nLab sink

Redirected from "wide cospan".
Contents

Contents

Definition

Given an object YY of a category CC, a sink or wide cospan to YY in CC is a family of morphisms of CC whose targets (codomains) are all YY, or equivalently, a family of objects in the over category C/YC / Y:

X 1 f 1 X 2 f 2 Y f 3 X 3 \array { X_1 \\ & \searrow^{f_1} \\ X_2 & \overset{f_2}\to & Y \\ & \nearrow_{f_3} \\ X_3 }

We do not, in general, require that this family be small; if it is so we would call it a “small sink”.

The dual concept is a family of morphisms of CC whose sources (domains) are all YY, or equivalently, a family of objects in the under category Y/CY / C:

X 1 f 1 Y f 2 X 2 f 3 X 3 \array { & & X_1 \\ & {}^{f_1}\nearrow \\ Y & \overset{f_2}\to & X_2 \\ & {}_{f_3}\searrow \\ & & X_3 }

Confusingly, this dual concept is called a source from YY in CC, even though the term ‘source’ has another meaning, one which we just used in the definition! One can of course say ‘domain’ instead of ‘source’ for this other meaning, but that leads to other confusions. Or one can say ‘cosink’ or wide span for a source in the sense dual to a sink or wide cospan, since a source from YY in CC is the same as a sink to YY in the opposite category C opC^{\mathrm{op}}.

Structured sinks

If U:CDU\colon C\to D is a functor, then a UU-structured sink is a collection of objects X iCX_i\in C together with a sink in DD of the form {U(X i)Y}\{U(X_i) \to Y\}. This notion figures in the definition of a final lift.

Examples

Last revised on January 1, 2024 at 02:34:22. See the history of this page for a list of all contributions to it.