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Stream: theory: category theory

Topic: Categories which admit enrichment in Hausdorff spaces

Eigil Rischel (Nov 04 2025 at 15:41):

Suppose $J$ is some class of limit shapes (for example, all countable limits). Let $C$ be a category which has all the limits in $J$. Suppose we are given an enrichment of $C$ over topological spaces (a topology on each homset so that composition is continuous in the product topology). Then we can ask whether the limits in $J$ are compatible with this enrichment, in the sense that the bijection $Hom(X, \lim_i Y_i) \cong \lim_i Hom(X,Y_i)$ is a homeomorphism (again, giving the right-hand side the topology of the limit in $\mathsf{Top}$. Call this a compatible enrichment.

Clearly every category admits at least one compatible enrichment, namely the indiscrete one. It's not hard to see that (using Zorn's lemma) there is a maximal compatible enrichment (in the sense of having the most open sets). I'm interested in categories whose maximal such enrichment is Hausdorff (that is, every hom-space is Hausdorff). This is clearly equivalent to the existence of just one compatible Hausdorff enrichment.

This has some non-trivial implications about $C$. For example, suppose $J$ is the set of all countable limits. Let $f_n: A \to \prod_i B_i$ be a sequence of maps into a product, writing $f_n^i : A \to B_i$ for the components. Suppose the sequences $f_n^i$ all stabilize at some point, $f_n^i = f_\infty^i$ when $n > N_i$. Then these converge, so in any compatible Hausdorff enrichment, $f_n$ also converges to $f_\infty$ (with components $f_\infty^i$). Let $g: \prod_i B_i \to C$ be any map. Then if $g f^n = h$ for all $n$ (that is, this composite is constant), we must also have $g f^\infty = h$. But this implication fails in the category of sets.

Clearly one condition that implies this if if every object can be written as a $J$-limit of $J$-cosmall objects (i.e those where a map out of a $J$-limit into them always factors over a projection). But, for example, I think the category of compact Hausdorff spaces has the above property but is not countably cogenerated in this sense.

Does anyone know of previous work studying this property, or an equivalent formulation in terms of "pure" category theory? (I was a bit vague above, but it's fine if $J$ is just given by the limits smaller than some regular cardinal.)

Morgan Rogers (he/him) (Nov 06 2025 at 09:27):

You said that the existence of a Hausdorff compatible enrichment implies that a maximal one must be Hausdorff... are you saying that there is a unique maximal one?

Eigil Rischel (Nov 07 2025 at 18:32):

My thinking was that the topology generated by the union of two compatible enrichments will itself be compatible, but I can't remember how I convinced myself of that (I guess if it's true, you don't even need Zorn's lemma)

Morgan Rogers (he/him) (Nov 08 2025 at 18:27):

Exactly, that would be a much nicer situation to be in. Even if it isn't true that there is a maximal one, it might be that the kinds of property you've mentioned are enough to guarantee the existence of an enrichment. This seems like a fun problem! What brought you to it?