When is an object determined by the number of maps from the other objects?

Question

Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$?

https://mathoverflow.net/a/269745 seems to gesture towards an (unpublished) answer — Martti Karvonen, Sep 29 '20 at 20:21
after a bit of digging, I discovered that a sufficient condition based on similar factorization properties is proven in A. Pultr: Isomorphism types of objects in categories determined by numbers of morphisms. Acta Sci. Math. Szeged35(1973), 155–160 — Martti Karvonen, Sep 29 '20 at 20:32
@martti : thanks for the very useful references. In fact I had a similar idea of proof based on factorization. So it seems that it is true for instance for $C = Set_f^G$ for any finite category $G$ (as well as for their dual). — Claudio Pisani, Sep 30 '20 at 18:13

varkor · Accepted Answer · 2022-07-13T21:06:36.877

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The first question is the main topic of two recent papers:

Reggio's Polyadic Sets and Homomorphism Counting, which is also a good reference for previous results in the literature (including that of Pultr mentioned in the comments). In particular, see Theorems 4.3, 5.10, and 5.12, the latter of which generalise beyond locally-finite categories.
Fujino–Matsumoto's Lovàsz's hom-counting theorem by inclusion-exclusion principle. In particular, see Theorems 2.11 and 2.12.

As far as I understand, the theorems give orthogonal sufficient conditions: it's unclear whether there is a general theorem subsuming both.

I don't believe there are answers for enriched categories yet.

edited Jul 13 '22 at 21:06

answered Jul 13 '22 at 14:47

varkor

8,685

One should also mention Lovàsz's hom-counting theorem by inclusion-exclusion principle by Shoma Fujino and Makoto Matsumoto (https://arxiv.org/abs/2206.01994). – Ivan Di Liberti Jul 13 '22 at 20:08
And Locally-finite extensive categories, their semi-rings, and decomposition to connected objects by the same authors (https://arxiv.org/abs/2207.05702). – Ivan Di Liberti Jul 13 '22 at 20:09
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@IvanDiLiberti: thanks, I forgot about that paper – I've updated my answer to include it. – varkor Jul 13 '22 at 21:07

When is an object determined by the number of maps from the other objects?

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