nLab continuous category

Idea

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.

Definition

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

Definition

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

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

Examples

• 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 $X$ is a stably locally compact locale? (or more generally a metastably locally compact locale?), then $Sh(X)$ is continuous and hence exponentiable. It does not suffice for $X$ to be locally compact (i.e. for its frame of opens to be a continuous poset).

  • Similarly, a Grothendieck (∞,1)-topos is a continuous $(\infty,1)$-category if and only if it is an exponentiable object, in the appropriate sense, in the $(\infty,1)$-category of $(\infty,1)$-toposes and geometric morphisms.

• In general, a locally small category $C$ is continuous if and only if it is a retract of a category $Ind(D)$ of ind-objects, where the functors exhibiting the retract preserve filtered colimits (Johnstone-Joyal 82, Theorem 2.8).

Wavy arrows

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

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

$$\hat{x} \xrightarrow{f} L(y) \xrightarrow{i} \hat{y} \xrightarrow{g} L(z)$$

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

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

Remark

• Continuous categories can be generalized to continuous algebras for any lax-idempotent 2-monad.

The setting of (∞,1)-categories

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

Related entries

• totally distributive category

• injective topos

• domain theory

• Scott topos

References

• 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)

• Jacob Lurie, Spectral Algebraic Geometry.