geometric category in nLab

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Definition

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

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

Makkai-Reyes call this a κ\kappa-logical category, while Shulman calls it a κ\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.

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).