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: enter image description here

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?

  • $\begingroup$ How do you prove that (compactly generated) is equivalent to ($\epsilon_X$ iso)? $\endgroup$ – Charles Rezk Aug 15 '19 at 16:11
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    $\begingroup$ @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? $\endgroup$ – Ivan Di Liberti Aug 15 '19 at 16:23
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    $\begingroup$ @IvanDiLiberti OK, I see it now. $\endgroup$ – Charles Rezk Aug 15 '19 at 17:10
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    $\begingroup$ @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. $\endgroup$ – Ivan Di Liberti Aug 15 '19 at 17:15
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    $\begingroup$ "Topologists don't want to know about it!" $\endgroup$ – 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.

The evil link and the good link are both available.

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


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