# Local presentability and representable presheaves over the category of topological spaces

Is the category of topological spaces locally presentable? n-lab claims that it is not locally FINITELY presentable, but how about for some larger cardinal? Here I really mean the 1-category of topological spaces and am not willing to identify it with simplicial sets. Essentially, I want to know if (after I fix appropriate Grothendieck universes) representable presheaves on Top are characterized by those presheaves which send colimits in Top to limits in Set, which would follow from local presentablility.

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–  Martin Brandenburg May 10 '10 at 17:38
$Top$ is co-wellpowered, cocomplete, and has a generating set (a point). Thus by SAFT every continuous functor $Top^{op} \to Set$ is representable. –  Martin Brandenburg May 10 '10 at 17:45
It seems to me that the only thing which is not obvious to check is that there is set(!) of topological spaces, such that every(!) topological space can be optained as a colimit of these spaces. Sounds impossible. In "joy of cats" it is remarked that every topological space is a quotient of a zero-dimensional hausdorff space. –  Martin Brandenburg May 10 '10 at 18:29
Well, to make sense of presheaves on Top, we have to use a Grothendieck universe of sets out of which to build our topological spaces anyhow, so, in essence, zero-dimensional Hausdorff spaces will be a set. How is this colimit constructed? In "joy of cat" it is just remarked. –  David Carchedi May 10 '10 at 22:11
How does the SAFT imply it is representable? Perhaps I am missing something... –  David Carchedi May 10 '10 at 22:14

The category of topological spaces is not locally $\lambda$-presentable for any $\lambda$. The reason for this is the existence of spaces which aren't $\lambda$-presentable (a.k.a. $\lambda$-small) for any $\lambda$ (in a locally presentable category every object is $\lambda$-presentable for some $\lambda$). An example of such a space is the Sierpinski space; a proof of this can be found in Mark Hovey's book on model categories, on page 49.

There is a convenient category of topological spaces which is locally presentable, the category of $\Delta$-generated spaces. This category contains most of the spaces usually studied by algebraic topologists (for example, the geometric realization of any simplicial set is a $\Delta$-generated space). Daniel Dugger has some expository notes on this here. A proof that the category of $\Delta$-generated spaces is locally presentable can be found this paper of L. Fajstrup and J. Rosický.

The second question was already answered in the comments: if $G\colon \mathbf{Top}^{\mathrm{op}} \rightarrow \mathbf{Set}$ is continuous, then it has a left adjoint $F$ by the special adjoint functor theorem. Therefore we have natural isomorphisms

$G(X) \cong \mathbf{Set}(\ast,GX) \cong \mathbf{Top}^{\mathrm{op}}(F(\ast),X)=\mathbf{Top}(X,F(\ast))$,

which shows that $G$ is represented by $F(\ast)$.

Edit: added the missing op's mentioned in the comment.

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Is $Top$ supposed to be $Top^{op}$ everywhere? –  David Carchedi May 11 '10 at 1:36