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 empty category is a locally regular category which is not regular.
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).
The syntactic category of an extensional type theory with dependent sum types, identity types, and propositional truncations is a locally regular category. If the dependent type theory also has dependent product types, the syntactic category becomes locally cartesian closed. If the type theory also has a unit type its syntactic category becomes a regular category.
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.
Last revised on March 8, 2024 at 05:07:58. See the history of this page for a list of all contributions to it.