# nLab regular monomorphism

category theory

## Applications

#### Higher category theory

higher category theory

# Contents

## Idea

A monomorphism is regular if it behaves like an embedding.

The universal factorization through an embedding is the image.

## Definition

A regular monomorphism is a morphism $f:c\to d$ in some category which occurs as the equalizer of some parallel pair of morphisms $d\stackrel{\to }{\to }e$, i.e. for which a limit diagram

$c\stackrel{f}{\to }d\stackrel{\to }{\to }e$c \stackrel{f}{\to} d \stackrel{\to}{\to} e

exists.

From the defining universal property of the limit it follows directly that a regular monomorphism is automatically a monomorphism.

The dual concept is that of a regular epimorphism.

## Properties

### Relation to effective monomorphisms

A monomorphism $i:A\to B$ is an effective monomorphism if it is the equalizer of its cokernel pair: if the pushout

$\begin{array}{ccc}A& \stackrel{i}{\to }& B\\ i↓& & ↓{i}_{1}\\ B& \underset{{i}_{2}}{\to }& B{+}_{A}B\end{array}$\array{ A & \stackrel{i}{\to} & B \\ i \downarrow & & \downarrow i_1 \\ B & \underset{i_2}{\to} & B +_A B }

exists and $i$ is the equalizer of the pair of coprojections ${i}_{1},{i}_{2}:B\stackrel{\to }{\to }B{+}_{A}B$. Obviously effective monomorphisms are regular.

###### Proposition

In a category with finite limits and finite colimits, every regular monomorphism $i:A\to B$ is effective.

###### Proof

Suppose $i:A\to B$ is the equalizer of a pair of morphisms $f,g:B\to C$, and with notation as above, let $j:E\to B$ be the equalizer of the pair of coprojections ${i}_{1},{i}_{2}$. Since $f\circ i=g\circ i$, there exists a unique map $\varphi :B{+}_{A}B\to C$ such that $\varphi \circ {i}_{1}=f$ and $\varphi \circ {i}_{2}=g$. Then, since

$fj=\varphi {i}_{1}j=\varphi {i}_{2}j=gj$f j = \phi i_1 j = \phi i_2 j = g j

and since $i:A\to B$ is the equalizer of the pair $\left(f,g\right)$, there is a unique map $k:E\to A$ such that $j=ik$. Since ${i}_{1}i={i}_{2}i$, there is a unique map $l:A\to E$ such that $i=jl$. The maps $k$, $l$ are mutually inverse.

###### Lemma

In a category with equalizers and cokernel pairs, a regular monomorphism is precisely an effective monomorphism.

## Examples

###### Proposition

In Top, the monics are the injective functions, while the regular monos are the embeddings (that is, the injective functions whose sources have the topologies induced from their targets); these are in fact all of the extremal monomorphisms.

###### Proof

Use lemma 1.

If $i:X\to Y$ is a subspace embedding, then we form the cokernel pair $\left({i}_{1},{i}_{2}\right)$ by taking the pushout of $i$ against itself (in the category of sets, and using the quotient topology on a disjoint sum). The equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the subspace topology. Since monos in $\mathrm{Set}$ are regular, we get the function $i$ back with the subspace topology. This completes the proof.

###### Proposition

In Grp, the monics are (up to isomorphism) the inclusions of subgroups, and every monomorphism is regular

In contrast, the normal monomorphisms (where one of the morphisms $d\to e$ is required to be the zero morphism) are the inclusions of normal subgroups.

###### Proof

Let $K↪H$ be a subgroup. We need to define another group $G$ and group homomorphisms ${f}_{1},{f}_{2}:H\to G$ such that

$K=\left\{h\in H\mid {f}_{1}\left(h\right)={f}_{2}\left(h\right)\right\}\phantom{\rule{thinmathspace}{0ex}}.$K = \{h \in H | f_1(h) = f_2(h)\} \,.

To that end, let

$X:=H/K\coprod \left\{\stackrel{^}{K}\right\}:=\left\{hK\mid h\in H\right\}\coprod \left\{\stackrel{^}{K}\right\}$X := H/K \coprod \{\hat K\} := \{ h K | h \in H \} \coprod \{\hat K\}

be the set of cosets together with one more element $\stackrel{^}{K}$.

Let then

$G={\mathrm{Aut}}_{\mathrm{Set}}\left(X\right)$G = Aut_{Set}(X)

be the permutation group on $X$.

Define $\rho \in G$ to be the permutation that exchanges the coset $eK$ with the extra element $\stackrel{^}{K}$ and is the identity on all other elements.

Finally define group homomorphism ${f}_{1},{f}_{2}:H\to G$ by

${f}_{1}\left(h\right):x↦\left\{\begin{array}{cc}hh\prime K& \mathrm{if}x=h\prime K\\ \stackrel{^}{K}& \mathrm{if}x=\stackrel{^}{K}\end{array}$f_1(h) : x \mapsto \left\{ \array{ h h' K & if x = h' K \\ \hat K & if x = \hat K } \right.

and

${f}_{2}\left(h\right)=\rho \circ {f}_{1}\left(h\right)\circ {\rho }^{-1}\phantom{\rule{thinmathspace}{0ex}}.$f_2(h) = \rho \circ f_1(h) \circ \rho^{-1} \,.

It is clear that these maps are indeed group homomorphisms.

So for $h\in H$ we have that

${f}_{1}\left(h\right):\stackrel{^}{K}↦\stackrel{^}{K}\phantom{\rule{thinmathspace}{0ex}},$f_1(h) : \hat K \mapsto \hat K \,,

and

${f}_{1}\left(h\right):eK↦hK$f_1(h) : e K \mapsto h K

and

${f}_{2}\left(h\right):\stackrel{^}{K}↦eK↦hK↦\left\{\begin{array}{cc}\stackrel{^}{K}& \mathrm{if}h\in K\\ hK& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$f_2(h) : \hat K \mapsto e K \mapsto h K \mapsto \left\{ \array{ \hat K & if h \in K \\ h K & otherwise } \right. \,.
${f}_{2}\left(h\right):eK↦\stackrel{^}{K}↦\stackrel{^}{K}↦eK\phantom{\rule{thinmathspace}{0ex}}.$f_2(h) : e K \mapsto \hat K \mapsto \hat K \mapsto e K \,.

So we have ${f}_{1}\left(h\right)={f}_{2}\left(h\right)$ precisely if $h\in K$.

## In an $\left(\infty ,1\right)$-category

In the context of higher category theory the ordinary limit diagram $c\stackrel{f}{\to }d\stackrel{\to }{\to }e$ may be thought of as the beginning of a homotopy limit diagram over a cosimplicial diagram

$c\stackrel{f}{\to }{d}_{0}\stackrel{\to }{\to }{d}_{1}\stackrel{\to }{\stackrel{\to }{\to }}{d}_{2}\cdots \phantom{\rule{thinmathspace}{0ex}}.$c \stackrel{f}{\to} d_0 \stackrel{\to}{\to} d_1 \stackrel{\to}{\stackrel{\to}{\to}} d_2 \cdots \,.

Accordingly, it is not unreasonable to define a regular monomorphism in an (∞,1)-category, to be a morphism which is the limit in a quasi-category of a cosimplicial diagram.

In practice this is of particular relevance for the $\infty$-version of regular epimorphisms: with the analogous definition as described there, a morphism $f:c\to d$ is a regular epimorphism in an (∞,1)-category $C$ if for all objects $e\in C$ the induced morphism ${f}^{*}:C\left(d,e\right)\to C\left(c,e\right)$ is a regular monomorphism in ∞Grpd (for instance modeled by a homotopy limit over a cosimplicial diagram in SSet).

Warning. The same warning as at regular epimorphism applies: with this definition of regular monomorphism in an (∞,1)-category these may fail to satisfy various definitions of plain monomorphisms that one might think of.

## References

Revised on April 3, 2013 20:09:39 by Todd Trimble (67.81.93.26)