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)? $\endgroup$ Aug 15, 2019 at 16:11
  • 1
    $\begingroup$ @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? $\endgroup$ Aug 15, 2019 at 16:23
  • 1
    $\begingroup$ @IvanDiLiberti OK, I see it now. $\endgroup$ Aug 15, 2019 at 17:10
  • 1
    $\begingroup$ @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. $\endgroup$ Aug 15, 2019 at 17:15
  • 1
    $\begingroup$ "Topologists don't want to know about it!" $\endgroup$ Aug 17, 2019 at 14:16

2 Answers 2


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. $\endgroup$ Aug 15, 2019 at 17:36
  • $\begingroup$ Thanks for including both links! $\endgroup$ Aug 17, 2019 at 14:22
  • $\begingroup$ Perhaps we could adopt more precise language than "evil" and "good." I suggest "private" and "published," respectively. $\endgroup$ Aug 25, 2019 at 1:43
  • 6
    $\begingroup$ I used "evil" and "good" for a reason and they are precisely the words that express my attitude. $\endgroup$ Sep 1, 2019 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.


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.