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In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, for this fact? In particular, I'm interested in the setting where "algebraic" refers to the multisorted, potentially infinitary setting.

To be precise, this would entail showing that there is no monadic functor from $\mathbf{Cat} \to \mathbf{Set}/S$ for any set $S$.

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    $\begingroup$ One way to see it is that the category of categories is not a regular category, while model of algebraic theory (even multisorted and infinitary) are regular, even exact categories. $\endgroup$
    – Simon Henry
    Commented Mar 14, 2020 at 20:58

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This follows from two Facts:

1) A category monadic over Set/S is always an exact category. That is it has quotient by equivalence relation that are effective and universal. It is in particular a regular category. This is showed for example here.

2) The category of categories is not a regular category. An explicit example of regularity (and hence exactness) failing in the category of categories (or posets, or topological spaces) can be found in the example section of the nLab page.

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    $\begingroup$ This is strictly speaking the correct answer, but to be more informative one should add that the theory of small categories is an essentially algebraic theory. ncatlab.org/nlab/show/essentially+algebraic+theory $\endgroup$
    – Paul Taylor
    Commented Mar 16, 2020 at 16:01
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    $\begingroup$ of course, but given that the question quote Palmgren & Vickers' paper, I assume this was considered as known. $\endgroup$
    – Simon Henry
    Commented Mar 16, 2020 at 18:09

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