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Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we call $T$ varietal if $\mathscr T$ is locally small. A large equational theory may be presented by a class $\Omega$ of set-indexed (i.e. possibly infinitary) operations and a class $E$ of equations.

May the presentations of varietal large equational theories be neatly characterised? If not, is there a fundamental obstacle to doing so?

In the case that there is not a precise characterisation, I would also be interested in necessary, and sufficient, conditions for a large presentations to be varietal. One sufficient condition is that $\Omega$ be bounded in that there exists some cardinal $\kappa$ such that every operation has arity less than $\kappa$, but I would like to know if there are stronger conditions: for instance, one that implies the theory of sup-lattices, or of compact Hausdorff spaces, is varietal.

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    $\begingroup$ I think you're asking the following: We're given a presentation $(\Omega, E)$, where $\Omega$ is a specification of a class of function symbols for each arity $n \in CARD$ and $E$ is a class of equations in the language $\Omega$. We want to determine whether the category of free algebras $\mathcal T$ for this presentation is varietal, i.e. locally small. Right? One thing to note is that it's equivalent to ask whether the underlying set of each free algebra $T \in \mathcal T$ is small. Another theory one might like to treat would be compact Hausdorff spaces (in terms of ultrafilter convergence) $\endgroup$
    – Tim Campion
    Commented Sep 7, 2020 at 16:55
  • $\begingroup$ @TimCampion: yes, that's right. Compact Hausdorff spaces is another example I would be interested in: I'll update the question, thanks. $\endgroup$
    – varkor
    Commented Sep 7, 2020 at 17:05

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The following are equivalent:

  1. $\mathcal T$ is varietal

  2. Each free algebra $T \in \mathcal T$ is small

  3. For each arity, there are a small number of $E$-equivalence classes of words in the language $\Omega$.

$(1) \Rightarrow (2)$ holds because the underlying set of an algebra can be identified with morphisms from the free algebra on 1 generator. $(2) \Rightarrow (1)$ holds because there are a small set of functions between any two small sets. The equivalence between $(2)$ and $(3)$ is hopefully clear.

I think there may not be much more to say at this level of generality: In cases where condition (3) is hard to check directly, then I suspect one might need to use more information about how $(\Omega, E)$ is presented in order to get a more usable criterion.

For instance, if $(\Omega, E)$ gives rise to a varietal theory, then it must be equivalently axiomatizable in the form $(\Omega', E')$ where $\Omega'$ has a small number of operations of each arity. But this condition is not sufficient.

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  • $\begingroup$ "I think there may not be much more to say at this level of generality" — it's also not clear to me. However, it doesn't seem unreasonable that there might be at least sufficient syntactic conditions that cover examples like the powerset monad and CHS monad. It is perhaps helpful to rephrase the condition that $T$ be varietal in these ways, but these reformulations are very direct, and it's not clear that there aren't stronger results at this level of generality. $\endgroup$
    – varkor
    Commented Sep 7, 2020 at 18:08
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    $\begingroup$ It might be interesting o look at the old paper of J. Reiterman dml.cz/bitstream/handle/10338.dmlcz/106455/… $\endgroup$
    – Jiří Rosický
    Commented Sep 8, 2020 at 10:41

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