nLab
coprojection

Contents

Context

Limits and colimits

Category theory

Contents

Definition

Generally in category theory a coprojection is one of the canonical morphisms p ip_i into a (categorical) coproduct:

p i:X i jX j. p_i \colon X_i \to \coprod_j X_j \,.

or, more generally into a colimit

p i:X ilim jX j. p_i \colon X_i \to \underset{\rightarrow_j}{\lim} X_j \,.

Hence a coprojection is a component of a colimiting cocone under a given diagram.

Coprojections are also sometimes called coproduct injections or inclusions, though in general they are not monomorphisms (see below).

Properties

Monicity

In general, the coprojections of a coproduct need not be monomorphisms. However, they are in certain common situations, such as:

It is easy to find examples of categories in which the coprojections of coproducts are not monic, e.g. the projection ×AA\emptyset \times A\to A in SetSet is not epic if AA is nonempty, so when regarded as a coprojection in Set opSet^{op} it is not monic. It is somewhat trickier to find examples of closed monoidal categories with this property, but Chu spaces give an example; see this MO question.

Last revised on June 8, 2015 at 13:50:39. See the history of this page for a list of all contributions to it.