On the large cardinals foundations of categories

Solution 1:

Crossposted from MO.

Allow me to make some comments as someone who converted to the universeful approach recently; but take it with a pinch of salt, as I have only been studying category theory for 2½ years.

I should briefly mention the trigger that led me to the pro-universe camp: about 6 months ago, I started learning about quasicategories and became convinced that the theory relied on far too much machinery to admit a workable elementary approach that would allow us to consider entities like the quasicategory of all Kan complexes within an NBG-like framework – which is the the raison d'être for quasicategories in the first place! So I decided that I would have to start taking universes seriously – and for this purpose the universes would not merely be for bookkeeping but for doing actual mathematics in.

If all we wanted to do was to ensure that sets of large cardinality and high complexity exist, it would probably be enough to just have a chain $M_0 \in M_1 \in M_2 \in \cdots$ of (transitive) models of set theory (say, ZFC), but I find this very unsatisfactory. For instance, consider theorems that assert that some object with some universal property exist: there is no guarantee that a universal object in $M_0$ remains universal in $M_1$. Indeed, one such theorem says that, for every set $X$, there exists a set $\mathscr{P}(X)$ and a binary relation $[\in]_X \subseteq X \times \mathscr{P}(X)$ such that, for every binary relation $R \subseteq X \times Y$, there is a unique map $r : Y \to \mathscr{P}(X)$ such that $\langle x, y \rangle \in R$ if and only if $\langle x, r(x) \rangle \in [\in]_X$. Of course, it is well-known that powersets need not be preserved when passing from one model of set theory to another. But if something as trivial as powersets is not preserved, what hope is there of preserving more complicated universal objects like the free ind-completion of a category, or even the free monoid on one generator (i.e. $\mathbb{N}$!)? For a category theorist, to work in a setting where universal objects have to be qualified by a universe parameter is simply untenable.

So, we have to find some kind of compromise: we need a chain of universes such that each universe is embedded in the next in as pleasant a way as possible, so that the mathematics in one universe agrees with the next as much as is feasible. The most ideal situation one could hope for goes something like this:

Let $\mathcal{C}$ be a category scheme, i.e. a definable function that assigns to each sufficiently nice universe $\mathbf{U}$ a category $\mathcal{C}(\mathbf{U})$ such that, for any universe $\mathbf{U}^+$ with a sufficiently nice embedding $\mathbf{U} \subseteq \mathbf{U}^+$, $\mathcal{C}(\mathbf{U})$ is a subcategory (in the strict sense) of $\mathcal{C}(\mathbf{U}^+)$.

We say that an inclusion $\mathbf{U} \subseteq \mathbf{U}^+$ is adequate for a category scheme $\mathcal{C}$ if the following conditions are satisfied:

1. $\mathcal{C}(\mathbf{U})$ is a full subcategory of $\mathcal{C}(\mathbf{U}^+)$: we do not get any new morphisms between objects in $\mathcal{C}(\mathbf{U})$ when passing to a larger universe.
2. The inclusion $\mathcal{C}(\mathbf{U}) \hookrightarrow \mathcal{C}(\mathbf{U}^+)$ preserves all limits and colimits that exist in $\mathcal{C}$ for $\mathbf{U}$-small diagrams: the most elementary kind of universal constructions are preserved when passing to a larger universe.
3. Moreover $\mathcal{C}(\mathbf{U})$ is closed in $\mathcal{C}(\mathbf{U}^+)$ under all limits and colimits for $\mathbf{U}$-small diagrams: so passing to a larger universe does not create new universal objects where none existed before.

So, when is $\mathbf{U} \subseteq \mathbf{U}^+$ adequate for the category scheme $\mathbf{Set}$, if $\mathbf{U} \in \mathbf{U}^+$? We take it as given that $\mathbf{U}$ and $\mathbf{U}^+$ are transitive models of ZFC. The first condition implies that this extension has no new subsets, and if there are no new subsets, there are no new functions either – so (1) holds if and only if the extension preserves powersets. By considering explicit constructions, one sees that equalisers are preserved, and as soon as we know (1), products are also preserved. We may then use the monadicity of $\mathscr{P} : \mathbf{Set}^\mathrm{op} \to \mathbf{Set}$ to deduce that colimits are preserved, so (1) implies (2). In particular, directed unions are preserved, so it follows that $\mathbf{U}$ embeds as an initial segment of the cumulative hierarchy of $\mathbf{U}^+$. What about (3)? Well, take a $\mathbf{U}$-set $I$ and a map $X : I \to \mathbf{U}$ in $\mathbf{U}^+$. By replacement in $\mathbf{U}^+$, we can form the disjoint union $\coprod_{i \in I} X(i)$, and if (3) holds, the cardinal of this set is in $\mathbf{U}$. Thus, $\mathbf{U}$ must actually be embedded as a Grothendieck universe in $\mathbf{U}^+$. Conversely, if $\mathbf{U}$ is embedded as a Grothendieck universe in $\mathbf{U}^+$, then (1), (2), and (3) are easily verified.

But are Grothendieck universes enough? For instance, it would be good if the following were true:

Let $\kappa$ be the smallest (uncountable strongly) inaccessible cardinal, let $\mathbb{B}$ be a $\mathbf{V}_{\kappa}$-small category, and let $\mathcal{M}$ be the category scheme obtained by defining $\mathcal{M}(\mathbf{U})$ to be the free ind-completion of $\mathbb{B}$ relative to $\mathbf{U}$ for each Grothendieck universe $\mathbf{U}$. Suppose $\mathcal{M}(\mathbf{V}_{\kappa})$ admits a cofibrantly generated model structure. Then, for all Grothendieck universes $\mathbf{U}$, there is a (unique) cofibrantly generated model structure on $\mathcal{M} (\mathbf{U})$ extending the one on $\mathcal{M}(\mathbf{V}_{\kappa})$.

However, I do not know if this is true. What I do know is that embeddings of Grothendieck universes are adequate for $\mathcal{M}$; in fact, locally presentable categories and adjunctions between them are very well behaved under this kind of universe enlargement.