**Question.** Is there a large [colimit-sketch](https://ncatlab.org/nlab/show/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](https://zbmath.org/?q=an:0395.18005), see also Burroni's [1970 CR note](http://gallica.bnf.fr/ark:/12148/bpt6k5619186c/f24.item)) a large mixed sketch based on filter convergence (see [here](https://mathoverflow.net/questions/277329/)) 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](https://webusers.imj-prg.fr/%7Erene.guitart/docpp/guitart08tteaet.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](https://doi.org/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](https://arxiv.org/abs/2106.11115).



Is there a large colimit-sketch for topological spaces?
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](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space), and $I$ is the [indiscrete space](https://en.wikipedia.org/wiki/Trivial_topology) 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.


