I see, thanks. What Kolmogoroff/Alexander had is more like cellular (co)homology, I suppose. (Have only glanced at some of these old articles).

We should put this into an `!include`

-entry *cohomology – early references* so that it’s easer to include the pointers also at *singular cohomology* and elsewhere.

The probability-theory Kolmogorov also invented (cochain) cohomology?!

Yes, and cup product too. (But not the singular cohomology, that was done by Eilenberg much later.)

]]>I didn’t know that the term originates with Whitney, interesting.

Trying to look at the article, I noticed that the doi link

for Whitney 37 does Not work. Even though that’s indeed the link given by Project Euclid here.

Probably there is a single wrong character in the doi. But for the moment I have fixed it by instead linking to the url of the Euclid page.

Further up, in the first line of the References-section, I changed “The original references” to

The original references on cochain cohomology and ordinary cohomology:

Finally, I made the author name *Andrei Kolmogoroff* of the first articles listed there redirect to the existing entry *Andrey Kolmogorov*.

Hoping that this is correct? The probability-theory Kolmogorov also invented (cochain) cohomology?!

]]>Added the original papers due to Kolmogoroff and Alexander.

]]>Added the original reference for the term “cohomology”.

]]>I am not expert on the algebraic geometry involved, but I gather (and this seems to be confirmed by what it says in the entry *motivic homotopy theory*, search for “Chow group” there) that Chow groups are realized as homotopy hom-sets in the $\infty$-topos over the Nisnevich site.

(This is traditionally stated, instead, in the $\mathbb{A}_1$-localization of that $\infty$-topos, which is not itself an $\infty$-topos anymore. But since this localization is a left adjoint, it equivalently gives a corresponding statement in the Nisnevich $\infty$-topos.)

]]>Do the chow groups in algebraic geometry have an interpretation as coming from hom groupoids in an $(\infty, 1)$-category?

]]>If we see cohomology arise from $Hom_{\mathbf{H}}(X, A)$, then there’s a single dependency, but we might reflect on different classes that the variables range over. $\mathbf{H}$ might belong to tangent $(\infty, 1)$-toposes, cohesive $(\infty, 1)$-toposes, etc. $A$ might be a general object, stable object, etc.

With the fully general definition, you might think the term ’cohomology’ would vanish. I’m unlikely to say that when we take the powerset of a set we’re considering cohomology with coefficients in the subobject classifier.

]]>I would argue that a “notion” of cohomology is a syntactic or linguistic construct, hence form a non-contractible set.

]]>Why should the type of notions of cohomology be a set? [Added: Clearly it’s a contractible type, hence ’the’.]

]]>Now you may say: Ah, but these two infinitudes are really just two notions of cohomology. To which the slogan replies: Sure, but once you start passing from the concrete particulars to the general abstract this way, you should go all the way and realize that in this abstract sense then even these two notions unify: Both are about hom-spaces in an $\infty$-topos!

But that is irrelevant to the question of how to *count* the number of “notions of cohomology” that the one unifying perspective is being compared to, since for the slogan to have any teeth there must be *more* than one of them.

Seriously, algebraic geometers commonly speak of one cohomology theory per site, and algebraic topologists speak of one cohomology theory per spectrum. In this sense there are uncountably infinitely many cohomology theories, all subsumed by the slogan.

Now you may say: Ah, but these two infinitudes are really just two notions of cohomology. To which the slogan replies: Sure, but once you start passing from the concrete particulars to the general abstract this way, you should go all the way and realize that in this abstract sense then even these two notions unify: Both are about hom-spaces in an $\infty$-topos!

(in $T \infty Grpd$ in the second case!).

]]>I have changed “Thousands” to “Thousand and one”, which is a closer estimate.

]]>]]>Etale cohomology, various versions of algebraic K-theory, the concept of ”arithmetic vs. geometric” cohomology theories, absolute Hodge cohomology, Hodge cohomology, Amitsur cohomology, archimedean cohomology, Andre-Quillen cohomology, Betti cohomology, Borel-Moore homology, cdh cohomology, Cech cohomology, Chow groups, arithmetic Chow groups, Arakelov Chow groups, group cohomology and continuous group cohomology, crystalline cohomology, crystalline Deligne cohomology, de Rham cohomology, Deligne cohomology, Deligne- Beilinson cohomology, smooth Deligne cohomology, Eichler cohomology, elliptic Bloch groups, equivariant Deligne cohomology, etale K-theory, etale motivic cohomology, flat cohomology, Fontaine-Messing cohomology, Friedlander- Suslin cohomology, Galois cohomology, Hyodo-Kato cohomology, Lawson homology, cohomology of Lie algebras, ”log” versions of Betti, de Rham, crystalline and etale cohomology, Milnor K-theory, Kato homology, Monsky- Washnitzer cohomology, morphic cohomology, motivic cohomology, nonabelian cohomology, Nisnevich cohomology, p-adic etale cohomology, parabolic cohomology, rigid cohomology, syntomic cohomology, rigid syntomic cohomology, relative log convergent cohomology, Rost’s cycle modules, singular cohomology of arithmetic schemes, Suslin homology, Tate cohomology, unramified cohomology, Weil-etale cohomology, Zariski cohomology, and various theories with compact support. Also, various notions of motives and of mixed motives, and various other kinds of algebraic cycle groups. In addition, many of the theories come with a choice of coefficients. One could also extend the list to theories occurring in other areas of mathematics, there would then be at least a few hundreds of them. (Andreas Holmstrom, Questions and speculation on cohomology theories in arithmetic geometry)

The slogan doesn’t say that *all* notions of cohomology are hom-spaces in an $\infty$-topos, only that “thousands” of them are, so I think it’s okay. Although are there really literally “thousands” of kinds of cohomology? I doubt I could come up with more than “dozens” myself, and I might have to stretch to reach the plural.

Well…

]]>Zoran also objected, in discussions here. But two people is not overwhelming opposition.

]]>I don’t know that it “is contested”. It has been contested, in that a person on MO somewhere may have said he isn’t sure if it’s right.

]]>You mentioned elsewhere that this view is contested. Of course the flip is always possible, where, if someone present a cohomology which can’t be seen in this way, we declare it to be not a cohomology, or we improve it so as to fit. The $\infty$-topos account becomes constitutive of what it is to be a cohomology.

[Such a flip is described here.]

At an earlier stage, Čech homology was declared no longer a homology theory for failing the Eilenberg-Steenrod exactness axiom. We have strong homology claiming to be its improvement. But on presenting this case of Čech homology to Barry Mazur once, he remarked “Yes, but it has made its comeback in étale cohomology”. Does anyone have a view on that?

]]>I have re-instantiated the statement of the slogan (all cohomology is hom-spaces in some $\infty$-topos), which I had introduced originally in rev #45 and then apparently removed (maybe unintentionally, I forget) in rev #97.

Anyway, it’s back, right before the Definition section.

]]>I think it is really meant to point to relation to “over spaces”, since that’s the special aspect of this particular statement. (I may have written that, way back, I forget and don”t have time to dig through the entry’s history now, which is always tedious.) Best way to fix it for the time being would be to add pointer to that section in Lurie’s HTT. It should be somewhere in section 7 (sorry, no time to check).

]]>Presumably the author of that sentence had a location on nLab in mind when they wrote it. Perhaps they meant at (infinity,1)-category of (infinity,1)-sheaves.

]]>This is discussed somewhere towards the end of “Higher Topos Theory”.

]]>A sentence on the page cohomology:

There is an equivalence between $(\infty,1)$-sheaves on $X$ and topological spaces over $X$, as described in detail at (∞,1)-sheaves and over-spaces

But there is no such entry. Where is that detailed description given?

]]>