nLab
regular monomorphism

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 $d\overrightarrow{\to }e$, i.e. for which a limit diagram

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

Examples

• 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 Grp, the monics are (up to isomorphism) the inclusions of subgroups, and every monomorphism is regular, though this is more difficult to prove than in the preceding cases. (See exercise 7H of Adamek, Herrlich, Strecker, Abstract and Concrete Categories.) 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.

• 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.

In an $(\mathrm{\infty }\mathrm{,1})$-category

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

$$c\stackrel{f}{\to }{d}_0\overrightarrow{\to }{d}_1\overrightarrow{\overrightarrow{\to }}{d}_2\cdots .$$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 for instance 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 $\mathrm{\infty }$-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. But the idea is that the only sensible notion of monomorphism in an (∞,1)-category is in fact that of regular monomorphism.

Mike Shulman: I think that that last sentence is false, cf. monomorphism in an (∞,1)-category.