# nLab monomorphism in an (infinity,1)-category

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

The notion of monomorphism in an $(\infty,1)$-category is the generalization of the notion of monomorphism from category theory to (∞,1)-category theory. It is the special case of the notion of n-monomorphisms for $n = 1$. In an (∞,1)-topos every morphism factors by an effective epimorphism (1-epimorphism) followed by a monomorphism through its 1-image.

The dual concept is that of an epimorphism in an (∞,1)-category.

There is also the concept regular monomorphism in an (∞,1)-category, but beware that this need not be a special case of the definition given here.

There are also a notions of (homotopy) monomorphism in model categories and derivators.

## Definition

For $C$ an (∞,1)-category, a morphism $f : Y \to Z$ is a monomorphism if regarded as an object in the (∞,1)-overcategory $X_{/Z}$ it is a (-1)-truncated object.

Equivalently this means that the projection

$C_{/f} \to C_{/Z}$

is a full and faithful (∞,1)-functor. This is in Higher Topos Theory after Example 5..5.6.13.

Equivalently this means that for every object $X \in C$ the induced morphism

$C(X,f) : C(X,Y) \to C(X,Z)$

of ∞-groupoids is such that its image in the homotopy category exhibits $C(X,Y)$ as a direct summand in a coproduct decomposition of $C(X,Z)$.

So if $C(X,Y) = \coprod_i C(X,Y)_{i \in \pi_0(C(X,Y))}$ and $C(X,Z) = \coprod_{j \in \pi_0((C(X,Z))} C(X,Z)_j$ is the decomposition into connected components, then there is an injective function

$j : \pi_0(C(X,Y)) \to \pi_0(C(X,Z))$

such that $C(X,f)$ is given by component maps $C(X,Y)_i \to C(X,Z)_{j(i)}$ which are each an equivalence.

## Properties

###### Definition

For $Z$ an object of $C$, write $Sub(Z)$

$Sub(Z) \simeq \tau_{\leq -1} C_{/Z} \,.$

for the category of subobjects of $C$.

This is partially ordered under inclusion.

###### Proposition

If $C$ is a presentable (∞,1)-category, then $Sub(Z)$ is a small category.

This appears as HTT, prop. 6.2.1.4.

###### Proposition

Monomorphisms are stable under (∞,1)-pullback: if

$\array{ A &\to& B \\ {}^{\mathllap{f'}}\downarrow && \downarrow^{\mathrlap{f}} \\ C &\to& D }$

is a pullback diagram and $f$ is a monomorphism, then so is $f'$.

This is a special case of the general statement that $k$-truncated morphisms are stable under pullback. (HTT, remark 5.5.6.12).

## References

The definition appears after example 5.5.6.13 in

with further discussion in section 6.2.

For model categories, see

Revised on May 13, 2015 19:15:18 by Adeel Khan (93.131.11.231)