nLab
orthogonal subcategory problem

Contents

Contents

Idea

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.

Here, an object XX is said to be orthogonal to a map f:ABf:A\to B if every morphism AXA\to X factors uniquely through ff. For example, in a category of presheaves, X:C opSetX:C^{op}\to Set is orthogonal to the canonical map colim iC(,a i)C(,colim ia i)colim_i C(-,a_i)\to C(-,\colim_i a_i) if it preserves the limit lim ia i\lim_i a_i in C opC^{op}.

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).

Properties

Lemma

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.

Proof

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:

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
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…

Theorem in the setting of locally presentable categories

Theorem

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

References

Last revised on August 19, 2019 at 16:16:35. See the history of this page for a list of all contributions to it.