Local presentability and representable presheaves over the category of topological spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:57:16Z http://mathoverflow.net/feeds/question/24123 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24123/local-presentability-and-representable-presheaves-over-the-category-of-topologica Local presentability and representable presheaves over the category of topological spaces David Carchedi 2010-05-10T16:38:58Z 2010-05-13T22:02:00Z <p>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.</p> http://mathoverflow.net/questions/24123/local-presentability-and-representable-presheaves-over-the-category-of-topologica/24185#24185 Answer by Daniel Schäppi for Local presentability and representable presheaves over the category of topological spaces Daniel Schäppi 2010-05-11T01:01:26Z 2010-05-11T02:52:50Z <p>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.</p> <p>There is a convenient category of topological spaces which <em>is</em> 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 <a href="http://www.uoregon.edu/~ddugger/delta.html" rel="nofollow">here</a>. A proof that the category of $\Delta$-generated spaces is locally presentable can be found <a href="http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html" rel="nofollow">this paper</a> of L. Fajstrup and J. Rosický.</p> <p>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</p> <p>$G(X) \cong \mathbf{Set}(\ast,GX) \cong \mathbf{Top}^{\mathrm{op}}(F(\ast),X)=\mathbf{Top}(X,F(\ast))$,</p> <p>which shows that $G$ is represented by $F(\ast)$.</p> <p>Edit: added the missing op's mentioned in the comment.</p>