# nLab coprojection

Contents

### Context

#### Limits and colimits

limits and colimits

category theory

# Contents

## 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:

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.

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