related by the Dold-Kan correspondence
An elegant Reedy category is a Reedy category such that the following equivalent conditions hold
Both the following conditions hold:
For every monomorphism of presheaves on , every nondegenerate element of remains nondegenerate in .
Every element of a presheaf is a degeneracy of some nondegenerate element in a unique way.
The principal theorem about elegant Reedy categories is that the Reedy model structure on presheaves (i.e. contravariant diagrams) over an elegant Reedy category coincides with the injective model structure. This is not true for presheaves valued in any model category, only well-behaved ones. We clarify the necessary conditions by building up to this theorem in stages, adding hypotheses on the codomain of the presheaves as necessary.
If is elegant, then every codegeneracy map (i.e. morphism in ) is a split epimorphism.
Let be a codegeneracy map; then the span has an absolute pushout, consisting of say and with . This absolute pushout is preserved by , so must be the image under or of some map ; WLOG say it is , so we hav . Now we have and such that , and our absolute pushout is preserved by , so there must be a zigzag of elements in relating to . At one end of that zigzag, there must be a such that ; hence is split epi.
Let be elegant and a codegeneracy in . Let be any category, and a monomorphism in . Then the following naturality square is a pullback:
This depends only on the fact that is split epi in . Let be a section of it, and let be the pullback of and , with projections and with , and an induced map such that and .
We claim that is an inverse of , making it an isomorphism. On the one hand we have . On the other, to show that it suffices to show that and . For the first, since is monic, it suffices to show , and for that we have
And for the second, we have
Let be elegant, a category with pullback-stable colimits, and a monomorphism in . Then for any object , the following square is a pullback, where denotes the Reedy latching object at :
The map is by definition the colimit in of a diagram whose objects are morphisms of the form , for a codegeneracy. By the Lemma 2, each of these pulls back along to , forming the corresponding diagram whose colimit is , and by assumption the pullback preserves the colimit.
Let be elegant and let be an infinitary-coherent category. Then for any and , the map is a monomorphism.
We use the terminology from the page κ-ary exact category. Consider the sink with target consisting of all morphisms indexed by nonidentity codegeneracies with domain . By assumption, for any two such and there is an absolute pushout and . By absoluteness, is the pullback . Thus, the images of these absolute pushouts form the kernel of this sink.
Now is the colimit of the diagram whose objects are indexed by such and whose morphisms are for a codegeneracy with . In this case, by the universal property of pullback, we have a unique map from to , where is the absolute pushout of and . Thus, a cocone under the above kernel is also a cocone under this diagram, and the converse is easy to see. Hence, is the quotient of the above kernel.
However, in any infinitary-regular category, the quotient of the kernel of a sink is exactly the extremal-epic / monic factorization of that sink. Therefore, the induced map is monic.
If is elegant and is a Grothendieck topos, then for any and monomorphism in , the induced map is monic.
is a pullback. In other words, is the intersection of the subobjects and of . But in any coherent category, the pushout of two subobjects over their intersection is their union, and hence in particular a subobject of their common codomain.
By definition, they have the same weak equivalences, so it suffices to show that their classes of cofibrations coincide. But every Reedy cofibration in any Reedy model structure is an injective (i.e. objectwise) cofibration, and the converse is Theorem 1.
The most common application is when . Thus, for instance, every simplicial presheaf on an elegant Reedy category is Reedy cofibrant.
The simplex category is an elegant Reedy category.
Joyal’s disk categories are elegant Reedy categories.
Every direct category is a Reedy category with no degeneracies, hence trivially an elegant one.
If is any presheaf on an elegant Reedy category , then the opposite of its category of elements is again an elegant Reedy category. This is fairly easy to see from the fact that is equivalent to the slice category .
Every EZ-Reedy category? is elegant.
Note that unlike the notion of Reedy category, the notion of elegant Reedy category is not self-dual: if is elegant then will not generally be elegant.
Elegant Reedy categories are useful to model homotopy type theory.
Benno van den Berg and Ieke Moerdijk, W-types in homotopy type theory, PDF