Contents

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 $r\left(f\right):r\left(a\right)\to r\left(b\right)$ 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

${\mathrm{hom}}_{C}\left(f,i\left(X\right)\right):{\mathrm{hom}}_{C}\left(B,i\left(X\right)\right)\stackrel{\sim }{\to }{\mathrm{hom}}_{C}\left(A,i\left(X\right)\right)$\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

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

is an isomorphism, so that ${\mathrm{hom}}_{{\Sigma }^{\perp }}\left(r\left(f\right),-\right)$ is a natural isomorphism between representables. By the Yoneda lemma, this means $r\left(f\right)$ 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 ${\mathrm{hom}}_{C}\left(\sigma ,k\right)$ 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 $\mathrm{sat}\left(\Sigma \right)$. An easy argument due to Gabriel-Zisman is that if $C$ is finitely cocomplete, then $\left(C,\mathrm{sat}\left(\Sigma \right)\right)$ satisfies the axioms for a calculus of fractions, so that the localization $C\left[\left(\mathrm{sat}\left(\Sigma \right){\right)}^{-1}\right]$ can be constructed. An easy argument then establishes the following sharpened lemma:

Lemma

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

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

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

References

Revised on June 24, 2010 04:21:06 by Urs Schreiber (134.100.32.31)