Is there a large colimit-sketch for topological spaces?

Question

Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?

In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}$ in $\mathcal{E}$ such that topological spaces can be seen as those functors $\mathcal{E} \to \mathbf{Set}$ which map the cocones in $\mathcal{S}$ to colimit cocones? Since the forgetful functor $\mathbf{Top} \to \mathbf{Set}$ creates colimits, I don't see any a priori reason why this cannot be true.

Let me explain a bit where this question comes from, since I also have some reference requests about related questions. Burroni has given in Esquisses des catégories à limites et des quasi-topologies (zBMATH review, see also Burroni's 1970 CR note) a large mixed sketch based on filter convergence (see here) whose models are topological spaces. Maybe this thesis contains other sketches as well, perhaps even a colimit-sketch, thus answering my question, but I could not find it online anywhere, and Ehresmann told me that, in fact, there is no digital copy. I was quite surprised to find out that there is actually a limit-sketch for topological spaces. This is explicitly mentioned by Guitart in Toute théorie est algébrique et topologique (Proposition 25, pdf), without proof, but the construction comes out of Edgar's description of topological spaces in The class of topological spaces is equationally definable (doi:10.1007/BF02945113), which in turn is based on Kelley's characterization of topological spaces in terms of net convergence (General topology, Chapter 2, Theorem 9): A topological space can be described as a set $X$ together with a monomorphism $C(P,X) \to X^P \times X$ for every directed set $P$ satisfying four axioms (think of $C(P,X)$ as the set of $P$-indexed convergent nets including their limits). These axioms can be written down in categorical language. For example, the axiom "subnets converge to the same point" is the following: for every cofinal map $Q\to P$ there is a morphism $C(P,X) \to C(Q,X)$ such that $$\begin{array}{cc} C(P,X) & \rightarrow & X^P \times X \\ \downarrow && \downarrow \\ C(Q,X) & \rightarrow & X^Q \times X \end{array}$$ commutes. I have written down all the details of that sketch, but I wonder (Side question) if this limit-sketch is written down somewhere else already? As a byproduct we get the notion of a topological space object internal to any complete category (not just a topos as studied by Macfarlane and Stout for instance). Let me also mention that I found out that $\mathbf{Top}^{\mathrm{op}}$ is the category of models of a large limit-sketch, which means, however, that $\mathbf{Top}$ is the category of $\mathbf{Set}^{\mathrm{op}}$-valued models of a large colimit-sketch. Please let me know if you know any other references about this topic apart from those already mentioned.

*Edit. Details of the sketches mentioned above can now be found in Large limit sketches and topological space objects.

Answer 1

The answer is no. I think this example illustrates why pure colimit sketches are rarely studied; one generally allows limits into the sketch before generalizing to allow colimits into the sketch. That is, I think this example illustrates the point that pure colimit-sketches are just not that useful. (This is after already noting that categories like $Grp$ and such can't be so sketched because they fail to have disjoint coproducts).

Here is the argument. Suppose that $Top$ were the category of models of a colimit sketch $(\mathcal E, \mathcal S)$. Then because colimits commute with colimits (and colimits in functor categories are computed levelwise), the representables $Hom(E,-): Set^{\mathcal E} \to Set$ would restrict to a strong generating class of colimit-preserving functors $Top \to Set$. This contradicts the following

Claim: Every colimit-preserving functor $F: Top \to Set$ carries the canonical map $S \to I$ to an isomorphism.

Here and in the following, $S$ is the Sierpinski space, and $I$ is the indiscrete space on two points, and $1,2$ are the discrete spaces on 1 and 2 elements respectively. And $C_2$ denotes the group with two elements. Note that $2$ and $I$ each have a natural $C_2$ action, both with quotient $1$, and that the map $2 \to I$ is $C_2$-equivariant.

Proof: Let $X = F(1)$. Because $F$ preserves coproducts, we have $F(2) = X \amalg X$. Moreover, the $C_2$ action on $X \amalg X$ is free, so we may write $F(2) = C_2 \times X$ as a $C_2$-set. Now, because $F$ preserves epimorphisms, we have that $F(I)$ is a $C_2$-equivariant quotient of $F(2)$ which has $F(1)$ as a further $C_2$-equivariant quotient. Therefore $F(I) = (C_2 \times X_0) \amalg X_1$ as a $C_2$-set, for some decomposition $X = X_0 \amalg X_1$.

Now consider the pushout $I = S \cup_2 S$, using the two bijections $\phi,\psi: 2 \rightrightarrows S$. Because $F$ preserves epimorphisms and because the map $F(2) \to F(I)$ factors through $F(S)$, we have $F(S) = (C_2 \times X_{00}) \amalg X_{01} \amalg X_1$ for some decomposition $X_0 = X_{00} \amalg X_{01}$, and $F(\phi)$ is the obvious quotient map. Moreover, because $\psi = \phi \sigma$, (where $\sigma$ denotes the action of the nontrivial element of $C_2$), we have that $F(\psi)$ differs from $F(\phi)$ only in that it applies $\sigma$ to the elements of the $C_2 \times X_{00}$ factor.

Observe, then that $F(\psi) = \tau F(\phi)$, where $\tau$ is this residual $C_2$ action (even though there is no such factorization before applying $F$ -- $S$ does not have a nontrivial $C_2$-action even though $F(S)$ does). Therefore, the pushout $F(I) = F(S) {}_{F(\phi)}\cup_{F(\psi)} F(S)$ is isomorphic to the pushout $F(S) {}_{F(\phi)} \cup_{F(\phi)} F(S)$. But since $F(\phi)$ is an epimorhism, pushing it out against itself yields an isomorphism. That is, either map $F(S) \rightrightarrows F(I)$ is a bijection as claimed.

Answer 2

Isbell shows in Function spaces and adjoints (1975) that every cocontinuous functor $\mathbf{Top} \to \mathbf{Set}$ is a coproduct of copies of the forgetful functor. (I have found an alternative proof for this within a more general theory, details will be added here when it's online.) But a necessary condition for a category to be modelled by a (possibly large) colimit sketch is that the cocontinuous functors to $\mathbf{Set}$ are jointly conservative. By picking any bijective continuous map which is not an isomorphism, we get a contradiction.

*Edit. An alternative proof can be found in my new paper Large limit sketches and topological space objects (2021), Theorem 8.7.