Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((fi)i∈I, A), sometimes denoted by (fi,A)I or (Aifi A)I consisting of an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai)i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.

Untitled

In Set, a sink (Aifi A)I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪i∈I fi[Ai]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s