Paths and cylinders
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Stable homotopy theory
Where an ordinary category has a hom-set, an (∞,1)-category has an ∞-groupoid of morphisms between any two objects, a hom-space.
There are several ways to present an (∞,1)-category by an ordinary category equipped with some extra structure: for instance may be a category with weak equivalences or a model category or even a simplicial model category. In all of these presentations, given two objects , there is a way to construct a simplicial sets that presents the hom-∞-groupoid . This simplicial set – or rather its homotopy type – is called the derived hom space or homotopy function complex and denoted or similarly.
There are many ways to present an (∞,1)-category by category theoretic data, and for each of these there are corresponding tools for explicitly computing the derived hom spaces.
The most basic data is that of a category with weak equivalences. Here the derived hom spaces can be constructed in terms of zig-zags of morphisms by a process called simplicial localization. This we discuss below in For a category with weak equivalences.
Particularly useful extra structure on a category with weak equivalences that helps with computing the derived hom spaces is the structure of a model category. Using this one can construct simplicial resolutions of objects – called framings – that generalize cylinder objects and path objects, and then construct the derived hom spaces in terms of direct morphisms between these resolutions. This we discuss below in For a model category.
Still a bit more helpful structure on top of a bare model category is that of a simplicial model category. Here, after a choice of cofibrant and fibrant resolutions of opjects, the derived hom spaces are given already by the sSet-hom objects. This we discuss below in For a simplicial model category.
For a category with weak equivalences
Let be a category with weak equivalences.
Fix . For , define a category
whose objects are zig-zags of morphisms in of length
such that each morphism going to the left, , is a weak equivalence, an element in ;
morphisms between such objects are collections of weak equivalences for all such that all triangles and squares commute.
Write for the nerve of this category, a simplicial set.
The hammock localization of is the simplicially enriched category with objects those of and hom-objects given by the colimit over the length of these hammock hom-categories
The Kan fibrant replacement of this simplicial set is the derived hom-space between and of the -category modeled by .
For a model category
The derived hom spaces of a model category may always be computed in terms of simplicial resolutions with respect to the Reedy model structure . These resolutions are often called framings (Hovey). These constructions are originally due to (Dwyer-Hirschhorn-Kan).
Let be any model category.
There is an adjoint triple
Let be a model category.
For any object, a simplicial frame on is a factorization of into a weak equivalence followed by a fibration in the Reedy model structure .
A right framing in is a functor with a natural isomorphism such that is a simplicial frame on .
Dually for cosimplicial frames.
This appears as (Hovey, def. 5.2.7).
For cofibrant and fibrant, there are weak equivalences in
(where in the middle we have the diagonal of the bisimplicial set ).
This appears as (Hovey, prop. 5.4.7).
Either of these simplicial sets is a model for the derived hom-space .
Let be a model category, let be the full subcategory of spanned by the cosimplicial objects whose coface and codegeneracy operators are weak equivalences, and let be the full subcategory of spanned by the simplicial objects whose face and degeneracy operators are weak equivalences.
- is the right half of an adjoint homotopical equivalence of homotopical categories, and is the left half of an adjoint homotopical equivalence of homotopical categories.
- The functor admits a right derived functor.
- The induced functor is the derived hom-space functor.
For a simplicial model category
We describe here in more detail properties of hom-objects in a simplicial model category for the case that the domain objects are cofibrant and the codomain objects are fibrant.
The crucial axiom used for this is the axiom of an enriched model category which says that
the tensor operation
is a Quillen bifunctor;
or equivalently that for a cofibration and a fibration the induced morphism
is a fibration, which is acyclic if either or is.
First of all the first statement directly implies that for the initial object and any object, the simplicial set is the terminal simplicial set, because for any simplicial set
where we use that the tensor Quillen bifunctor is required to respect colimits and that the empty colimit is the initial object. (All equality signs here denote isomorphisms, to distinguish them from weak equivalences.)
Similarly one has for all that .
Using this, the second equivalent form of the enrichment axiom has as a special case the following statement.
We apply the enriched model category axiom to the cofibration and the fibration to obtain a fibration
The right hand is the pullback of the terminal simplicial set to itself, hence is itself the point. So we have a fibration and is a fibrant object in the standard model structure on simplicial sets, hence a Kan complex. .
In a simplicial model category , for cofibrant and a fibration, the morphism of simplicial sets is a Kan fibration that is a weak homotopy equivalence if is acyclic.
Dually, for a cofibration and fibrant, the morphism is a cofibration of simplicial sets.
This is as before. Explicity, consider the first case, the second one is the formal dual of that:
We enter the enrichment axiom with the morphisms and and find for the required pullback that
and hence that is, indeed, a fibration, which is acyclic if is.
Let be a simplicial model category.
Then for a cofibant object and
a weak equivalence between fibrant objects, the enriched hom-functor
is a weak homotopy equivalence of Kan complexes.
Similarly, for a fibrant object and a weak equivalence between cofibrant objects, the morphism
is a weak homotopy equivalence of Kan complexes.
The second case is formally dual to the first, so we restrict attention to the first one.
By the above, the axioms of an enriched model category ensure that the above statement is true when is in addition a fibration. So we reduce the situation to that case.
This is possible because both and are assumed to be fibrant. This allows to apply the factorization lemma that is described in great detail at category of fibrant objects. By this lemma, for every morphism between fibrant objects there is a commutative diagram
Since is assumed a weak equivalence it follows by 2-out-of-3 that is also a weak equivalence.
Therefore by the above properties of simpliciall enriched categories we obtain a span of acyclic fibrations of Kan complexes
By the Whitehead theorem every weak equivalence of Kan complexes is a homotopy equivalence, hence there is a weak equivalence
that is homotopic to our . Therefore this is also a weak equivalence.
Let be a model category. We discuss how its simplicial function complexes from prop. 2 are related to the simplicial localization from def. 1 and def. 2.
Suppose now that is a cofibrant replacement functor and a fibrant replacement functor, a cosimplicial resolution functor and a simplicial resolution functor in the model category .
There are natural weak equivalences between the following equivalent realizations of this SSet hom-object:
The top row weak equivalences are those of prop. 2
In a category of fibrant objects
There is also an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a category of fibrant objects. This is described in (Cisinksi 10) and (Nikolaus-Schreiber-Stevenson 12, section 3.6.2).
Hom-spaces of equivalences
For a simplicial model category and an object, the delooping of the automorphism ∞-group
has the homotopy type of the component on of the nerve of the subcategory of weak equivalences:
The equivalence is given by a finite sequence of zig-zags and is natural with respect to sSet-enriched functors of simplicial model categories that preserve weak equivalences and send a fibrant cofibrant model for again to a fibrant cofibrant object.
This is Dwyer-Kan 84, 2.3, 2.4.
For a model category, the simplicial set is a model for the core of the (∞,1)-category determined by .
That core, like every ∞-groupoid is equivalent to the disjoint union over its connected components of the deloopings of the automorphism -groups of any representatives in each connected component.
For some original references by William Dwyer and Dan Kan see simplicial localization. For instance
- William Dwyer, Dan Kan, A classication theorem for diagrams of simplicial sets, Topology 23 (1984), 139-155.
The theory of framings is due to
and in parallel section 5 of
and in sections 16, 17 of
A useful quick review of the interrelation of the various constructions of derived hom spaces is page 14, 15 of
Discussion of derived hom spaces for categories of fibrant objects is in
and section 3.6.2 of