be a cofibrant replacement for the projective model structure on functors, so that for any diagram the diagram is a projectively cofibrant diagram (see there for more details). Then the homotopy colimit is presented by the ordinary colimit on :
This is sometimes called the Quillen formula for computing homotopy colimits. Analogously with a fibrant replacement functor for the injective model structure, we have
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.
Then we also have the coend expression
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 itself).
we have a -adjunction
The crux now is that as varies, the left adjoint here is a left Quillen bifunctor
From the fact that this is a Quillen bifunctor and using the observation that for the trivial weight the weighted colimit reduces to an ordinary colimit, follows the above Bousfield-Kan-type formula for the homotopy colimit.
A general way of obtaining resolutions that compute homotopy (co)limits involves bar constructions. (…)
As a special case of enriched homotopy theory, we may consider model categories or homotopical categories that are enriched over a notion of -category as presentations for -categories. (Here we allow to also be of the form (n,r), with the obvious convention that and .) For example:
simplicial sets are models for -groupoids (-categories), so simplicial model categories are presentations for -categories. Of course, arbitrary model categories are also presentations for -categories, but simplicial model categories are easier to work with, and in particular easier to construct homotopy limits in.
If is a category enriched over -categories and we are considering it to be an -category (which happens to be strict at the bottom level), it is natural to define a “weak equivalence” in the underlying ordinary category of to be a morphism that is an -category-theoretic equivalence. We call this the natural or trivial homotopical structure on . In certain cases (such as 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 has a natural homotopical structure, locally-defined homotopy limits and -limits coincide. For this is trivial. For it is proven in (Gambino 10) (particularly section 6). For it is proven in (among other places) Lurie’s book, section 4.2.4. The case 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 enriched over -categories with a larger class of weak equivalences than just the -categorical equivalences. Then presents an -category (its “homotopy -category”) obtained by formally turing these weak equivalences into -categorical equivalences. Homotopy limits in with this homotopical structure should then present -limits in its homotopy -category. In the case this is also essentially in Lurie’s book; for other values of it may not be in the literature.
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 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 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 is equivalent to the category of graded -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.
For a combinatorial simplicial model category as above and for any simplicially enriched category there is the projective and the injective global model structure on functors on the enriched functor category .
In the projective model structure the fibrations and the local equivalences are objectwise those of
In the injective model structure the cofibrations and the local equivalences are objectwise those of .
and in the form
both themselves left Quillen bifunctors.
for the functor that sends everything to the identity on the singleton set. This is the tensor unit in the monoidal category .
With the above assumptions and ingredients, the homotopy colimit over is given either by
This is disucssed for instance in section 4 of (Gambino 10).
A cofibrant replacement of the terminal object in the projective global model structure on functors is the the fat simplex-functor that assigns to the nerve of opposite category of the undercategory of under
For instance prop 14.8.8 in
Notice that if takes values in cofibrant objects of , then it is itself cofibrant as an object of . In that case no further cofibrant replacement of is necessary and it therefore follows with the general formula and the above proposition that the homotopy colimit over is given by the formulas
This is famously the formula introduced and used by Bousfield and Kan (but there originally missing the necessary condition that be objectwise cofibrant). See Bousfield-Kan map.
In this simple case, we have the following simple observation:
For as above, the terminal functor is already cofibrant in .
For as above, a functor is cofibrant in if
it sends both morphisms and to cofibrations
it sends (and hence also and ) to cofibrant objects in .
Since a coend over a tensor product where the first factor in the integrand in the tensor unit is just an ordinary colimit over the remaining , it follows that if is of the form of the above observation, then the ordinary colimit over already computes the homotopy pushout:
The dual version of this statement (for homotopy limits and homotopy pullbacks) is discussed in more detail in the examples below.
Here we consider special cases of homotopy pullback in more detail.
satisfies * is fibrant for all ; * and either or is a fibration;
Conversely this means that on an arbitrary pullback diagram can be computed by finding a natural transformation whose component morphisms are weak equivalences and such that satisfies the above conditions.
One way to compute this using the above prescription by noticing that the generalized universal bundle provides a fibrant replacement of the pullback diagram in that we have
with all vertical morphisms weak equivalences and with the left bottom horizontal morphism a fibration.
More on that in the further examples below.
If is a pointed object, with point , then for a homotopy pullback of the form
the sequence is called a fibration sequence. The object is the homotopy kernel or homotopy fiber of . 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
As a special case of the above general example we get the following.
Let Grpd equipped with the canonical model structure. Write for a group regarded as a discrete monoidal groupoid (elements of are the objects of the groupoids and all morphisms are identities) write and for the corresponding one-object groupoid (single object, one morphism per element of ). Write for the terminal groupoid (one object, no nontrivial morphism). Notice that there is a unique functor . Then we have
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 , where is the action groupoid of acting on itself by multiplication from one side. So we have a weak equivalence of pullback diagrams
and the homotopy limit in question is weakly equivalent to the ordinary limit over the lower diagram. That is directly seen to be which we just write as :
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 is the datum of a toplogical space equipped with an action of . Then, whereas , see equivariant cohomology.
The above example generalizes straightforwardly to the case where the trivial inclusion is replaced by any inclusion of any subgroup of pretty much literally by replacing by throughout.
To see this, we again build a fibrant replacement of the pullback diagram. Following the constructions at generalized universal bundle consider first the groupoid given by the pullback diagram
As at generalized universal bundle one proves that the left vertical morphism is a fibration.
Now, notice (which was implicit in the above example) that since is a path object in a category of fibrant objects we have a section of . In the above pullback diagram this induces a morphism making the obvious diagram commute. Now, the latter morphism, being the pullback of an acyclic fibration is an acyclic fibration, so its right inverse is a weak equivalence. This way we obtain the morphism of pullback diagrams
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:
This example, too, is important at geometric function theory.
Every simplicial set is the homotopy colimit over its cells.
be the corresponding bisimplicial set which in degree is the the constant simplicial set on the set of -simplices.
For the standard homotopical structure on , the homotopy colimit over is equivalent to the origianal :
in the standard model structure on simplicial sets.
where is a cofibrant resolution in the Reedy model structure and in . But by the discussion at Reedy model structure – simplex category we have that
is always cofibrant in ;
a cofibrant resolution of the point in is given by .
It follows that the hocolim is given by
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.
This kind of argument has many immediate generalizations. For instance for the injective model structure on simplicial presheaves over any small category , 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 is the hocolim over its simplicial diagram of component presheaves.
For the following write for the fat simplex.
The fat simplex is Reedy cofibrant.
By the discussion at homotopy colimit, the fat simplex is cofibrant in the projective model structure on functors . By the general properties of Reedy model structures, the identity functor is a left Quillen functor, hence preserves cofibrant objects.
For a Reedy cofibrant object, the Bousfield-Kan map
is a weak equivalence in .
(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 and are indeed cofibrant in . Clearly the functor is objectwise a weak equivalence in , hence is a weak equivalence.
Let 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 can equivalently be computed after their restriction to
See (Dugger, example 18.2).
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.
Define a simplicial topological space by setting
and using the obvious face and degeneracy maps: face maps act by mapping components of the coproducts of 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 . The degeneracy maps similarly introduce identity morphisms.
Let be a category and Top a functor. Then the homotopy colimit of is equivalent to the geometric realization of simplicial topological spaces of :
This is an application of the bar-construction method.
See for instance (Dugger, part 1) for an exposition.
See higher homotopy van Kampen theorem for details.
the cotower category, a colimit of shape is called a sequential colimit. For a combinatorial model category it is easy to characterize the cofibrant objects in the projective model structure on functors : 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 .
Dually for sequential limits of a tower diagram.
Here we discuss some properties of homotopy limits and colimits in such model categories of simplicial presheaves.
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 preserves finite homotopy limits.
In particular, if and have the same underlying category but 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 -presheaves. Here the morphism is -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.
Regarded as a functor between the corresponding local injective model structures on simplicial sheaves on both sides
this functor preserves homotopy pullbacks.
This appears as theorem 1.5 in
The classical references are
More recently one has:
An introduction is
A general overview via universal properties is in the
Georges Maltsiniotis lectures, Sevilla (2008)
I, localizers, (pdf);
IV, summary on derivators (pdf)
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
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 Forum 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.
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