The n-Category Café

I imagine you thought of this after our little discussion about ‘cartesian closed structures’ for categories, over on the category theory mailing list.

Interesting question! As you note, the easiest kind of property-like structure on categories is ‘having all limits of a specified sort’.

Can we boost this up a notch… in the right way? There’s an obvious wrong way: for a 2-category to have all limits still seems pretty property-like.

Hmm, but is it property-like structure or property-like stuff? You didn’t mention ‘property-like stuff’, so I’m not sure it even makes sense, but I think it does.

Anyway: is there something for 2-categories that comes ‘after’ having limits, something more like a structure???

The first sort of example that I can feel happy about supposes a category enriched in groups (abelian groups, I suppose, thought of as 1-object groupoids, of course) and the 2-functor leaves objects alone and sends each hom-group to the free abelian monoid on the hom-group’s underlying set — this is naturally a rig with product extending the group multiplication, and naturally you use the product for vertical composition. Then the image of a group in its rig is still the set of units — isomorphisms — in that rig.

Someone else can think about more X-oidal variations.

Here’s an example which is somewhat tautological, but perhaps also illuminating. Let $C$ be the 2-category whose objects are pseudomonic functors, whose morphisms are squares commuting up to (specified) isomorphism, and whose 2-morphisms are pairs of natural transformations making the evident cylinder commute. Then I claim that the forgetful 2-functor $C\to Cat$, which takes each pseudomonic functor to its target, forgets structure-like stuff. In other words, one way to put structure-like stuff on a category is to describe a way of putting property-like structure on the objects of that category.

This 2-category $C$ has some interesting sub-2-categories, such as the 2-category of property-like monads (on varying base categories). The corresponding 3-category to $C$, whose objects are pseudomonic 2-functors, also contains a sub-3-category of property-like 2-monads, which contains further subcategories of lax-idempotent 2-monads and colax-idempotent 2-monads.

Hi,

I don’t think this is the right place to ask this question. This is a kind of meta-post, where I might be asking where this post actually belongs. I am wondering about approximation and idealization. Specifically, I am wondering if anyone has seen some work on the following. In the semantics of programming languages we find Domains as a place to talk about iteration and approximation. We can define a Scott Topology on the Domain and now our Domain-maps are continuous maps. Next,we can see our Domains as categories and turn the continuous Domain-maps into continuous functors. If we push the idea further, we have continuous functors and a notion of approximation which is now over categories. Lambek ponders the existence of _the_ category of Sets. What about _approximations_ to the category of sets. For that matter, what might it look like to approximate any well-known category like that of manifolds or FDVec?
Naturally, any thoughts on this would be most appreciated.

I have been thinking about this since it was posted…and it seems interesting.

My opinion is, looking at a mathematical gadget as consisting of stuff equipped with structure such that some properties hold, a mathematical gadget itself may be “property-like”.

For example, we can talk about “THE” Euclidean 3-dimensional space (up to isomorphism). Or more generally, if we fix some positive integer n, we can talk about “THE” Euclidean n-dimensional space (up to some isomorphism).

But isn’t this adjective “Property-like” merely a generalization of Bourbaki’s “Univalent theory”? If so, why can’t we borrow the adjective “Univalent”?

Just my thoughts before I have to run to class…