A Characterization of Categories with a Conservative Forgetful Functor to SET

Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the "higher-order" categories of topological spaces, smooth manifolds, schemes, etc. or some "relational" categories like posets.

Is there any intrinsic characterization of the difference in behavior here? Do we know that certain classes of categories always have conservative forgetful functors to $\bf{Set}$?

I'd be particularly interested in a model-theoretic description (i.e. something like the fact that the first examples are categories of models of an algebraic first order theory while the later ones are not), but I'd also be interested in a description in terms of categorical properties.

EDIT: As pointed out in the answers, it's often possible to give a conservative faithful functor to $\bf{Set}$. However, these aren't the "underlying set" forgetful functors; they include extra information. Maybe the question becomes more interesting if we require the functor to be "canonical". This doesn't make much sense in terms of categories alone, but perhaps it does in terms of model theory: given a (perhaps higher-order) theory $\mathbb{T}$, when is the forgetful functor $\text{MOD-}\mathbb{T}(\bf{Set}) \rightarrow \bf{Set}$ conservative?

Solution 1:

A conservative functor $\mathsf{Top} \to \mathsf{Set}$ has been described in the comments. In a similar vein, for schemes we can take $X \mapsto |X| \amalg \mathcal{P}(\mathrm{Stalks}(X))$. This should also work for manifolds. Actually, for manifolds the much more formal $X \mapsto \coprod_{Y ~ \mathrm{conn}} \mathrm{Hom}(Y,X)$ works too, where the coproduct is over connected $Y$. So there are more categories conservative over $\mathsf{Set}$ than you might think.

It might be interesting to ask which categories admit a representable conservative functor to $\mathsf{Set}$. Or at least "familialy representable": a set of objects $S_\alpha$ such that $\coprod_\alpha \mathrm{Hom}(S_\alpha,-)$ is conservative is called a strong generator for a category. Categories with strong generators are too general to admit a good classification, but I'm pretty sure that topological spaces and schemes don't have strong generators (although manifolds do: the connected manifolds). Also, see the nlab article for other sorts of "generators" (the nlab prefers the term "separator"). Generating hypotheses are important auxiliary hypotheses in theorems which often rule out categories of a "topological" nature.

One question which has received attention in the categorical literature is when a category admits a faithful functor to $\mathsf{Set}$, or is "concrete". Famously, the homotopy category of topological spaces does not. Necessary and sufficient conditions are known characterizing when a category is concrete -- if the category has finite limits, then the condition is equivalent to being regularly well-powered (i.e. every object has a small set of regular subobjects). But this question doesn't really get at the distinction you want: the usual functor $\mathsf{Top} \to \mathsf{Set}$ is faithful, for example.

Stronger notions which may be related to the distinction you want include accessible categories, topological categories, and algebraic categories.

As to the edit, if $\mathbb{T}$ is a first-order theory, I can think of several reasonable categories of models to consider:

• $\mathrm{Mod}(\mathbb{T})$, the category of models and homomorphisms. In this case, it seems to me that one easy criterion for the forgetful functor to be conservative is that there be no relation symbols (except equality), but only function symbols (including constants) in the language. The converse -- if the forgetful functor is conservative then $\mathbb{T}$ can be axiomatized over a language with no relation symbols -- might be true, I don't know. EDIT Actually, this converse probably fails. For example, let $\mathbb{T}$ be the theory with two binary relations $R,S$ and the axiom $\forall x,y \, R(x,y) \Leftrightarrow \neg S(x,y)$. Then the forgetful functor is conservative, but it seems very unlikely that this theory can be axiomatized in a signature with no relations.

• $\mathrm{Elem}(\mathbb{T})$, the category of models and elementary embeddings. In this case, it seems to me that the forgetful functor to $\mathsf{Set}$ is always conservative.

• For just about any reasonable class of formulas $\Phi$, the category $\Phi-\mathrm{Elem}(\mathbb{T})$, the category of models and homomorphisms which preserve the satisfaction of formulas in $\Phi$. $\mathrm{Mod}(\mathbb{T})$ is the case when $\Phi$ is just the atomic formulas (those built up from variables, function symbols, and relation symbols, but no logical connectives or quantifiers). As long as $\Phi$ contains the atomic formulas and their negations (so that the morphisms of $\mathrm{Mod}(\mathbb{T})$ are all embeddings or at least strong homomorphisms, I believe that the forgetful functor to $\mathsf{Set}$ is conservative.

As for higher-order logic, I really don't know because I'm not familiar enough with it.

Solution 2:

Freyd proved in "On the concreteness of certain categories" that if $C$ is locally small than there is a conservative functor $C \to \text{Set}$.

Solution 3:

The nicest kind of forgetful functors are those which are monadic, which is a reasonable categorical description of categories of models of some sort of theory. Monadic functors are always conservative. And a nice characterization is known of categories monadic over $\text{Set}$ (this is Borceux, Handbook of Categorical Algebra, Volume 2, Theorem 4.4.5):

Theorem: A category $C$ is monadic over $\text{Set}$ iff it

• has finite limits,
• is (Barr) exact, and
• has a regular projective generator $P$, meaning a projective object $P$ such that the coproduct $\coprod_X P$ exists for every set $X$ and such that, for every object $Y \in C$, the natural map $\coprod_{f : P \to Y} P \to Y$ is a regular epimorphism.

With these hypotheses, the monadic forgetful functor is $\text{Hom}(P, -)$.

For example, $\text{Top}$ is not Barr exact, so it is not monadic over $\text{Set}$ with respect to any forgetful functor. On the other hand, the category of compact Hausdorff spaces is monadic over $\text{Set}$; the corresponding monad is the ultrafilter monad, which you can think of as an infinitary algebraic theory whose operations are given by taking limits wrt ultrafilters.