### Context

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Definitions and terminology

A **split monomorphism** in a category $C$ is a morphism $m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ such that the composite $r \circ m$ equals the identity morphism $1_A$. Then the morphism $r$, which satisfies the dual condition, is a **split epimorphism**.

We say that: * $r$ is a **retraction** of $m$, * $m$ is a **section** of $r$, * $A$ is a **retract** of $B$, * the pair $(r,m)$ is a **splitting** of the idempotent $m \circ r\colon B \to B$.

A split monomorphism in $C$ can be equivalently defined as a morphism $m\colon A \to B$ such that for every object $X\colon C$, the function $C(m,X)$ is a surjection in $\mathbf{Set}$; the preimage of $1_A$ under $C(m,A)$ yields a retraction $r$.

Alternatively, it is also possible to define a split monomorphism as an **absolute monomorphism**: a morphism such that for every functor $F$ out of $C$, $F(m)$ is a monomorphism. From the definition as a morphism having a retraction, it is obvious that any split monomorphism is absolute; conversely, that the image of $m$ under the representable functor $C(1,A)$ is a monomorphism reduces to the characterization above.

## Properties

Any split monomorphism is automatically a regular monomorphism (it is the equalizer of $m\circ r$ and $1_B$), and therefore also a strong monomorphism, an extremal monomorphism, and (of course) a monomorphism.

## In higher category theory

In higher category theory, we may still consider the notion of “split monomorphism”, i.e. a morphism $m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ with $r \circ m$ being equivalent to the identity of $A$. However, in a higher category, such a morphism $m$ will not necessarily be a “monomorphism”, that is, it need not be $(-1)$-truncated.

In general, we can say that in an $(n,1)$-category, a “split monomorphism” will be $(n-2)$-truncated. Thus: * in a (0,1)-category (a poset), a split mono is $(-2)$-truncated, i.e. an isomorphism; * in a 1-category, a split mono is $(-1)$-truncated, i.e. a monomorphism; * in a (2,1)-category, a split mono is $0$-truncated, i.e. a discrete morphism; * in an (∞,1)-category, a split mono is not necessarily truncated at any finite level.