nLab continuous category

Continuous categories

Continuous categories


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 CC be a category and Ind(C)Ind(C) its category of ind-objects. We assume that CC has filtered colimits, which is equivalently to say that the restricted Yoneda embedding ()^:CInd(C)\hat{(-)} : C\to Ind(C) has a left adjoint colim\colim.


A category CC with filtered colimits is a continuous category if colim:Ind(C)C\colim: Ind(C) \to C has a left adjoint.

If CC is a poset, then Ind(C)=Idl(C)Ind(C) = Idl(C) 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 (,1)(\infty,1)-categories essentially verbatim.


Wavy arrows

If CC is continuous, with L:CInd(C)L:C\to Ind(C) the left adjoint of colim\colim, and x,yCx,y\in C, we define a wavy arrow xyx\rightsquigarrow y to be a morphism x^L(y)\hat{x} \to L(y) in Ind(C)Ind(C). This is a categorification of the way-below relation on a continuous poset: when CC is a poset we have a wavy arrow xyx\rightsquigarrow y just when xyx\ll y. But unlike in the posetal case, it is not clear how to define wavy arrows unless CC is continuous (whereas \ll can be defined in any poset with directed joins). However, see totally distributive category.

Since colimx^=x\colim \hat{x} = x and colimL(y)=y\colim L(y)=y, the functor colim\colim assigns to every wavy arrow a “straight” arrow xyx\to y in CC. Moreover, wavy arrows can be composed: the composite of f:x^L(y)f:\hat{x} \to L(y) and g:y^L(z)g:\hat{y} \to L(z) is the composite

x^fL(y)iy^gL(z) \hat{x} \xrightarrow{f} L(y) \xrightarrow{i} \hat{y} \xrightarrow{g} L(z)

where ii is the adjunct of the identity y=colimy^y = \colim \hat{y} (or of the identity colimL(y)=y\colim L(y) = y). This composition is associative. Thus, we almost have a category whose objects are those of CC and whose morphism are wavy arrows — but it does not have identities.

However, if C˜(x,y)\tilde{C}(x,y) denotes the set of wavy arrows, then the composition defines a map of profunctors C˜ CC˜C˜\tilde{C} \otimes_C \tilde{C} \to \tilde{C} which is in fact an isomorphism. Combined with the map from wavy arrows to straight ones, this makes C˜\tilde{C} into an idempotent comonad on CC in the bicategory Prof.


The setting of (∞,1)-categories

Continuous (∞,1)-categories are introduced under the name of compactly assembled ∞-categories in Lurie SAG, §21.1.2.


Last revised on October 14, 2022 at 16:18:02. See the history of this page for a list of all contributions to it.