nLab orthogonal subcategory problem

Idea

The orthogonal subcategory problem for a class of morphisms $\Sigma$ in a category $C$ asks whether the full subcategory $\Sigma^\perp$ of objects orthogonal to $\Sigma$ is a reflective subcategory.

Here, an object $X$ is said to be orthogonal to a map $f:A\to B$ if every morphism $A\to X$ factors uniquely through $f$. For example, in a category of presheaves, $X:C^{op}\to Set$ is orthogonal to the canonical map $colim_i C(-,a_i)\to C(-,\colim_i a_i)$ if it preserves the limit $\lim_i a_i$ in $C^{op}$.

This problem is related to the problem of localization. Suppose $\Sigma^\perp$ is indeed a reflective subcategory; let $r: C \to \Sigma^\perp$ be the reflector (the left adjoint to the inclusion $i: \Sigma^\perp \to C$).

Properties

Lemma

If $f: a \to b$ belongs to $\Sigma$, then $r(f): r(a) \to r(b)$ is an isomorphism in $\Sigma^\perp$.

Proof

By definition of orthogonality, for every object $X$ in $\Sigma^\perp$, $f$ induces an isomorphism of hom-sets

$$\hom_C(f, i(X)): \hom_C(B, i(X)) \stackrel{\sim}{\to} \hom_C(A, i(X))$$

Since $r \dashv i$, this means that for all $X$ in $\Sigma^\perp$ the map

$$\hom_{\Sigma^\perp}(r(f), X): \hom_{\Sigma^\perp}(r(B), X) \to \hom_{\Sigma^\perp}(r(A), X)$$

is an isomorphism, so that $\hom_{\Sigma^\perp}(r(f), -)$ is a natural isomorphism between representables. By the Yoneda lemma, this means $r(f)$ is an isomorphism.

This lemma can be sharpened. First, given a category $C$, there is a Galois connection between classes of morphisms $\Sigma$ and classes of objects $K$, induced by the predicate that $\hom_C(\sigma, k)$ is a bijection for all $\sigma \in \Sigma$ and all $k \in K$. The induced monad on the collection of morphism classes $\Sigma$ (partially ordered by inclusion) may be called “saturation”, denoted $sat(\Sigma)$. An easy argument due to Gabriel-Zisman is that if $C$ is finitely cocomplete, then $(C, sat(\Sigma))$ satisfies the axioms for a calculus of fractions, so that the localization $C[(sat(\Sigma))^{-1}]$ can be constructed. An easy argument then establishes the following sharpened lemma:

Lemma

If $\Sigma^\perp \hookrightarrow C$ is reflective, then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical equivalence

$$C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp$$

Lemma

If $\Sigma^\perp \hookrightarrow C$ is reflective, then the reflection $r: C \to \Sigma^\perp$ exhibits a canonical equivalence

$$C[(sat(\Sigma))^{-1}] \simeq \Sigma^\perp$$

To be connected with things like small object argument, Bousfield localization, and others…

Theorem in the setting of locally presentable categories

Theorem

If $\Sigma$ is a set of morphisms in a locally presentable category $\mathcal{A}$ then the orthogonality subcategory $\Sigma^\top$ is reflective in $\mathcal{A}$.

References

• Gabriel-Zisman

• Peter Freyd, Max Kelly, Categories of continuous functors I, Jour. Pure Appl. Alg. 2 (1972), 169-191. (web)

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994). But see the errata.