A locally regular category is a relative of a regular category in which the condition of finite limits is weakened to finite connected limits. It is so named because every slice category of a locally regular category is a regular category, although the converse is not quite true.
A category $C$ is locally regular if
It has finite connected limits — equivalently, it has pullbacks and equalizers;
It has (extremal epi, mono) factorizations which are stable under pullback; and
Every span factors as an extremal epi followed by a jointly-monic span.
Every regular category is locally regular. Factorizations of spans may be obtained by factorizations of single morphisms into a binary product.
The category $LH$ of topological spaces (or locales) and local homeomorphisms is locally regular, but not regular. Its slice categories are precisely the sheaf toposes of spaces (or locales).
A locally regular category is regular if and only if it has a terminal object.
The factorization axiom for spans implies, by induction, a similar factorization property for nonempty finite cosinks. However, similar factorizations for empty cosinks (i.e. “supports”) do not necessarily exist.
Every locally regular category gives rise to a tabular allegory of binary relations (where we define a “binary relation” to mean a jointly-monic span). For composition of relations, we require pullbacks and stable factorizations of spans. For intersection of binary relations, we require equalizers.
Conversely, every tabular allegory has a locally regular category of maps (left adjoints). So locally regular categories are essentially the same as tabular allegories.