nLab corestriction

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 $S\subset A$, 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.

As factorization through the image

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 $T$ is, if it exists, the unique morphism $f|^T \colon A \to T$ 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.

As the formal dual of restriction

Alternatively, one may consider a notion of corestriction which is 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.

References

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)