orthogonal subcategory problem



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

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



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


By definition of orthogonality, for every object XX in Σ \Sigma^\perp, ff induces an isomorphism of hom-sets

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

Since rir \dashv i, this means that for all XX in Σ \Sigma^\perp the map

hom Σ (r(f),X):hom Σ (r(B),X)hom Σ (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 hom Σ (r(f),)\hom_{\Sigma^\perp}(r(f), -) is a natural isomorphism between representables. By the Yoneda lemma, this means r(f)r(f) is an isomorphism.

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


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

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

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


Revised on January 29, 2017 10:29:33 by Todd Trimble (