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.

  • $\begingroup$ related: mathoverflow.net/questions/13516/… $\endgroup$ May 10 '10 at 17:38
  • 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. $\endgroup$ May 10 '10 at 17:45
  • $\begingroup$ 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. $\endgroup$ May 10 '10 at 18:29
  • $\begingroup$ 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. $\endgroup$ May 10 '10 at 22:11
  • $\begingroup$ How does the SAFT imply it is representable? Perhaps I am missing something... $\endgroup$ 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.

  • $\begingroup$ Is $Top$ supposed to be $Top^{op}$ everywhere? $\endgroup$ May 11 '10 at 1:36
  • $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Oct 26 '15 at 18:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.