Given a category $C$, I'll say that a set $J$ of families $\{f\_i\colon A\to B\_i\mid i\in I\}\;$ is a *co-coverage* if their opposites $\{f\_i^{op}\colon B\_i\to A\mid i\in I\}\;$ form a coverage on $C^{op}$. In this case, each family in $J$ will be called a *co-covering family* and the pair $(C,J)$ a *co-site*.


Let $T$ be a (possibly multisorted) algebraic theory. For each sort $t\in T$ and $n\in\mathbb{N}$, we have a product cone of projections $\pi\_i\colon t^n\to t$, which we declare should be preserved by co-presheaves. Thus we "want" them to be co-covering families. 


One could attempt to declare that such families $\{\pi\_i\colon t^n\to t\mid 1\leq i\leq n\}\;$ are the co-covering families (one family for each $t,n$), but this choice rarely seems to form a co-coverage. 


**Question:** What are some sufficient conditions under which the evident "product sketch" for an algebraic theory forms a co-coverage?


**Question:** What are some sufficient conditions under which the category $T-\mathbf{Alg}$ is a topos?



Which algebraic theories are co-sites?
In the following I'll speak of coverages on $T^\mathrm{op}$ rather than co-coverages on $T$.


Let me take the liberty of loosening your question. It seems a little funny to me to to ask that the finite coproduct cocones in $T^\mathrm{op}$ form a coverage on the nose. After all, the point of a coverage is to present a notion of sheaf, or equivalently a Grothendieck topology. And every set of cocones generates a Grothendieck topology by just throwing in all the additional sieves that the axioms of a Grothendieck topology tell you have to be in there. So let's just let $J$ be the Grothendieck topology on $T^\mathrm{op}$ generated by the finite coproduct cocones. In general there's no reason for this site to be subcanonical.


But what I think you're really driving at is the question:



> 
> When do the $J$-sheaves $T \to \mathsf{Set}$ coincide with the $T$-algebras (= the finite-product-preserving functors $T \to \mathsf{Set}$)?
> 
> 
> 


Note that the $J$-sheaves are always contained in the $T$-algebras. If they coincide, $T\mathrm{-Alg}$ is a Grothendieck topos on the site $(T^\mathrm{op},J)$ (and in particular $J$ is subcanonical). The converse is also true: to see this, note that the proof of Giraud's theorem actually shows that any set of generators for a topos can be taken as a subcanonical site of definition (and the Grothendieck topology is uniquely determined). So we can reformulate this question as:



> 
> When is $T\mathrm{-Alg}$ a (necessarily Grothendieck) topos?
> 
> 
> 


In the single-sorted case, Johnstone has given a [syntactic characterization](http://link.springer.com/article/10.1007%2FBF01188056?LI=true). That article is behind a paywall, but the jist of it is this. All of the hypotheses of Giraud's theorem are satisfied by $T\mathrm{-Alg}$ for any $T$, except for disjointness and pullback-stability of coproducts; Johnstone translates these conditions into the following:


**Theorem** (Johnstone):
If $T$ is single-sorted, then $T\mathrm{-Alg}$ is a topos if and only if $T$ is degenerate (i.e. the terminal category) or else the following criteria are satisfied:


1. $T$ has no pseudo-constants.
2. Every operation of $T$ is sufficiently unary.


A pseudo-constant is a unary operation $u$ such that $u(x) = u(y)$ for ally $x,y$. The definition of "sufficiently unary" is distributed throughout the paper in such a way that I can't easily extract it. But the idea is that $T\mathrm{-Alg}$ is the topos of $M$-sets when $T$ is the free category with finite products on a monoid $M$ -- i.e. if all the operations of $T$ are unary. Johnstone's perspective seems to be that you can't stray too far from this case and still have a topos.


I'm sure these conditions could all be formulated more categorically, and you could probably do it if you took some time with Johnstone's paper. Or maybe it would become clear just thinking about what disjoint and universal coproducts mean. I don't know how easy it would be to extend the results to the many-sorted case, although the basic observation -- that if $T$ is the free theory on a category then $T\mathrm{-Alg}$ is a topos -- remains true.


When $T$ is freely generated by a category $C$ with a sketched terminal object $1$, you will get a topos if the terminal object is strict, because in this case $T$ is just freely generated under finite products by $C \setminus \{1\}$. But if 1 is not strict, then if we assume that $T\mathrm{-Alg}$ is a topos and in particular $J$ is subcanonical, then by the Yoneda lemma the initial object of $T\mathrm{-Alg}$ is not strict, a contradiction (the initial object in any topos is strict).


Johnstone gives the example where $T$ is the theory of a *Jónsson-Tarski algebra*, i.e. a set equipped $X$ with a bijection with $X \times X$. The Jónsson-Tarski algebras form a topos, which is kind of cool.




---


**Added:** An example which has gained relevance of late with the advent of [condensed / pyknotic mathematics](https://ncatlab.org/nlab/show/condensed+mathematics) is the (large) site $\mathcal C$ of free compact Hausdorff spaces. This is the Kleisli category for the ultrafilter monad on $Set$. So it is a full subcategory of compact Hausdorff spaces, where the objects are the Stone-Cech compactifications of discrete spaces. Sheaves on $\mathcal C$ are called *condensed* or *pyknotic* sets (depending on how one deals set-theoretically with the largeness of the site $\mathcal C$). They form a topos (modulo set-theoretic details). And a presheaf $F: \mathcal C^{op} \to Set$ is a sheaf precisely if $F$ preserves finite products. That is, $\mathcal C$ is a Lawvere theory whose algebras form a topos.


