The notion of continuous category is a categorification of the notion of continuous poset. It can be further categorified to a notion of continuous (∞,1)-category.
Let be a category and its category of ind-objects. We assume that has filtered colimits, which is equivalently to say that the restricted Yoneda embedding has a left adjoint .
A category with filtered colimits is a continuous category if has a left adjoint.
If is a poset, then is its category of ideals. Thus, a poset is a continuous category exactly when it is a continuous poset. This definition can be extended to -categories essentially verbatim.
Any finitely accessible category, and hence any locally finitely presentable category, is a continuous category.
A Grothendieck topos is a continuous category if and only if it is an exponentiable object in the 2-category of Grothendieck toposes and geometric morphisms, i.e. an exponentiable topos.
If is a stably locally compact locale? (or more generally a metastably locally compact locale?), then is continuous and hence exponentiable. It does not suffice for to be locally compact (i.e. for its frame of opens to be a continuous poset).
Similarly, a Grothendieck (∞,1)-topos is a continuous -category if and only if it is an exponentiable object, in the appropriate sense, in the -category of -toposes and geometric morphisms.
In general, a locally small category is continuous if and only if it is a retract of a category of ind-objects, where the functors exhibiting the retract preserve filtered colimits (Johnstone-Joyal 82, Theorem 2.8).
If is continuous, with the left adjoint of , and , we define a wavy arrow to be a morphism in . This is a categorification of the way-below relation on a continuous poset: when is a poset we have a wavy arrow just when . But unlike in the posetal case, it is not clear how to define wavy arrows unless is continuous (whereas can be defined in any poset with directed joins). However, see totally distributive category.
Since and , the functor assigns to every wavy arrow a “straight” arrow in . Moreover, wavy arrows can be composed: the composite of and is the composite
where is the adjunct of the identity (or of the identity ). This composition is associative. Thus, we almost have a category whose objects are those of and whose morphism are wavy arrows — but it does not have identities.
However, if denotes the set of wavy arrows, then the composition defines a map of profunctors which is in fact an isomorphism. Combined with the map from wavy arrows to straight ones, this makes into an idempotent comonad on in the bicategory Prof.
Continuous (∞,1)-categories are introduced under the name of compactly assembled ∞-categories in Lurie SAG, §21.1.2.
Jiří Adámek, F. William Lawvere, Jiří Rosický, Continuous categories revisited , TAC 11 no.11 (2003) pp.252-282. (abstract)
Peter Johnstone, Andre Joyal, Continuous categories and exponentiable toposes, JPAA 25 (1982), doi (free PDF)
Mathieu Anel, Damien Lejay, Exponentiable Higher Toposes , arXiv:1802.10425 (2018). (abstract)
Last revised on October 14, 2022 at 16:18:02. See the history of this page for a list of all contributions to it.