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category theory

# Contents

## Definition

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

$\array { X_1 \\ & \searrow^{f_1} \\ X_2 & \overset{f_2}\to & Y \\ & \nearrow_{f_3} \\ X_3 }$

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 $C$ whose sources (domains) are all $Y$, or equivalently, a family of objects in the under category $Y / C$:

$\array { & & X_1 \\ & {}^{f_1}\nearrow \\ Y & \overset{f_2}\to & X_2 \\ & {}_{f_3}\searrow \\ & & X_3 }$

Confusingly, this dual concept is called a source from $Y$ in $C$, 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 $Y$ in $C$ is the same as a sink to $Y$ in the opposite category $C^{\mathrm{op}}$.

### Structured sinks

If $U\colon C\to D$ is a functor, then a $U$-structured sink is a collection of objects $X_i\in C$ together with a sink in $D$ of the form $\{U(X_i) \to Y\}$. 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.