A monomorphism is regular if it behaves like an embedding.
effective epimorphism $\Rightarrow$ regular epimorphism $\Leftrightarrow$ covering
effective monomorphism $\Rightarrow$ regular monomorphism $\Leftrightarrow$ embedding .
The universal factorization through an embedding is the image.
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
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.
Warning. (CassidyHebertKelly) use `regular monomorphism' in a more general way: for them, a regular monomorphism is by definition the joint equalizer or an arbitrary family of parallel pairs of morphisms with common domain. This concept is sometimes called strict monomorphism, dual to the more commonly used strict epimorphism.
A monomorphism $i: A \to B$ is an effective monomorphism if it is the equalizer of its cokernel pair: if the pushout
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.
In a category with finite limits and finite colimits, every regular monomorphism $i: A \to B$ is effective.
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
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.
In a category with equalizers and cokernel pairs, a regular monomorphism is precisely an effective monomorphism.
In Set, or more generally in any pretopos, every monomorphism is regular.
Similarly, in Ab, and more generally any abelian category, every monomorphism is regular.
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.
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.
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.
We follow exercise 7H of (AdamekHerrlichStrecker).
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
To that end, let
be the set of cosets together with one more element $\hat K$.
Let then
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
and
It is clear that these maps are indeed group homomorphisms.
So for $h \in H$ we have that
and
and
So we have $f_1(h) = f_2(h)$ precisely if $h \in K$.
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
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.