corestriction

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

Then for any morphism $f \colon A\to B$, the *restriction* $f|_S \colon S \to B$ of $f$ onto $S$ is the precomposition $f|_S \coloneqq f \circ i_S$ of $f$ by $i_S$. (For a different representative of the subobject, $i_{\tilde{S}} \colon \tilde{S}\to A$ there is a unique isomorphism $b \colon S\to\tilde{S}$ such that $i_{\tilde{S}}\circ b = i_S$, hence $f_{\tilde{S}} = f\circ b$.)

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

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

For a monomorphism $i_T \colon T \hookrightarrow B$, the *corestriction* $f|^T \colon A \to T$ of $f$ onto $S$ is, if it exists, the unique morphism $f|^T \colon A \to S$ such that there is a decomposition $f = i_T \circ f|^T$.

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

Alternatively, one may consider the notion of corestriction to be the formal dual to the notion of a restriction: if $p^U \colon B\to U$ is an epimorphism, then the co-restriction of a morphism $f \colon A\to B$, in this dual sense, is the postcomposition $p^U \circ f \colon \colon A\to U$ of $f$ by $p^U$, 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*.

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

- Gabriella Böhm, Definition 3.1 and Remarks 3.2 in:
*Hopf algebroids*, in:*Handbook of Algebra*, Volume 6, 2009, Pages 173-235 (arXiv:0805.3806, doi:10.1016/S1570-7954(08)00205-2)

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.