nLab
corestriction

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Definition

The notion of co-restriction of a function, or more generally of a morphism, is either the formal dual of that of restriction, or something similar. For the latter, recall:

Given any subset SAS\subset A, we may consider the corresponding inclusion i S:SAi_S \colon S \hookrightarrow A as a function. More generally, for a subobject SAS\subset A (an equivalence class of monomorphisms in some ambient category) we may consider any representative monomorphism i S:SAi_S \colon S\hookrightarrow A.

Then for any morphism f:ABf \colon A\to B, the restriction f| S:SBf|_S \colon S \to B of ff onto SS is the precomposition f| Sfi Sf|_S \coloneqq f \circ i_S of ff by i Si_S. (For a different representative of the subobject, i S˜:S˜Ai_{\tilde{S}} \colon \tilde{S}\to A there is a unique isomorphism b:SS˜b \colon S\to\tilde{S} such that i S˜b=i Si_{\tilde{S}}\circ b = i_S, hence f S˜=fbf_{\tilde{S}} = f\circ b.)

This way, restriction of a morphism amounts to replacing its domain by one of its subobjects.

As factorization through the image

Similarly, the standard notion of corestriction (e.g. Böhm 08, Def. 3.1) of a morphism f:ABf \colon A \longrightarrow B is to instead replace the *co*domain BB by a subobject:

For a monomorphism i T:TBi_T \colon T \hookrightarrow B, the corestriction f| T:ATf|^T \colon A \to T of ff onto SS is, if it exists, the unique morphism f| T:ASf|^T \colon A \to S such that there is a decomposition f=i Tf| Tf = i_T \circ f|^T.

The corestriction exists if and only if TT contains the image of ff, that is i Im(f)i_{Im(f)} factors through i Ti_{T}. In particular, by the universal property of the image, the corestriction onto the image S=Im(f)S = Im(f) always exists and this is sometimes understood as the corestriction of ff, by default.

As the formal dual of restriction

Alternatively, one may consider the notion of corestriction to be the formal dual to the notion of a restriction: if p U:BUp^U \colon B\to U is an epimorphism, then the co-restriction of a morphism f:ABf \colon A\to B, in this dual sense, is the postcomposition p Uf::AUp^U \circ f \colon \colon A\to U of ff by p Up^U, which surely always exists.

This is the notion of corestriction considered e.g. in Andreotti 57, page 14, where the factorization through the image (above) is instead called coastriction.

References

The notion of corestriction is well known, while rarely made explicit in print. One may find it e.g. in

See also:

  • A. Andreotti, page 14 (paragraph 2-14) in: Généralités sur les catégories abéliennes (suite) Séminaire A. Grothendieck, Tome 1 (1957) Exposé no. 2 (numdam:SG_1957__1__A2_0)

Last revised on December 16, 2020 at 04:59:45. See the history of this page for a list of all contributions to it.