### An Exact Square from a Reedy Category

#### Posted by Emily Riehl

I first learned about exact squares from a blog post written by Mike Shulman on the $n$-Category Café.

Today I want to describe a family of exact squares, which are also homotopy exact, that I had not encountered previously. These make a brief appearance in a new preprint, A necessary and sufficient condition for induced model structures, by Kathryn Hess, Magdalena Kedziorek, Brooke Shipley, and myself.

**Proposition.** If $R$ is any (generalized) Reedy category, with $R^+ \subset R$ the direct subcategory of degree-increasing morphisms and $R^- \subset R$ the inverse subcategory of degree-decreasing morphisms, then the pullback square:
$\array{
iso(R) & \to & R^- \\
\downarrow & \swArrow id & \downarrow \\
R^+ & \to & R}$
is (homotopy) exact.

In summary, a Reedy category $(R,R^+,R^-)$ gives rise to a canonical exact square, which I’ll call the *Reedy exact square*.

## Exact squares and Kan extensions

Let’s recall the definition. Consider a square of functors inhabited by a natural transformation $\array{A & \overset{f}{\to} & B\\ ^u\downarrow & \swArrow\alpha & \downarrow^v\\ C& \underset{g}{\to} & D}$ For any category $M$, precomposition defines a square $\array{M^A & \overset{f^\ast}{\leftarrow} & M^B\\ ^{u^\ast}\uparrow & \swArrow \alpha^\ast & \uparrow^{v^\ast}\\ M^C& \underset{g^\ast}{\leftarrow} & M^D}$ Supposing there exist left Kan extensions $u_! \dashv u^\ast$ and $v_! \dashv v^\ast$ and right Kan extensions $f^\ast \dashv f_\ast$ and $g^\ast \dashv g_\ast$, the mates of $\alpha^*$ define canonical Beck-Chevalley transformations: $u_! f^\ast \Rightarrow g^\ast v_!\quad and \quad v^\ast g_\ast \Rightarrow f_\ast u^\ast.$ Note if either of the Beck-Chevalley transformations is an isomorphism, the other one is too by the (contravariant) correspondence between natural transformations between a pair of left adjoints and natural transformations between the corresponding right adjoints.

**Definition.** $\array{A & \overset{f}{\to} & B\\
^u\downarrow & \swArrow\alpha & \downarrow^v\\
C& \underset{g}{\to} & D}$ is an *exact square* if, for any $M$ admitting pointwise Kan extensions, the Beck-Chevalley transformations are isomorphisms.

Comma squares provide key examples, in which case the Beck-Chevalley isomorphisms recover the limit and colimit formulas for pointwise Kan extensions.

The notion of homotopy exact square is obtained by replacing $M$ by some sort of homotopical category, the adjoints by derived functors, and “isomorphism” by “equivalence.”

## The proof

In the preprint we give a direct proof that these Reedy squares are exact by computing the Kan extensions, but exactness follows more immediately from the following characterization theorem, stated using comma categories. The natural transformation $\alpha \colon v f \Rightarrow g u$ induces a functor $B \downarrow f \times_A u \downarrow C \to v \downarrow g$ over $C \times B$ defined on objects by sending a pair $b \to f(a), u(a) \to c$ to the composite morphism $v(b) \to v f(a) \to g u(a) \to g(c)$. Fixing a pair of objects $b$ in $B$ and $c$ in $C$, this pulls back to define a functor $b \downarrow f \times_A u \downarrow c \to vb \downarrow gc.$

**Theorem.**
A square
$\array{A & \overset{f}{\to} & B\\
^u\downarrow & \swArrow\alpha & \downarrow^v\\
C& \underset{g}{\to} & D}$
is exact if and only if each fiber of $b \downarrow f \times_A u \downarrow c \to v b \downarrow g c$ is non-empty and connected.

See the nLab for a proof. Similarly, the square is homotopy exact if and only if each fiber of this functor has a contractible nerve.

In the case of a Reedy square $\array{ iso(R) & \to & R^- \\ \downarrow & \swArrow id & \downarrow \\ R^+ & \to & R}$ these fibers are precisely the categories of Reedy factorizations of a fixed morphism. For an ordinary Reedy category $R$, Reedy factorizations are unique, and so the fibers are terminal categories. For a generalized Reedy category, Reedy factorizations are unique up to unique isomorphism, so the fibers are contractible groupoids.

## Reedy diagrams as bialgebras

For any category $M$, the objects in the lower right-hand square
$\array{
M^{iso(R)} & \leftarrow & M^{R^-} \\
\uparrow & \swArrow id & \uparrow \\
M^{R^+} & \leftarrow & M^R}$
are *Reedy diagrams* in $M$, and the functors restrict to various subdiagrams. Because the indexing categories all have the same objects, if $M$ is bicomplete each of these restriction functors is both monadic and comonadic. If we think of the $M^{R^-}$ as being comonadic over $M^{iso(R)}$ and $M^{R^+}$ as being monadic over $M^{iso(R)}$, then the Beck-Chevalley isomorphism exhibits $M^R$ as the category of bialgebras for the monad induced by the direct subcategory $R^+$ and the comonad induced by the inverse subcategory $R^-$.

There is a homotopy-theoretic interpretation of this, which I’ll describe in the case where $R$ is a strict Reedy category (so that $iso(R)=ob(R)$), though it works in the generalized context as well. If $M$ is a model category, then $M^{iso(R)}$ inherits a model structure, with everything defined objectwise. The Reedy model structure on $M^{R^-}$ coincides with the injective model structure, which has cofibrations and weak equivalences created by the restriction functor $M^{R^-} \to M^{iso(R)}$; we might say this model structure is “left-induced”. Dually, the Reedy model structure on $M^{R^+}$ coincides with the projective model structure, which has fibrations and weak equivalences created by $M^{R^+} \to M^{iso(R)}$; this is “right-induced”.

The Reedy model structure on $M^R$ then has two interpretations: it is right-induced along the monadic restriction functor $M^R \to M^{R^-}$ and it is left-induced along the comonadic restriction functor $M^R \to M^{R^+}$. The paper A necessary and sufficient condition for induced model structures describes a general technique for inducing model structures on categories of bialgebras, which reproduces the Reedy model structure in this special case.

## Re: An exact square from a Reedy category

Hahaha! I can’t believe I didn’t notice when writing my paper on Reedy categories that the connectedness of the category of Reedy factorizations is exactly the condition for that square to be exact. Nice.

Presumably this also generalizes to what I called “c-Reedy categories”?