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

### Context

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

(∞,1)-category theory

# Contents

## Idea

The notion of monomorphism in an $\left(\infty ,1\right)$-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.

## 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

${C}_{/f}\to {C}_{/Z}$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

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

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

So if $C\left(X,Y\right)={\coprod }_{i}C\left(X,Y{\right)}_{i\in {\pi }_{0}\left(C\left(X,Y\right)\right)}$ and $C\left(X,Z\right)={\coprod }_{j\in {\pi }_{0}\left(\left(C\left(X,Z\right)\right)}C\left(X,Z{\right)}_{j}$ is the decomposition into connected components, then there is an injective function

$j:{\pi }_{0}\left(C\left(X,Y\right)\right)\to {\pi }_{0}\left(C\left(X,Z\right)\right)$j : \pi_0(C(X,Y)) \to \pi_0(C(X,Z))

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

## Properties

###### Definition

For $Z$ an object of $C$, write $\mathrm{Sub}\left(Z\right)$

$\mathrm{Sub}\left(Z\right)\simeq {\tau }_{\le -1}{C}_{/Z}\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{Sub}\left(Z\right)$ is a small category.

This appears as HTT, prop. 6.2.1.4.

###### Proposition

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

$\begin{array}{ccc}A& \to & B\\ {}^{f\prime }↓& & {↓}^{f}\\ C& \to & D\end{array}$\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 $f\prime$.

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.

Revised on November 29, 2012 18:55:53 by Urs Schreiber (82.169.65.155)