The idea here is to create an enormous list of well-known categories and their properties. As the list gets large we’ll have to figure out a good structure for it, but right now I’ll just do a few entries in alphabetical order. Andrew Stacey solicits input about making this a real database; see the Forum.
We can import new entries by looking at category:category, which should have all articles dedicated to specific categories.
Ab: abelian groups as objects, homomorphisms as morphisms. Since this category consists of models of a Lawvere theory it has all small limits and colimits. It is the prototypical abelian category and also a closed symmetric monoidal category under the tensor product of abelian groups.
Alg: algebras as objects, homomorphisms as morphisms. This is actually many different categories, depending on a choice of ground field or more generally commutative ring. Since any one of these categories consists of models of a Lawvere theory, it has all small limits and colimits.
Bicat: bicategories($=$weak 2-categories) as objects, 2-functors as its 1-cells, pseudonatural transformations between its 1-cells as its 2-cells, and modifications between its 2-cells as its 3-cells. BiCat is a tricategory. BiCat is sometimes denoted $Hom$.
CartSp: spaces $\mathbb{R}^n$ as objects, smooth maps as morphisms.
Cat: small categories as objects, functors as morphisms. Since this category consists of models of a finite limits theory, it has all small limits and colimits. It is cartesian closed, and its internal homs make it into a 2-category.
Diff: smooth manifolds as objects, smooth maps as morphisms. This category has finite but not infinite products, and all small coproducts. It does not have equalizers or coequalizers, and is not cartesian closed.
Diffeological spaces as objects, smooth maps as morphisms. This category has all small limits and colimits, it is cartesian closed, and it has a weak subobject classifier. So, it is a quasitopos.
Grp: groups as objects, homomorphisms as morphisms. Since this category consists of models of a Lawvere theory it has all small limits and colimits.
$Mod_R$: (right) $R$-modules as objects, $R$-module homomorphisms as morphisms. There is one such category for any ring $R$ (or more generally, any small Ab-enriched category. This is also the category of models of a Lawvere theory and is thus complete and cocomplete. It is an abelian category and even Grothendieck category. If $R$ is commutative, then $Mod_R$ is a closed symmetric monoidal category under the tensor product of $R$-modules.
Rel: sets as objects, relations as morphisms. This is an allegory and therefore a dagger category. The empty set is an zero object, and disjoint union plays the role of biproduct. This category does not have equalizers or coequalizers.
Ring: rings as objects, ring homomorphisms as morphisms. Since this category consists of models of a Lawvere theory it has all small limits and colimits. (maybe we should have unital rings $Ring_1$ separately; the difference is important sometimes).
SimpSet: simplicial sets as objects, simplicial maps as morphisms. This is a presheaf topos.
Set: sets as objects, functions as morphisms. This has the following properties:
At least assuming classical logic, these properties suffice to characterize $Set$ uniquely up to equivalence among all categories. Note, however, that the definitions of “locally small” and “(co)complete” presuppose a notion of small and therefore a knowledge of what a set is.
See also Understanding Set.
Top: topological spaces as objects, continuous functions as morphisms. This has all small limits and colimits, but is not cartesian closed — a problem that may be addressed by introducing a convenient category of topological spaces.
$k$-Vect: vector spaces as objects, linear maps as morphisms. This is actually many different categories, depending on a choice of field $k$; in fact, it is a special case of $Mod_R$ when $R$ is a field. Since any one of these categories consists of models of a Lawvere theory, it has all small limits and colimits.
Rafael Borowiecki: I could add something of the order 100 categories here, but i know very little of their properties. Should i add them anyway?
It would also be helpful to have a list of properties to fill in for each category to make it a real database. As a start i suggest: size, concrete, complete, enrichment, topos. This list could also get very large resulting in an even larger table (written as a list). For instance one could also add functoriality (is it a category of functors?, notice that the Yoneda lemma do not hold for any category), cartesian closedness, monoidality, model structure.
I don’t know where to put this request but the logging of changes should be automated. Simply add a smaller box one might fill in if one wishes below the editing box and hope that not every edit will also edit this changes box. It could be limited to size.
Toby: I also could add hundreds of categories. At this stage, I'm inclined to add only categories that we have pages for. On the other hand, often people only create pages because there are links to them, so perhaps people should also add categories that they think are important and conspicuously missing? As long as the page isn't flooded with a huge list, it will be manageable. And maybe we could even manage a huge list? I'd like to hear what John thinks. (Sorry to be so indecisive!)
Automatic logging is not an option with Instiki, which is why we use latest changes instead. You could request that it be added as a feature to Instiki, but (unless you can program it) don't hold your breath. However, note that there is a page Recently Revised; it's disabled now, but that should be temporary. (Try this one to see how it's supposed to work.) When it comes back, it will take care of much of the need for automatic logging, requiring people only to report major changes that others need to know about.
Rafael: I was wondering what happened to Recently Revised. Now i know it is not improved to Latest changes and will be back. Thx for the information.
John Baez: I don’t see any use of listing categories here unless one includes information about their basic categorical properties: for starters, which sorts of limits and/or colimits they have:
whether they admit interesting monoidal structures, whether they are cartesian (or monoidal) closed, and whether they are topoi, quasitopoi or allegories. In short: the basic properties one instantly wants to know about any category one meets.
My idea is that this article should be a quick place to look up basic categorical properties of categories, including categories that may not have their own entry yet. The entries here should not become full-fledged articles on the categories involved, since we already have or can create those.
By the way: people have been talking to Rafael on Recently Revised for many weeks now, but he doesn’t seem to reply to their comments there, which is making some people annoyed.
Toby: I can see that a list of categories with properties could be useful.
John, you don't mean Recently Revised; you mean latest changes. (Your link will break when Recently Revised itself comes back.)
Rafael, I don't know if you've been checking latest changes, but I often log your edits there after I see them. But I don't see why anyone would get upset that you don't read their commentary if they leave it on the wrong page.
Rafael: I am sorry to have missed the logging but i only recently by accident noticed that you also log at Latest Changes. I thought that all changes were in Recently Revised. Since Toby is logging for me, thanks Toby, i will try to remember to do it myself so others don’t have to do it for me.
John: Sorry, I meant latest changes. It’s a good way to discuss what’s going on with various new entries.
Andrew: Actually, it’s a very bad way to discuss what’s going on, but an okay way to announce what’s going on. But that’s another battle for another day. What I’m really here for is to ponder the format of this page a little. At the moment, we can’t do a proper database-based page of all this stuff so we have to think of the next best thing. Is the current format most useful? Or would a table with loads of ‘X’s be more appropriate? The categories would, of course, be linked to their pages where one could find out more. For example, here we would merely record that, say, Rel didn’t have all limits and colimits. On the actual page there would be an example showing why not. One would need a few “codes” for the entries along the lines of “Yes”, “No”, “Don’t Know”, “Sort of Yes”, “Sort of No”. That sort of thing. One could then search by name using the browser search. What you wouldn’t get, that you would get from a database, would be the ability to, say, only show complete categories. But how likely is that to be a request? I’d expect the main use to be “I don’t remember if Frolicher spaces are locally cartesian closed or not.”. I would think that a table format would be more useful than this list format, but I may be in a minority here (I usually am) and I’d rather not go to the effort of putting it in that format only to find no-one else likes it.
Zoran Škoda: I agree with John that the main purpose is to list many properties of well known categories, rather than listijng definitions of exotic categories. Those categories which already have their page on the nlab could be expanded on those pages rather than here. For some not only existence of limits but also a construction of limits can be useful.
Toby: I agree with Andrew about everything from ‘Actually,’ to ‘it.’. However, making a table with specific columns can be somewhat limiting, so we would need to make clear up top that adding new columns is fine. Blank entries in the table should default to ‹Nobody has bothered to put the answer here yet.›, which should be distinguished from ‹This is an unsolved problem.›. Other useful entries besides ‘Yes’ or ‘No’ might be ‘if and only if $R$ is noetherian’, ‘equivalent to the ultrafilter principle’, and the like. Some standard abbreviations might be useful here; tables can be hard to deal with when they get too wide.
Rafael: I have been thinking why the database is not growing. As for the existence of (co)limits, it must be because most existence results are hard to find, or at least hard enough to not search for them systematically. So, if some of you reading this could present methods (even theorems that assert the existence under some conditions or characterization theorems) to prove such existences, one could check it oneself and include it in the database. The point is that i don’t know any method except the one John Baez used (but i do know some useful but limited theorems such that toposes are finitely complete and finitely cocomplete). You can see where this is going? A lot of theorems (except for the methods) that imply properties of categories from other categories or other properties are needed. Then, we have to assume that the database will grow and a table would be much to cumbersome. I remember someone at nLab could write the database in Ruby, is this offer still on?