# 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

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

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

$\array{ A & \stackrel{i}{\to} & B \\ i \downarrow & & \downarrow i_1 \\ B & \underset{i_2}{\to} & B +_A B }$

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 $\phi: B +_A B \to C$ such that $\phi \circ i_1 = f$ and $\phi \circ i_2 = g$. Then, since

$f j = \phi i_1 j = \phi i_2 j = g j$

and since $i: A \to B$ is the equalizer of the pair $(f, g)$, there is a unique map $k: E \to A$ such that $j = i k$. Since $i_1 i = i_2 i$, there is a unique map $l: A \to E$ such that $i = j l$. 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 $(i_1, i_2)$ 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 $Set$ 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 $K \hookrightarrow H$ be a subgroup. We need to define another group $G$ and group homomorphisms $f_1, f_2 : H \to G$ such that

$K = \{h \in H | f_1(h) = f_2(h)\} \,.$

To that end, let

$X := H/K \coprod \{\hat K\} := \{ h K | h \in H \} \coprod \{\hat K\}$

be the set of cosets together with one more element $\hat K$.

Let then

$G = Aut_{Set}(X)$

be the permutation group on $X$.

Define $\rho \in G$ to be the permutation that exchanges the coset $e K$ with the extra element $\hat K$ and is the identity on all other elements.

Finally define group homomorphism $f_1,f_2 : H \to G$ by

$f_1(h) : x \mapsto \left\{ \array{ h h' K & if x = h' K \\ \hat K & if x = \hat K } \right.$

and

$f_2(h) = \rho \circ f_1(h) \circ \rho^{-1} \,.$

It is clear that these maps are indeed group homomorphisms.

So for $h \in H$ we have that

$f_1(h) : \hat K \mapsto \hat K \,,$

and

$f_1(h) : e K \mapsto h K$

and

$f_2(h) : \hat K \mapsto e K \mapsto h K \mapsto \left\{ \array{ \hat K & if h \in K \\ h K & otherwise } \right. \,.$
$f_2(h) : e K \mapsto \hat K \mapsto \hat K \mapsto e K \,.$

So we have $f_1(h) = f_2(h)$ precisely if $h \in K$.

## In an $(\infty,1)$-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

$c \stackrel{f}{\to} d_0 \stackrel{\to}{\to} d_1 \stackrel{\to}{\stackrel{\to}{\to}} d_2 \cdots \,.$

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 $f^* : C(d,e) \to C(c,e)$ 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)