Colimit density and monads

Question

Let $C$ be a cocomplete category, and suppose that it has an object that is colimit dense. Is $C$ automatically monadic over $Set$? And if not, is there an explicit counterexample?

Answer 1

I guess I'll go out on a limb and assume arsmath means, when he says an object $c$ of $C$ is colimit-dense, that the full subcategory containing $c$ is a dense subcategory in the usual sense.

There's some categorical lore which is helpful: a necessary condition for a category $C$ to be monadic over $Set$ (that is, for there to exist some monadic functor $U: C \to Set$) is that $C$ be a regular category. I think this result is proved somewhere in Francis Borceux's Handbook of Categorical Algebra, but the quickest reference I can find to hand is here, Proposition 1, part (2) (here $Set$ is a regular category, and any monad $T: Set \to Set$ preserves epis because every epi in $Set$ is already split, by the axiom of choice).

Now I claim the category $Pos$ of posets is not regular, hence cannot be monadic over $Set$. It is not regular because the pullback of a regular epi in $Pos$ need not be a regular epi; see the explicit example described here. On the other hand, it isn't too hard to see that $Pos$ is cocomplete and the two-element chain $\{0 \leq 1\}$ is colimit dense in $Pos$. So this gives an explicit counterexample.

Answer 2

I just want to add an argument that $\mathsf{Cat}$ is also a counterexample (partly because I suspect that Todd chose to use $\mathsf{Pos}$ instead on account of it not being obvious that $\mathsf{Cat}$ has a dense generating object :). The same examples that show $\mathsf{Pos}$ is not regular should show that $\mathsf{Cat}$ is not regular. So the question is how to find a dense generating object for $\mathsf{Cat}$.

In fact $\sum_n \Delta^n$, the coproduct of the dense generating set of finite linear orders / simplices is a dense generator in $\mathsf{Cat}$ / $\mathsf{sSet}$. This is not immediate, as Zhen Lin points out above -- the coproduct of the objects of a dense generating set need not even be a generating object in general, as evidenced by sheaves over any nontrivial space (with the representables as the dense generating family).

Nonetheless, $\sum_n\Delta^n$ is a dense generator in $\mathsf{Cat}$ or in $\mathsf{sSet}$ because every representable is a retract of $\sum_n \Delta^n$, and a subcategory is dense iff its closure under retracts is dense (since if $i: C \to \tilde C$ is a full subcategory such that every object of $\tilde C$ is a retract of an object of $\tilde C$, then $\mathsf{Hom}(i,1): [\tilde C^\mathrm{op}, \mathsf{Set}] \to [C^\mathrm{op}, \mathsf{Set}]$ is an equivalence, i.e. $i$ is a Morita equivalence). Actually, in $\mathsf{Cat}$, it's clear that the three-element set $\{[0],[1],[2]\} = \{1,2,3\}$ consisting of the simplices of dimension $\leq 2$ / ordinals $\leq 3$ is dense, but these objects are all retracts of the single simplex $[2]$ / ordinal 3. So this object gives an even simpler dense generator for $\mathsf{Cat}$. The (nerve of) the ordinal $\omega$ would also do for a dense generator in either category.

As Todd observed, by the theorem discussed by Vitale in the notes he linked to,

A category is monadic over $\mathsf{Set}$ if and only if it is exact and contains a regular projective generating object.

(here's the link again), it follows (once we observe that the representable simplicial sets are projective, so their coproduct is too, and that a dense generator is a regular generator) that $\mathrm{Hom}(\sum_n \Delta^n, 1): \mathsf{sSet} \to \mathsf{Set}$ is monadic, surprising as that seems! And categories and simplicial sets are just certain types of $M$-set where $M$ is the endomorphism monoid of $\sum_n \Delta^n$ (via full, reflective inclusions)! I haven't thought about how to characterize the image of these inclusion functors.

In any 2-valued Grothendieck topos (i.e. a topos category such that every non-initial object has a point), the coproduct of a dense generating set similarly contains every member of the generating set as a retract, and so serves as a dense generating object. If this topos is a presheaf category where the base category has a terminal object, then the representables are a projective dense generating family, so their coproduct is a projective dense generating object, so 2-valued presheaf toposes where 1 is projective are monadic over $\mathsf{Set}$.