nLab
elegant Reedy category

Definition

Definition

An elegant Reedy category is a Reedy category $R$ such that the following equivalent conditions hold

• For every monomorphism $A\hookrightarrow B$ of presheaves on $R$ and every $x\in R$, the induced map $A_x{⨿}_{L_{x}A}{L}_{x}B\to B_x$ is a monomorphism.

• Every span of codegeneracy maps in $R_{-}$ has an absolute pushout in $R_{-}$.

• Both the following conditions hold:

  1. For every monomorphism $A\hookrightarrow B$ of presheaves on $R$, every nondegenerate element of $A$ remains nondegenerate in $B$.

  2. Every element of a presheaf $R$ is a degeneracy of some nondegenerate element in a unique way.

In particular, if $R$ is elegant, then every codegeneracy map is a split epimorphism.

Properties

Model structures

Theorem

If $R$ is an elegant Reedy category and $M$ is a model category in which the cofibrations are exactly the monomorphisms, then the Reedy model structure and the injective model structure on $M^{R^{\mathrm{op}}}$ coincide.

In particular, this implies that every $M$-valued presheaf on an elegant Reedy category is Reedy cofibrant.

Examples

• The simplex category $\Delta$ is an elegant Reedy category.

• Joyal’s disk categories ${\Theta }_n$ are elegant Reedy categories.

• Every direct category is a Reedy category with no degeneracies, hence trivially an elegant one.

• If $X$ is any presheaf on an elegant Reedy category $R$, then the opposite of its category of elements $(\mathrm{el}X{)}^{\mathrm{op}}$ is again an elegant Reedy category. This is fairly easy to see from the fact that ${\mathrm{Set}}^{\mathrm{el}X}$ is equivalent to the slice category ${\mathrm{Set}}^{R^{\mathrm{op}}}/X$.

Note that unlike the notion of Reedy category, the notion of elegant Reedy category is not self-dual: if $R$ is elegant then $R^{\mathrm{op}}$ will not generally be elegant.

Related concepts

• generalized Reedy category

• Reedy model structure

References

• Julie Bergner and Charles Rezk, Reedy categories and the $\Theta$-construction, arXiv:1110.1066