nLab geometric category

Definition

An infinitary coherent category or geometric category is a regular category in which the subobject posets $Sub(X)$ have all small unions which are stable under pullback.

More generally, for $\kappa$ a regular cardinal, a $\kappa$-geometric category, or $\kappa$-coherent category, is a regular category with unions for $\kappa$-small families of subobjects, stable under pullback. (For $\kappa = \omega$ this reduces to the notion of coherent category, called a pre-logos by Freyd–Scedrov.)

Makkai-Reyes call this a $\kappa$-logical category, while Shulman calls it a $\kappa$-ary regular category. See also (Butz-Johnstone, p. 12).

Properties

See familial regularity and exactness for a general description of the spectrum from regular categories through finitary and infinitary coherent categories.

Well-poweredness

Frequently, geometric categories are additionally required to be well-powered. If a geometric category is well-powered, then its subobject posets are complete lattices, hence also have all intersections. Moreover, by the adjoint functor theorem for posets, it is a Heyting category.

However, since geometric logic does not include implication or infinite conjunction, this categorical structure should not necessarily be expected to exist in a category called “geometric” (and when they do exist, they are not preserved by geometric functors). A requirement of well-poweredness is also inconsistent with the spectrum of familial regularity and exactness.

Note, however, that if a geometric category has a small generating set, then it is necessarily well-powered. In particular, this applies to the syntactic category of any (small) geometric theory, and also to any Grothendieck topos.

Related concepts

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

References

Around lemma A1.4.18 in

• Peter Johnstone, Sketches of an Elephant

• Michael Makkai, Gonzalo Reyes, First Order Categorical Logic, Lecture Notes in Math. 611 (Springer-Verlag, 1977).

• Casten Butz, Peter Johnstone, Classifying toposes for first order theories, BRICS Report Series RS-97-20

• Mike Shulman, Exact completions and small sheaves, TAC 27(7), 2012 (link).