## compact elements and continuous functors

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Hi,

I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:

A functor F:C→D is continuous if it preserves all small (weighted) limits that exist in C, i.e. if for every small category diagram A:E→C in C there is an isomorphism F(limA)≃lim(F∘A).F(\lim A) \simeq \lim (F\circ A).

I found this on the n-Lab. What I am interested in is a notion of compact element which is defined for Domains as this:

if k is less than the sup of any directed subset D, then there is an element x in D such that k is less than x.

Any reference would be good. My intuitions are telling me that if a category is compact, in this sense, it should have a kind of finite presentation. This would be an abstraction from a dcpo of groups where the compact elements are finitely generated. I realize this might not make sense so any help would be great.

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The equivalent of compact elements could be the following: an object $X$ is called finitely presentable when $\mathrm{Hom}(X,-)$ preserves directed colimits. The equivalent of a Scott domain would then be a locally finitely presentable category: one in which every object is a colimit of finitely presentable ones. The nLab page doesn't have many pointers, but Adamek and Rosicky have a nice book about the topic (albeit not oriented towards domains). – Chris Heunen Apr 8 2011 at 1:35
Hi Chris -- I was in the middle of writing my answer and didn't see your comment, which contains much of what I said. – Todd Trimble Apr 8 2011 at 1:41
Thanks very much. I might be able to start working on this a bit more now. – Ben Sprott Apr 8 2011 at 7:50
Hey, It looks like I am interested in yet a higher level of abstraction. When I asked about finitely present categories, I was being specific. I mean that categories themselves could be the elements of the domain. I apologize for the roughness with which I am speaking. I think the intuition I am working on is borrowed from the fact that the set of endomaps of a domain is also a domain. This is, itself, a path of abstraction. If the categories are the elements of the domain, then the compact objects which Todd talks aobut can be abstracted up into compact categories. – Ben Sprott Apr 11 2011 at 5:57
I also think that the n-categorical framework is at work here too. – Ben Sprott Apr 11 2011 at 5:58

## 2 Answers

Your intuitions look good. The analogue of compact element for general categories is the notion of finitely presentable object:

• An object $c$ of a category $C$ is finitely presentable if the hom-functor $\hom_C(c, -): C \to Set$ preserves directed colimits (colimits of diagrams $D \to C$ where $D$ is a directed poset).

These coincide with finitely presentable algebras when $C$ is a category of algebras of a finitary algebraic theory. This is part of the very beautiful theory of locally finitely presentable categories (and the more general locally presentable categories); an excellent reference for this is the book by Adamek and Rosicky, Locally Presentable and Accessible Categories.

"Locally finitely presentable" means a category which is cocomplete such that every object is a directed colimit of finitely presentable objects (for example, every group can be presented as a directed colimit of finitely presentable groups). Locally finitely presentable posets are the same as algebraic lattices.

Edit: Finn is right that finitely presentable objects are also called "compact objects".

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Have you tried nLab again?

I'm not sure continuous functors are what you're looking for, incidentally. It seems more likely that a 'Scott-continuous functor' should be one that preserves filtered colimits.

Replacing the preorders (and metric spaces) of domain theory with (enriched) categories is not a new idea. Have a look at Categories for fixpoint semantics (1978) by Daniel Lehmann, and Solving recursive domain equations with enriched categories (1994) by Kim Ritter Wagner.

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Functors that preserve filtered colimits are often called "finitary". – Mike Shulman Apr 8 2011 at 6:27
Hey,    It looks like I am interested in yet a higher level of abstraction.  When I asked about finitely present categories, I was being specific.  I mean that categories themselves could be the elements of the domain.  I apologize for the roughness with which I am speaking.  I think the intuition I am working on is borrowed from the fact that the set of endomaps of a domain is also a domain.  This is, itself, a path of abstraction.  If the categories are the elements of the domain, then the compact objects which Todd talks aobut can be abstracted up into compact categories. – Ben Sprott Apr 11 2011 at 5:56
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