nLab
sink

Contents

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:

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:

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

  • Any cocone under a diagram is a sink; indeed a cocone is precisely a sink indexed by the objects of the domain of the diagram together with a commutativity condition for the arrows in the diagram.

Revised on August 7, 2017 16:23:21 by Mike Shulman (76.167.222.204)