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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$\begingroup$ related: mathoverflow.net/questions/13516/… $\endgroup$– Martin BrandenburgCommented May 10, 2010 at 17:38
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1$\begingroup$ $Top$ is co-wellpowered, cocomplete, and has a generating set (a point). Thus by SAFT every continuous functor $Top^{op} \to Set$ is representable. $\endgroup$– Martin BrandenburgCommented May 10, 2010 at 17:45
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$\begingroup$ 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. $\endgroup$– Martin BrandenburgCommented May 10, 2010 at 18:29
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$\begingroup$ 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. $\endgroup$– David CarchediCommented May 10, 2010 at 22:11
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$\begingroup$ How does the SAFT imply it is representable? Perhaps I am missing something... $\endgroup$– David CarchediCommented May 10, 2010 at 22:14
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1 Answer
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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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$\begingroup$ Is $Top$ supposed to be $Top^{op}$ everywhere? $\endgroup$ Commented May 11, 2010 at 1:36
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$\begingroup$ In fact the only presentable objects in Top are the discrete spaces. See Adamek and Rosicky's book Locally Presentable and Accessible Categories, Example 1.14(6). Top also fails to have a dense generator, see the same book, Example 1.24(7); I believe Top doesn't even have a strong generator and I think/hope the same example shows this. The point is, Top is about as far from being locally presentable as can be, and it's purely for size reasons: the Fajstrup-Rosicky paper Daniel linked to discusses a method for cutting down to a locally presentable category. $\endgroup$ Commented Oct 26, 2015 at 18:26