Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: representability of multirepresentable functors

Amar Hadzihasanovic (Apr 24 2021 at 13:00):

(This is more of a question-out-of-curiosity, not something I need for research)

One way of saying that a category has ($I$-indexed) products is to say that the functor $\prod_{i\in I} \mathrm{Hom}(-,X_i)$ is representable for all $I$-indexed families of objects $\{X_i\}_{i \in I}$. So this kind of representability is fairly common.

What happens if we ask, instead, that $\sum_{i\in I} \mathrm{Hom}(-,X_i)$ be representable?
I don't think I've really met this “in the wild”. Are there “reasonable” examples of categories where this holds?

These functors are called “multirepresentable” so the question can be rephrased as “what are examples of categories where multirepresentability implies representability?”.

Amar Hadzihasanovic (Apr 24 2021 at 13:06):

I'm also happy with no-go theorems like “these properties of a category are incompatible with the representability of multirepresentables”.

Mike Shulman (Apr 24 2021 at 14:34):

This can never hold unless $|I|=1$.

Suppose $S$ is a representing object. The natural isomorphism $\hom(-,S) \cong \sum_{i\in I} \hom(_,X_i)$ corresponds, by the Yoneda lemma, to an element of $\sum_{i\in I} \hom(S,X_i)$, which consists of a particular $i_S\in I$ and a morphism $g:S\to X_{i_S}$. The fact that this is an isomorphism means that every map $Y\to X_i$, for any $i$, must correspond to a map $Y\to S$, which by naturality corresponds to a map $Y\to X_{i_S}$; in other words, there are no maps $Y\to X_i$, for any object $Y$, unless $i=i_S$. Taking $Y=X_i$ for some $i\neq i_S$ we find a contradiction from its identity map, unless $i_S$ is the only element of $I$.

Amar Hadzihasanovic (Apr 24 2021 at 18:39):

Oh, that's right, thanks Mike!

There is something that still puzzles me, as I found the existence of non-trivial examples at least intuitively plausible. A logical interpretation of such a representing object would be a proposition $S$ whose each proof is a proof of exactly one of the $X_i$, which sounds like a propositions-as-types interpretation of a generalised exclusive or. This feels like a somewhat “classical” connective so I was expecting it to have quite degenerate categorical semantics... but at least some semantics.
The issue seems to be that having a natural isomorphism forces a “uniform” choice of which one of the $X_i$ is proven, which in turn would force all the other $X_j$ to “have no proofs in any context whatsoever”, which is impossible because every proposition can be proven from itself.

So I'd like to amend the question and ask if there are (weaker) ways to give consistent categorical semantics to the intuition above?

Mike Shulman (Apr 24 2021 at 19:14):

Amar Hadzihasanovic said:

A logical interpretation of such a representing object would be a proposition $S$ whose each proof is a proof of exactly one of the $X_i$, which sounds like a propositions-as-types interpretation of a generalised exclusive or.

That's not what it sounds like to me. For an exclusive or I would expect a proof to be something like a proof of one $X_i$ together with a refutation of all the others. (And that's exactly what it is, in a Chu construction.)

To me, "a proposition $S$ whose each proof is a proof of exactly one of the $X_i$" sounds like a description of a disjoint coproduct. This doesn't quite match your proposed universal property, but I would say the fault lies with the latter: it fails take into account that the hypotheses of such a proof might also involve disjunctions, so that a proof might involve case splits that choose a different one of the $X_i$'s to prove in each branch.

Amar Hadzihasanovic (Apr 25 2021 at 04:15):

Mike Shulman said:

That's not what it sounds like to me. For an exclusive or I would expect a proof to be something like a proof of one $X_i$ together with a refutation of all the others. (And that's exactly what it is, in a Chu construction.)

Yes, good point. I had (vaguely!) in mind some weaker form where instead of “evidence of $X_i$ and evidence of $\neg X_j$ for $j \neq i$” we ask for “evidence of $X_i$ and no evidence of $X_j$ for $j \neq i$”.

I think the kind of examples I was thinking of would have objects that have different “components” with the property that any morphism into them can access only a single component. But I had missed the obvious fact that the identity morphism could never satisfy this property, and that's crucial in using Yoneda in your counterargument.

Amar Hadzihasanovic (Apr 25 2021 at 04:19):

But now I think there may be examples that form semicategories, i.e. non-unital categories. One such toy example would have sets as objects, and functions whose image is a single element as morphisms. This is not a subcategory of $\mathbf{Set}$ because the only identity it can contain is the one on a single-element set, but it is closed under composition, so it is a sub-semicategory.

Amar Hadzihasanovic (Apr 25 2021 at 04:21):

Then the disjoint union of sets does satisfy the representability condition I was asking for. This is admittedly a “cheating” example because essentially it forces $\mathrm{Hom}(Y,X)$ to be equal to $\mathrm{Hom}(1,X) \simeq X$ for all nonempty $Y$.

Amar Hadzihasanovic (Apr 25 2021 at 04:22):

So then the isomorphism that I'm asking for is the “obvious” isomorphism $\sum_i \mathrm{Hom}(1,X_i) \simeq \mathrm{Hom}(1, \sum_i X_i)$.

Amar Hadzihasanovic (Apr 25 2021 at 04:28):

But there may be more interesting semicategories that can do this sort of “compartmentalisation” of incoming morphisms.

Mike Shulman (Apr 25 2021 at 15:24):

Perhaps you could make any extensive category into a semicategory by restricting the morphisms to factor through a connected object. (But I would prefer to just work with the extensive category, and modify the intuition to "any morphism with connected domain can only access a single component".)