nLab sink

Definition

Given an object $Y$ of a category $C$, a sink or wide cospan 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’ or wide span for a source in the sense dual to a sink or wide cospan, 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

• 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.

• A source with two morphisms is a span, a sink with two morphisms is a cospan.

• A terminal source is a dependent product, while an initial sink is a dependent sum.

Related pages

• final lift, topological concrete category, solid functor

• factorization structure for sinks

• wide pullback, wide pushout

• dependent product type, dependent sum type