nLab
sink

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Definition

Given an object YY of a category CC, a sink 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’ for a source in the sense dual to a sink, 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 May 19, 2021 at 00:07:01. See the history of this page for a list of all contributions to it.