on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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 $A_x \amalg_{L_x A} L_x B \to B_x$ is a monomorphism.
Every span of codegeneracy maps in $R_-$ has an absolute pushout in $R_-$.
Both the following conditions hold:
For every monomorphism $A\hookrightarrow B$ of presheaves on $R$, every nondegenerate element of $A$ remains nondegenerate in $B$.
Every element of a presheaf $R$ 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 $R$ is elegant, then every codegeneracy map (i.e. morphism in $R_-$) is a split epimorphism.
Let $f:x\to y$ be a codegeneracy map; then the span $y \xleftarrow{f} x \xrightarrow{f} y$ has an absolute pushout, consisting of say $g:y\to z$ and $h:y\to z$ with $g f = h f$. This absolute pushout is preserved by $R(z,-)$, so $1_z\in R(z,z)$ must be the image under $g$ or $h$ of some map $s:z\to y$; WLOG say it is $g$, so we hav $1_z = g s$. Now we have $s h:y\to y$ and $1_y$ such that $g s h = h = h 1_y$, and our absolute pushout is preserved by $R(y,-)$, so there must be a zigzag of elements in $R(y,z)$ relating $s h$ to $1_y$. At one end of that zigzag, there must be a $t:y\to x$ such that $f t = 1_y$; hence $f$ is split epi.
Let $R$ be elegant and $f:x\to y$ a codegeneracy in $R$. Let $M$ be any category, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$. Then the following naturality square is a pullback:
This depends only on the fact that $f$ is split epi in $R$. Let $s:y\to x$ be a section of it, and let $P$ be the pullback of $B_f$ and $\mu_x$, with projections $p:P\to A_x$ and $q:P\to B_y$ with $\mu_x p = B_f q$, and an induced map $\phi:A_y \to P$ such that $p \phi = A_f$ and $q\phi = \mu_y$.
We claim that $A_s p : P \to A_y$ is an inverse of $\phi$, making it an isomorphism. On the one hand we have $A_s p \phi = A_s A_f = A_{f s} = 1$. On the other, to show that $\phi A_s p = 1$ it suffices to show that $p \phi A_s p = p$ and $q \phi A_s p = q$. For the first, since $\mu_x$ is monic, it suffices to show $\mu_x p \phi A_s p = \mu_x p$, and for that we have
And for the second, we have
Let $R$ be elegant, $M$ a category with pullback-stable colimits, and $\mu:A\to B$ a monomorphism in $M^{R^{\mathrm{op}}}$. Then for any object $x\in R$, the following square is a pullback, where $L_x$ denotes the Reedy latching object at $x$:
The map $L_x B \to B_x$ is by definition the colimit in $M/B_x$ of a diagram whose objects are morphisms of the form $B_f : B_y \to B_x$, for $f$ a codegeneracy. By the Lemma 2, each of these pulls back along $\mu_x$ to $A_f : A_y \to A_x$, forming the corresponding diagram whose colimit is $L_x A \to A_x$, and by assumption the pullback preserves the colimit.
Let $R$ be elegant and let $M$ be an infinitary-coherent category. Then for any $x\in R$ and $A\in M^{R^{\mathrm{op}}}$, the map $L_x A \to A_x$ is a monomorphism.
We use the terminology from the page κ-ary exact category. Consider the sink with target $A_x$ consisting of all morphisms $A_f : A_y \to A_x$ indexed by nonidentity codegeneracies $f$ with domain $x$. By assumption, for any two such $f:x\to y$ and $f':x\to y'$ there is an absolute pushout $g:y\to z$ and $g':y'\to z$. By absoluteness, $A_z$ is the pullback $A_y \times_{A_x} A_y$. Thus, the images of these absolute pushouts form the kernel of this sink.
Now $L_x A$ is the colimit of the diagram whose objects are $A_y$ indexed by such $f:x\to y$ and whose morphisms are $A_g: A_{y'} \to A_{y}$ for $g:y\to y'$ a codegeneracy with $g f = f'$. In this case, by the universal property of pullback, we have a unique map from $A_{y'}$ to $A_z$, where $z$ is the absolute pushout of $f$ and $f'$. Thus, a cocone under the above kernel is also a cocone under this diagram, and the converse is easy to see. Hence, $L_x A$ 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 $L_x A \to A_x$ is monic.
If $R$ is elegant and $M$ is a Grothendieck topos, then for any $x\in R$ and monomorphism $\mu:A\to B$ in $M^{R^{\mathrm{op}}}$, the induced map $L_x B \sqcup_{L_x A} A_x \to B_x$ is monic.
Since Grothendieck toposes are infinitary-coherent, by Lemma 4 $L_x B\to B_x$ is monic. By assumption $A_x \to B_x$ is monic. And since Grothendieck toposes have pullback-stable colimits, by Lemma 3 the square
is a pullback. In other words, $L_x A$ is the intersection of the subobjects $L_x B$ and $A_x$ of $B_x$. 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.
If $R$ is elegant and $M$ is a Cisinski model category, then the Reedy model structure on $M^{R^{\mathrm{op}}}$ coincides with the injective model structure.
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 $M = SSet$. Thus, for instance, every simplicial presheaf on an elegant Reedy category is Reedy cofibrant.
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 $(el X)^{op}$ is again an elegant Reedy category. This is fairly easy to see from the fact that $Set^{el X}$ is equivalent to the slice category $Set^{R^{op}}/X$.
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 $R$ is elegant then $R^{op}$ will not generally be elegant.
Elegant Reedy categories are useful to model homotopy type theory.
Mike Shulman, The univalence axiom for elegant Reedy presheaves (arXiv:1307.6248)
Mike Shulman, Univalence for inverse diagrams and homotopy canonicity arXiv:1203.3253
Benno van den Berg and Ieke Moerdijk, W-types in homotopy type theory, PDF