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.

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

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

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

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

is an isomorphism, so that ${\mathrm{hom}}_{{\Sigma }^{\perp }}(r(f),-)$ 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 ${\mathrm{hom}}_C(\sigma ,k)$ 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 $\mathrm{sat}(\Sigma )$. An easy argument due to Gabriel-Zisman is that if $C$ is finitely cocomplete, then $(C,\mathrm{sat}(\Sigma ))$ satisfies the axioms for a calculus of fractions, so that the localization $C[(\mathrm{sat}(\Sigma ){)}^{-1}]$ 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

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

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

References

• Gabriel-Zisman

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