related by the Dold-Kan correspondence
A Reedy model structure is a global model structure on functors:
As opposed to the projective and injective model structure on functors this does not require any further structure on , but instead makes a strong assumption on .
If all three exist, then, in a precise sense, the Reedy model structure sits in between the injective and the projective model structure. As such, it has the advantage that both the cofibrations as well as the fibrations can be fairly explicitly described and detected in terms of cofibrations and fibrations in .
The basic idea is as follows. Given a diagram and an object , define its latching object to be
where the colimit is over the full subcategory of containing all objects except the identity . Dually, define its matching object to be
where the limit is over the full subcategory of containing all objects except . There are evident canonical, and natural, morphisms
Note that is the initial object and is the terminal object, since there are no objects of degree .
In the case , the latching object can be thought of as the object of degenerate -simplices sitting inside the object of all -simplices. When is an ordinal, then and , and dually for .
We now define a morphism in to be a cofibration or trivial cofibration if for all , the map
is a cofibration or trivial cofibration in , respectively, and to be a fibration or trivial fibration if for all , the map
is a fibration or trivial fibration in , respectively. Define to be a weak equivalence if each is a weak equivalence in .
One then verifies that this defines a model structure; the details can be found in (for instance) Hovey and Hirschhorn’s books. In particular, to factor a morphism in either of the two necessary ways, we construct the factorization inductively on , by factoring the induced morphism
in the appropriate way in .
For a suitable enriching category, there is a refinement of the notion of Reedy category to a notion of -enriched Reedy category such that if is a -enriched model category – in particular when it is a simplicial model category for SSet – the enriched functor category is itself a -enriched model category (see Angeltveit).
In the case that we do have extra assumptions on the codomain in that
the Reedy model structure, having the same weak equivalences as the global model structure on functors, presents similarly the (∞,1)-category of (∞,1)-functors , from into the (∞,1)-category presented by .
Any Reedy cofibration or fibration is, in particular, an objectwise one, but the converse does not generally hold.
An object is Reedy cofibrant if and only if each map is a cofibration in . In particular, this implies that each is cofibrant in .
In general the Reedy structure will not be the same as either, but will be a kind of mixture of both. If then the Reedy model structure coincides with the projective model structure, if it coincides with the injective model structure.
For a detailed comparison of Reedy and global projective/injective model structures see around example A.2.9.22 in HTT.
In addition to its existing for all , another advantage of the Reedy structure is the explicit nature of its cofibrations, fibrations, and factorizations.
If admits more than one structure of Reedy category, then will have more than one Reedy model structure. For instance, if is the walking arrow, then we can regard it as either the ordinal or its opposite , resulting in two different Reedy model structures on .
For a general Reedy category , the diagonal functor need not be either a right or a left Quillen functor (although, of course, it has left and right adjoints given by colimits and limits over ). One can, however, characterize those Reedy categories for which one or the other is the case, and in this case one can construct homotopy limits and colimits using the derived functors of these Quillen adjunctions.
If in addition is a -enriched model category for some symmetric monoidal model category , then so is
This appears as (Barwick, lemma 4.2, corollary 4.3).
The identity functors provide left Quillen equivalences
from the projective model structure on functors to the injective one.
See also HTT, remark A.2.9.23
The simplest nontrivial example is obtained for
the interval category.
We take the degree on the objects to be as indicated. Then and contains only the identity morphisms.
For a functor, i.e. a morphism in , we find
the latching object ;
the latching object ;
the matching object
the matching object
From this we find that for a natural transformation
it is a Reedy cofibration in if
it is a Reedy fibration in if
is a fibration
the universal morphism
is a fibration.
Notice that since fibrations are preserved by pullbacks and under composition with themselves, it follows that also is a fibration.
The cofibrant objects in are those arrows in for which and are cofibrant;
The fibrant objects in are those arrows in that are fibrations between fibrant objects in .
So in accord with the proposition above one finds that this Reedy model structure on coincides with the injective global model structure on functors on .
Let be the natural numbers regarded as a poset using the greater-than relation.
With the degree as indicated, this is a Reedy category with and containing only identity morphisms.
Now the functor category is the category of towers of morphisms in .
The analysis of the Reedy model structure on this involves just a repetition of the steps involved in the analysis of the arrow category in the above example. One finds:
a natural transformation is a fibration precisely if
the component is a fibration
all universal morphisms are fibrations.
the fibrant objects are the towers of fibrations on fibrant objects in .
A detailed discussion of the model structure on towers is for instance in (GoerssJardine, chapter 6)
By duality it follows that analogously there is a model structure on co-towers
in a model category , whose fibrations and weak equivalences are the degreewise ones, and whose cofibrations are those transformations that are a cofibration in degree 0 and where the canonical pushout-morphisms in each square are cofibrations.
a map is in precisely if it is an injection;
a map is in precisely if it is a surjection.
Dually, for the opposite category
a morphism is in precisely if the map underlying it is an injection;
a morphism is in precisely if the map underlying it is a surjection.
We record some useful facts.
Let be a model category
For a simplicial object and ,
More details on this are currently at generalized Reedy model structure.
If for instance in all monomorphisms are cofibrations, then every object in is cofibrant.
Every simplicial set regarded as a simplicial diagram
is Reedy cofibrant in .
The latching object of at is
The canonical map
identifies along in as a bunch of degenrate -cells. In total, is identified as the set of all degenerate -cells of .
Therefore is clearly an injection of sets, hence a monomorphism of (constant) simplicial sets. Monomorphisms are the cofibrations in .
The canonical cosimplicial simplicial set
is Reedy cofibrant in .
The latching object at is
This is . The inclusion is a monomorphism, hence a cofibration in (in fact these are the generating cofibrations).
where is a cofibrant resolution of the point in . Using the above observation, we can take this to be since this is cofibrant by the above observation and clearly is objectwise a weak equivalence in .
Therefore the hocolim is (up to equivalence) represented by the simplicial 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.
Let be a model category.
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.
The following proposition should be read as a warning that an obvious idea about simplicial enrichment of Reedy model structures over the simplex category does not work.
However, this does not in general harmonize with the Reedy model structure to make a simplicial model category.
the pushout-product is a cofibration in ;
and it is an acyclic cofibration if is;
it is not necessarily acyclic if is.
However, there are left Bousfield localizations of for which
the above -enrichment does constitute an -enriched model category;
the result model structure is Quillen equivalent to itself.
This is in fact a useful technique for replacing by a Quillen equivalent and -enriched model structure. More discussion of this point is at simplicial model category in the section Simplicial Quillen equivalent models.
The original text is
A review of Reedy model structures is in section A.2.9 of
Discussion of functoriality of Reedy model structures is in
The discussion of enriched Reedy model structures is in
The main statement is theorem 4.7 there.
The Reedy model structure on towers is discussed for instance in section 6 of
The Reedy model structure on categories of simplicial objects is discussed in more detail for instance in