nLab coprojection

Definition

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

$$p_i \colon X_i \to \coprod_j X_j \,.$$

or, more generally into a colimit

$$p_i \colon X_i \to \underset{\rightarrow_j}{\lim} X_j \,.$$

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:

• In a distributive category; see there for a proof.

• In a category with zero morphisms, since then they are split monomorphisms.

It is easy to find examples of categories in which the coprojections of coproducts are not monic, e.g. the projection $\emptyset \times A\to A$ in $Set$ is not epic if $A$ is nonempty, so when regarded as a coprojection in $Set^{op}$ 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.