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Following from the discussion begun here, I thought I’d air a few simple ideas on how one might construct a ’free co-completion’ internally to a 2-category, without referring to a category of sets (the idea being that one could then define $\mathsf{Set}$ to be the free co-completion of $1$). I’ll try to add details gradually, but I thought I’d just share the ideas anyway; maybe people can develop them or see problems quicker than I will.
So suppose that we have some $2$-category $C$ (a prototypical example being $\mathsf{CAT}$) with an object $c$ which we’ll think of as analogous to $\mathsf{Cat}$ in $\mathsf{CAT}$. Let $a$ be some other object of $C$. I wish to define the free co-completion of $a$ with respect to $c$.
To get some idea for what we should do, consider a functor $F : D \rightarrow \mathcal{A}$ between categories. How do we freely construct a colimit of $F$ in $\mathcal{A}$? Well, it seems to me that we should be able to proceed by first forming the category $\mathsf{Co-Cone}$ of co-cones of $F$, and then formally add an initial object. The category of co-cones of $F$ is of course a certain comma category, i.e. a 2-limit in $\mathsf{CAT}$. Adding an initial object is a certain 2-colimit in $\mathsf{CAT}$, if I am not mistaken. Thus, for an individual functor, it seems that we can construct a free colimit of it internally to a 2-category.
What we now need to do is to carry out this construction for all functors at once. In other words, we should apply some construction to the slice category $\mathsf{Cat} / \mathcal{A}$ (which is again a certain 2-limit) which on objects recovers the construction of the previous paragraph. I don’t see any significant problems with doing this, but I’ll stop here for now.
Thoughts very welcome!
It doesn’t seem to me to solve any problems if you define $Set$ but assuming from the start that you’re given $Cat$. I would want instead to characterize $Set$ as a free cocompletion with respect to some given class of “small weights” in $C$, either two-sided fibrations or proarrows in an equipment.
My thinking was that one is only allowed to use purely 2-categorical means to treat $\mathsf{Cat} / \mathcal{A}$, so that one could replace $\mathsf{Cat}$ by any sufficiently nice $C$ (or maybe any $C$ at all). The significance of $\mathsf{Cat}$ is only that in this case we should recover something which is exactly the classical free co-completion.
Also, my feeling was that in a categorical foundations, it is reasonable to assume that one has universes, i.e. a copy of $\mathsf{CAT}$ living inside $\mathsf{CAT}$, which we call $\mathsf{Cat}$ (and obviously there could be further levels if needed). Indeed, my thought is that one has to have some kind of universe if one is to get something like $\mathsf{Set}$: one cannot get something from nothing.
As a third thought, for me at least, it would be interesting to have a purely 2-categorical treatment of free co-completion, even if it does not fully solve the kind of problem we are discussing.
Thus I am very interested to understand your suggested characterisation. It may well improve significantly on what I have just described, and I would like to understand how! Would you be able to elaborate?
For what it’s worth, for many years I’ve wanted to explore the semantics of what I’ve called “epistemologies”, which essentially takes free cocompletion as basic. The basic set-up is a bicategory $\mathbf{B}$ (understood as playing a role of bicategory of enriched categories and bimodules/profunctors) for which the inclusion $i: Ladj(\mathbf{B}) \hookrightarrow \mathbf{B}$ of the subbicategory consisting of objects of $\mathbf{B}$, left adjoint 1-cells, and 2-cells between them, has a right biadjoint $p: \mathbf{B} \to Ladj(\mathbf{B})$, or better yet a Kock-Zoeberlein right biadjoint. I call a $\mathbf{B}$ with this property a potent bicategory. Here $Ladj(\mathbf{B})$ is understood as playing a role of enriched categories and functors, and the monad $p i$ plays the role of a free cocompletion.
Things get more and more interesting when we put more assumptions on $\mathbf{B}$. For example, if $\mathbf{B}$ has a compact closed structure (in the bicategorical sense), then the object $v = p i(1)$, the free cocompletion of the monoidal unit, really does play a role of base of enrichment, and one can develop formal enriched category theory in this context. If $\mathbf{B}$ is moreover a cartesian bicategory, then I think things get more interesting still.
I had wondered for a long time whether we could get an interesting semantics for potent compact closed cartesian bicategories by looking at the examples of “small complete small categories” in realizability toposes as possible bases of enrichment. Mike seemed to think this ought to work. I regret letting that discussion drop!
My point was that if you’re going to assume universes anyway, what’s the point of assuming $Cat$ but not assuming $Set$?
Todd: interesting, thanks! I definitely think just taking free co-completion as basic is reasonable and promising as well.
Mike: ah, I see, thanks for the clarification. My feeling was that there is a big difference between assuming $\mathsf{Cat}$ and assuming $\mathsf{Set}$ in a categorical foundations (where by this I mean here one in which categories are primitive). I.e. if one is axiomatising the 2-category of categories, I think it is reasonable to have ’copies’ of that same axiomatisation living inside one another. To me it is actually not at all obvious that one easily get $\mathsf{Set}$ given such a universe. If one is axiomatising $\mathsf{Set}$ as well, then I would think that it is probably not a categorical foundations (in the same sense as I clarified in the earlier paranthesis) any more. On the other hand, it could be that Todd’s approach is better than the one I suggested, where one takes free co-completion more or less as primitive. But I’d be interested to know what kind of characterisation you had in mind.
To me it makes much more sense to have $Set$ as an object of the 2-category of categories than it does to have $Cat$. For one thing, $Cat$ is really a 2-category, not a category, hence not really an object of the 2-category of categories but rather the 3-category of 2-categories.
I think it should be fairly easy to obtain $Set$ from $Cat$ as the discrete objects, e.g. those that are right orthogonal to $2\to 1$.
What I had in mind was a universal characterization of $Set$ relative to a class of small profunctors, as in section 10 of this paper.
To me it makes much more sense to have $Set$ as an object of the 2-category of categories than it does to have $Cat$. For one thing, $Cat$ is really a 2-category, not a category, hence not really an object of the 2-category of categories but rather the 3-category of 2-categories.
I see your point. I’m not completely sure I agree, because I think small categories as objects of a 1-category only are quite fundamental (Indeed the very thing we are discussing here, namely for formulating the notion of a diagram, is an example). I think if one had $\mathsf{Set}$, one would probably need to build $\mathsf{Cat}$. But probably one could indeed build a perfectly good theory with $\mathsf{Set}$ axiomatised one way or another, and maybe that it is indeed preferable.
I think it should be fairly easy to obtain $Set$ from $Cat$ as the discrete objects, e.g. those that are right orthogonal to $2\to 1$.
That’s interesting. My feeling was that it was not obvious how to do this, thanks for the suggestion of how to do it. I had not considered right orthogonality as part of the axiomatic setup I had in mind, but maybe it should be there.
An interesting question (for me) would be whether we can show (or what we need to be able to show) that $\mathsf{Set}$ defined in this way is the free co-completion of $1$. Maybe one would end up needing something like the construction I was getting at in #1.
What I had in mind was a universal characterization of $Set$ relative to a class of small profunctors, as in section 10 of this paper.
This looks interesting. I have not really digested it yet, but it may well be that this kind of thing would be much more interesting and useful than the construction I was getting at, which is after all rather simplistic.
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