For each $C$ locally finitely presentable category, the full subcategory of finitely presentable objects $C_{fp}$ is a dense generator, i.e. the natural functor $C \to \mathrm{PSh}(C_{fp})$ is a full embedding. One can think of this as "an object $A \in C$ is controlled by mappings from finitely presentable objects into it" (and it is literally the colimit of the corresponding canonical diagram)
Question 1. For which categories of algebras of finitary algebraic theories (i.e., finitary monads on $\mathrm{Set}$) is the subcategory of finitely presentable objects also a dense cogenerator, i.e. is the natural functor $C^{op} \to \mathrm{PSh}(C_{fp}^{op})$ a full embedding? That is, it is true that “the object $A \in C$ is controlled by mappings into finitely presentable objects” (and is the limit of the corresponding canonical diagram)?
In particular, is this true for the category of commutative rings?
If the class of such algebraic theories is very narrow, then the following question becomes especially interesting
Question 2. What are the natural small full subcategories in finitary algebraic categories, which often turn out to be dense generators and cogenerators?
The motivation for the questions is related to the algebra-geometry duality. If finitely presentable objects are a dense cogenerator, then all “affine spaces” are full embedded in the topos of presheaves on the “test” spaces (corresponding to finitely presentable objects), which gives them a chance, with a successful choice of site, to get into the topos of sheaves.
UPD. The cocomplete category with a small dense generator is total (corollary 6.5, "A survey of totality for enriched and ordinary categories", Max Kelly). Therefore, such an algebraic category $C$ must be cototal, but this happens quite rarely. In particular, this is not the case for the category of commutative rings. This is weird. Is the category of affine schemes not a full subcategory of the topos of sheaves on the Zariski site?
UPD2. Moreover, assuming Vopenka's principle, every cocomplete category having a (small) dense generator is locally presentable ("Adamek, Rosicky - Locally Presentable and Accessible Categories", Theorem 6.14). Then if some non-trivial algebraic category has a dense cogenerator, then it is a poset and this means that ZFC + Vopenka's principle is contradictory. This convinces me that, in fact, it is possible to prove in ZFC that no algebraic category has a dense cogenerator (except small: $1$ and ${\varnothing, 1}$, which are reflexive subcategories of $\mathrm{Set}$).
