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What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible?

More generally, what is an example of an accessible category $\mathcal C$ (not essentially small) such that one of the following related conditions holds?

  • $\mathcal C^{op}$ continuous (i.e. $\mathcal C$ has cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ has a left adjoint);

  • $\mathcal C^{op}$ is precontinuous (i.e. $\mathcal C$ has colimits and cofiltered limits and the colimit functor $Ind(\mathcal C^{op}) \to \mathcal C^{op}$ preserves limits);

  • $\mathcal C$ has finite colimits and cofiltered limits, and they commute;

  • $\mathcal C^{op}$ is finitely accessible.

The closest thing to an example I can think of is the category $Hilb$ of Hilbert spaces and contractive maps, which is a self-dual $\aleph_1$-accessible category. But I don't believe that finite limits commute with filtered colimits in $Hilb$.

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    $\begingroup$ I've accepted Ivan's answer, which addresses the title question and hence also the first and last bullet points, but I'd be very interested in seeing more examples. $\endgroup$
    – Tim Campion ♦
    Jan 2, 2020 at 23:11
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    $\begingroup$ I think you've made a slip-up in the last paragraph - $\mathbf{Hilb}$ is $\aleph_1$-accessible, but not locally $\aleph_1$-presentable, nor locally $\kappa$-presentable for any cardinal $\kappa$, on account of the fact that it is self-dual and not a preorder. $\endgroup$
    – Robert Furber
    Jan 3, 2020 at 18:15
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    $\begingroup$ @RobertFurber Thanks, fixed! That was a particularly pernicious error since the theorem you're referring to -- the fact that a category $C$ and its opposite can't both be locally presentable unless $C$ is a preorder -- is in some sense the big obstruction that the whole question is dancing around! $\endgroup$
    – Tim Campion ♦
    Jan 3, 2020 at 23:12

2 Answers 2

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In Accessible Categories: The Foundations of Categorical Model Theory by Makkai and Paré, there is the example of a finitely accessible self-dual category. Apparently the example is due to Isbell. This is the category of sets and partial monomorphisms. The example appears right after Prop. 3.4.4 and right before 3.4.5.

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  • $\begingroup$ Oh wow, thanks! I've been through this one before, too -- should have remembered! $\endgroup$
    – Tim Campion ♦
    Jan 2, 2020 at 22:57
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    $\begingroup$ @TimCampion, I relate so much to that. I keep rediscovering stuff I know since forever every day. Is it too early to complain about getting old? $\endgroup$
    – Ivan Di Liberti
    Jan 2, 2020 at 23:01
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    $\begingroup$ "So that as Plato had an imagination, that all knowledge was but remembrance; so Solomon giveth his sentence: that all novelty is but oblivion." $\endgroup$
    – fosco
    Jan 3, 2020 at 10:56
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Any locally presentable category where epimorphisms are stable under $\lambda$-codirected limits is equivalent to a complete lattice (see http://www.tac.mta.ca/tac/volumes/33/10/33-10.pdf, 3.10).

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  • $\begingroup$ Thanks! So in particular, that means that the opposite of a locally presentable category can never be precontinuous, giving a negative answer to the second bullet point. $\endgroup$
    – Tim Campion ♦
    Jan 5, 2020 at 17:41

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