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I asked a closely related question on Mathematics Stackexchange but got no answer.

Let $\mathbf1$ be a category with exactly one object and one morphism, and, for any category $\mathcal C$, write $\mathcal C^{\mathcal C}$ for its category of endofunctors.

I previously asked if the equivalence $\mathcal C^{\mathcal C}\sim\mathcal C$ implies that $\mathcal C\sim\mathbf1$, but got no complete answer.

In the meantime I realized that Theorem 3 in Complete lattices and the generalized Cantor theorem by Roy O. Davies, Allan Hayes and George Rousseau, published in Proc. Amer. Math. Soc. 27 (1971), 253–258, link, shows that the implication $\mathcal C^{\mathcal C}\sim\mathcal C\implies\mathcal C\sim\mathbf1$ does hold for posets.

But the theorem in question is much stronger! Indeed it shows that, for a poset $X$, the existence of an injective morphism $\text{End}(X)\to X$ implies that $X$ is a singleton, and similarly for a surjective morphism $X\to\text{End}(X)$.

This suggests the following questions:

Let $\mathcal C$ be a category.

Question 1 Does the existence of a fully faithful functor $\mathcal C^{\mathcal C}\to\mathcal C$ imply $\mathcal C\simeq\mathbf1$?

Question 2 Does the existence of an essentially surjective functor $\mathcal C\to\mathcal C^{\mathcal C}$ imply $\mathcal C\simeq\mathbf1$?

Of course a positive answer to any of these questions would also solve the previously asked question.

[Edit: The comments refer to a previous version of the question.]

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  • $\begingroup$ Why the downvote? $\endgroup$
    – Pierre-Yves Gaillard
    Commented Jul 31, 2019 at 12:58
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    $\begingroup$ I didn't downvote, but I could imagine a downvote because you've basically just said "here are some properties; what implications hold among them?" without indicating why the properties, or the implications among them, are interesting, or what partial results you've got. $\endgroup$
    – LSpice
    Commented Jul 31, 2019 at 13:22
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    $\begingroup$ @LSpice - Thanks for your comment. The main partial result is that (as indicated in the post) the properties are equivalent for posets. I had previously asked if (P2) implies (P1). Apparently nobody knows. So I thought that answers to closely related questions might shed some light on the question asked before, and that, asking many such related questions was a way of maximizing the probability of getting some of them answered. I wish I had better ideas about how to tackle the main question. $\endgroup$
    – Pierre-Yves Gaillard
    Commented Jul 31, 2019 at 13:49
  • $\begingroup$ I missed that motivation, since, to be honest, I just started skimming once I saw a big yellow block of questions all of the same form. It might make sense to move it higher, so that people see why you're asking the questions before, not after, you ask them. $\endgroup$
    – LSpice
    Commented Jul 31, 2019 at 13:54
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    $\begingroup$ I didn't downvote either, but it's very often the case that asking a whole bunch of questions results in the uncomfortable situation where no one answer answers them all, so that no answer should be the accepted answer. This is perhaps not fatal, as long as someone is willing to collate the individual answers later into a summary comprehensive answer (perhaps made CW to preserve decorum) which can then be accepted. But putting lots of questions into a post is probably better avoided in general. $\endgroup$
    – Todd Trimble
    Commented Jul 31, 2019 at 15:30

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