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Stream: theory: category theory

Topic: A nerve-realization adjunction

Patrick Nicodemus (Dec 22 2025 at 20:36):

Let $X,Y$ be topological spaces.
Let $f : X \to Y$ be a continuous function.
Let $\tau(Y)$ denote the topology of $Y$, regarded as a poset category.

Write $J_f : \tau(Y) \to \mathbf{Top}/X$ for the functor which sends the open set $U$ to the inclusion $f^{-1}(U)\subset X$. Then the nerve-realization adjunction induced by $J_f$ between $\mathbf{Psh}(Y)$ and $\mathbf{Top}/X$ restricts to the familiar adjunction between $\mathbf{Sh}(Y)$ and $\mathbf{Etale}(X)$ which is usually written as the composition of the equivalance $\mathbf{Sh}(X) \simeq \mathbf{Etale}(X)$ and an unrelated adjunction $\mathbf{Sh}(X) \iff \mathbf{Sh}(Y)$.

I realized this recently and found it interesting. I just wanted to share this.

I have written up a proof here in some personal notes.
https://github.com/patrick-nicodemus/research/blob/main/geometry/manifolds.typ, line 192

An important theorem here, somehow enabling this nice correspondence seems to be that the pullback functor $\mathbf{Top}/Y\to\mathbf{Top}/X$ is cocontinuous. As a result, it preserves pointwise left Kan extensions; so the usual functor $\tau(Y) \to \mathbf{Top}/Y$ (whose associated "realization" is the etale space functor for sheaves on Y) can be composed with the pullback functor to give a realization from sheaves on Y to etale spaces on X. (It is interesting that the pullback functor sends etale spaces to etale spaces.)

fosco (Dec 23 2025 at 18:24):

A particular case gives you (admittedly, with a certain amount of work)

Patrick Nicodemus said:

the equivalence $\mathbf{Sh}(X) \simeq \mathbf{Etale}(X)$

just choose $f=id$!

Patrick Nicodemus (Dec 23 2025 at 18:30):

Yes, this is something I maybe should have made more explicit: part of what makes this interesting to me is that it generalizes this nerve realization adjunction which is pretty well known.

fosco (Dec 23 2025 at 18:39):

This is highly unrelated and directed to French native speakers.

I remember have been scolded by algebraic geometers in my alma mater because it's not supposed to be an "etale space" but an "*étalé space", accent on both e, and it makes a difference (also: étale cohomology, accent only on the first e). Is it correct? :smile:

Damiano Mazza (Dec 23 2025 at 18:46):

I'm not a native French speaker but I can confirm: étalé space, étale cohomology.

I remember a conversation about this some time ago on this Zulip. The general consensus, if I am not mistaken, seemed to be "write it and pronounce it however you want" (unless you're writing/speaking French, in which case you are obviously going to stick to the correct spelling and pronounciation). I don't think French native speakers would disagree with this, but maybe I don't know enough algebraic geometers! :smile:

John Baez (Dec 23 2025 at 19:21):

The nLab says this:

Grammar and spelling

In French, the verb 'étaler' means, roughly, to spread out; '-er' becomes '-é' to make a past participle. So an 'espace étalé' is a space that has been spread out over $B$. On the other hand, 'étale' is a (relatively obscure, distantly related) nautical adjective that can be translated as 'calm' or 'slack'.

To quote from the Wiktionnaire français:

'étale' _qualifie la mer qui ne monte ni ne descend à la fin du flot ou du jusant_
('flot' = 'flow' and 'jusant' = 'ebb').

• Further reference

Pronunciation

The French pronunciation of "étalé" is three syllables; that of "étale" is two syllables, with the final "e" silent. Many English-speaking mathematicians follow the respective French pronunciations; see for example this nForum thread.

Others use the two-syllable pronunciation (corresponding to French "étale") for both written forms "étale" and "étalé". This group included [[Grothendieck]] himself, when speaking English. (See Colin McLarty, from 4:45 to 6:05 and particularly at 5:26.)

John Baez (Dec 23 2025 at 19:23):

I write "etale", since as an American my job is to simplify things.

fosco (Dec 23 2025 at 19:36):

ah, it wasn't a dream that there was something on an nLab page then