The n-Category Café

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.