Isbell shows in Function spaces and adjoints 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 on limit sketches, Theorem 8.7.

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.

Isbell shows in Function spaces and adjoints 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.

Isbell shows in Function spaces and adjoints 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 on limit sketches, Theorem 8.7.