# CG spaces from the perspective of sheaves over compact Hausdorff spaces

A compactly generated space is a space $$X$$ such that $$f : X \rightarrow Y$$ is continuous if and only if $$K \rightarrow X \stackrel{f}{\rightarrow} Y$$ is continuous for each compact hausdorff space $$K$$.

Let $$I$$ be the category of compact hausdorff topological spaces with continuous maps. I think being compactly generated is equivalent to the canonical map $$\epsilon_X : \int_{K \in I} [K, X]_{\text{Top}} \times K \rightarrow X$$ being an isomorphism. $$\epsilon_X$$ is the counit of an adjunction between functors $$L : [ I^{op}, \text{Set}]_{\text{Cat}} \rightarrow \text{Top}$$ sending $$S$$ to $$\int_{K \in I} S(K) \times K$$ and $$R : \text{Top} \rightarrow [ I^{op}, \text{Set}]_{\text{Cat}}$$ sending a space $$X$$ to the presheaf $$S : I^{op} \rightarrow \text{Set}$$ sending $$K$$ to $$[K, X ]_{\text{Top}}$$. $$L$$ is the left Kan extension of the inclusion $$\iota : I \rightarrow \text{Top}$$ along the yoneda embedding $$Y : I \rightarrow [I^{op}, \text{Set}]_{\text{Cat}}$$, as in the diagram below:

That $$L$$ is left adjoint to $$R$$ is a famous theorem, which was discussed here.

If I am not mistaken, this adjunction is related to Peter Scholze and Dustin Clausen's condensed mathematics. Surprisingly, there is no information lost if we take instead of $$I$$ the smaller category of profinite sets. There is a fully faithful functor from compactly generated spaces to condensed sets (sheaves on the étale site of a point) that arises in this way.

My question is, has anyone taken this perspective to get a streamlined, categorical proof that the category of compactly generated spaces is cartesian closed? We might start by observing that the category of presheaves on $$I$$ has all small limits and colimits, $$L$$ preserves colimits, and $$R$$ preserves limits. Might this allow us to carry over the cartesian closed-ness of $$[I^{op}, \text{Set}]_{\text{Cat}}$$ to the category of compactly generated spaces?

• How do you prove that (compactly generated) is equivalent to ($\epsilon_X$ iso)? – Charles Rezk Aug 15 '19 at 16:11
• @CharlesRezk I think he is actually right. His definition of compactly generated sounds to me like "$X$ is the colimits of the canonical diagram induced by $I$", this is equivalent to say that the codensity monad of I in Top fixes $X$, which leads to $\epsilon_X$ is an isomorphism. Does this make any sense to you? – Ivan Di Liberti Aug 15 '19 at 16:23
• @IvanDiLiberti OK, I see it now. – Charles Rezk Aug 15 '19 at 17:10
• @DeanYoung you might want to give a look to A4.3.1 in Sketches of an Elephant, where inheritance of cc by reflective subcategories is characterized. – Ivan Di Liberti Aug 15 '19 at 17:15
• "Topologists don't want to know about it!" – theHigherGeometer Aug 17 '19 at 14:16

The following paper seems worth looking at, although I am not certain whether the proofs therein are related to yours:

Martín Escardó, Jimmie Lawson, Alex Simpson: Comparing Cartesian closed categories of (core) compactly generated spaces. Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145.

• Thanks. I'll read this and see if it sheds any light here. – Dean Young Aug 15 '19 at 17:36
• Thanks for including both links! – theHigherGeometer Aug 17 '19 at 14:22
• Perhaps we could adopt more precise language than "evil" and "good." I suggest "private" and "published," respectively. – Ben Wieland Aug 25 '19 at 1:43
• I used "evil" and "good" for a reason and they are precisely the words that express my attitude. – Andrej Bauer Sep 1 '19 at 18:42

I found an answer using Ivan Di Liberti's very helpful reference in the comments. I figured I would post it here in case anyone has interest in this approach to compactly generated spaces in the future.

For a cardinal $$\kappa$$, let $$I_{\kappa}$$ be the category of compact hausdorff spaces with cardinality less than or equal to $$\kappa$$. The cardinal is to deal with size issues. We make $$I_{\kappa}$$ into a site where covers are jointly surjective collections of maps. $$I_{\kappa}$$ induces an adjunction between $$\text{Sh}( I_{\kappa}^{op})$$, the category of sheaves of sets on $$I_{\kappa}^{op}$$, and $$\text{Top}$$. The left adjoint $$L$$ sends $$F$$ to $$\int_{K \in I_{\kappa}} F(K) \times K$$, and the right adjoint $$R$$ sends $$X$$ to the sheaf sending $$K$$ to $$[K, X]_{\text{Top}}$$. $$L$$ is a left Kan extension of $$I_{\kappa} \rightarrow \text{Top}$$ along the inclusion $$I_{\kappa} \rightarrow \text{Sh}( I_{\kappa}^{op})$$.

This adjunction between $$L$$ and $$R$$ is idempotent; it factors as $$\text{Sh}( I_{\kappa}^{op}) \leftrightarrow \text{CG}_{\kappa} \leftrightarrow \text{Top}$$, where $$\text{CG}$$ is what I call $$\kappa$$-compactly generated spaces. $$\text{CG}_{\kappa}$$ is a reflexive subcategory of $$\text{Sh}( I_{\kappa}^{op})$$ and a coreflexive subcategory of $$\text{Top}$$. What Ivan Di Liberti realized is that this may allow us to apply proposition 4.3.1 in The Elephant. Let $$L' : \text{Sh}( I_{\kappa}^{op}) \leftrightarrow \text{CG} : R'$$ be the reflexive adjunction. To show that $$\text{CG}_{\kappa}$$ inherits a cartesian closed structure from $$\text{Sh}(I_{\kappa}^{op})$$, it suffices to show that $$L'$$ preserves products.

$$L'$$ sends a representable sheaf $$[-, K]$$ to $$K$$. It preserves products of compact hausdorff spaces; the canonical map $$L'([-, K] \times [-, K']) \rightarrow L'([-, K]) \times L'([-, K'])$$ is an isomorphism. To see this, note that it is a bijection (one can show that it is an isomorphism after composing with the forgetful functor to set, and this can be made very canonical). Now the famous lemma about a continuous map from a compact space to a hausdorff space being open finishes the job.

This result can be extended to work for the general case. $$\text{Sh}(I^{op}_{\kappa})$$ is a sheaf topos. It is therefore locally presentable (see Borceux, Handbook of Categorical Algebra, prop. 3.4.16, page 220). So there is a small set of $$\lambda$$-small objects which generates this category under $$\lambda$$-filtered colimits. Every sheaf is a $$\lambda$$-filtered colimit of representables here. $$L'$$ and products commute with $$\lambda$$-filtered colimits, reducing to the previous paragraph.

Now it seems that, if we take the large colimits $$\text{colimit } \text{Sh}(I_{\kappa}^{op})$$ and $$\text{colimit } \text{CG}_{\kappa}$$, I think we obtain compactly generated spaces, along with the desired product preserving reflexive adjunction.