Limits, colimits and universes

Question

For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and morphisms are actually sets.

However, what if instead of sets and classes we want to adopt the foundational system with Grothendieck universes? Do we even need the smallness condition anymore?

For instance, in the framework of classes (without universes) we say that a category $\mathsf{C}$ is complete (resp., cocomplete) if limit (resp., colimits) of all diagrams indexed by small categories exist. What are the appropriate versions of these notions in the framework of universes?

Answer 1

Probably it is not a very common terminology, but I'm used to the following natural definitions:

Definition 1. Let $\mathcal{A}$ and $\mathcal{B}$ be categories. The category $\mathcal{A}$ is called $\mathcal{B}$-complete ($\mathcal{B}$-cocomplete) iff for every functor $\mathcal{F}\colon\mathcal{B}\to\mathcal{A}$ there exists a projective (inductive) limit of $\mathcal{F}$.

Definition 2. Let $\mathcal{U}$ be a Grothendieck universe, $\mathcal{A}$ be a category. The category $\mathcal{A}$ is called $\mathcal{U}$-complete ($\mathcal{U}$-cocomplete) iff it is $\mathcal{B}$-complete ($\mathcal{B}$-cocomplete) for every $\mathcal{U}$-small category $\mathcal{B}$.

(Note, that the definitions 1 and 2 don't disagree with each other, because from the set-theoretic point of view, a universe cannot be a category).

Thus the framework of universes allows us to consider different types of small-completeness. All results concerning limits/completeness presented in the class-theoretic framework can easily be transferred (and even generalized) to the framework of universes. As an example, every $\mathcal{U}$-small $\mathcal{U}$-complete category is a complete preorder. By the Grothendieck's axiom, every category is $\mathcal{U}$-small for some universe $\mathcal{U}$, therefore categories which have all limits are not very interesting (they are preorders).

Another picture arises when we talk about the preservation (reflection etc) of limits (colimits). For instance, every right adjoint functor preserves all limits (is $\mathcal{U}$-continuous for every Grothendieck universe $\mathcal{U}$). Another example: fully faithful functors reflect all limits. It is worth to note, that these statements are invariant under changes of foundations (universes or classes).

Answer 2

As explained by Joel in his answer here the existence of a Grothendieck universe $\mathcal{U}$ is exactly equivalent to the existence of one strongly inaccessible cardinal $\kappa$, with the Grothendieck universe being $V_\kappa$ (the universe of sets of hereditary cardinality less than $\kappa$), and this relationship has been studied in set theory for over a century.

Over at his answer here Joel explains that a category theorist may actually want an elementary chain of universes $$V_\gamma\prec V_\delta\prec\dots\prec V_\lambda\prec\dots$$ where $\gamma<\delta<\lambda<\dots$ are all strongly inaccessible cardinals, $V_\gamma,V_\delta,V_\lambda,\dots$ are the universes hereditarily below these cardinals, and the relation $\prec$ indicates that truth is preserved from one universe to the next so the whole chain of universes will ultimately agree on the truth of a given statement.

As Joel shows at his second linked answer (which I highly recommend along with the first), the consistency strength of such a chain is strictly above the existence of one Grothendieck universe but below the existence of a Mahlo cardinal.

To address your question more directly, we can formalize the notion of 'smallness' for a diagram relative to any of the $V_\kappa$ for $\kappa$ inaccessible, and if we need larger universes outside $V_\kappa$ we can use the next member up in the chain, so on and so forth.