The n-Category Café

“ In the case of Penrose’s impossible figures, such as the tribar, I would only talk of an impossibility of patching the locally consistent data into a consistent whole.”

Oh, that is neat!

Being a great believer in pilot projects, I’d like to see a 1-2 page description of the carrying in arithmetic case, targeted to the bright primary school student. You guys can easily target the bright high school student that Urs mentions; but it will take some hard thinking to make the case clearly to a more naive audience. And it would force one to focus on only the most fundamental concepts. That would be a valuable contribution.

I have listened many times while hiking to one of Roger Penrose’s talks about twistor theory and as he says once while on TV he was caught unprepared to explain what cohomology was: What you see locally (form each of the tribar legs) is the precise non local measure of the impossibility of a whole (attributed to Penrose), I think that’s what he said.

Thanks for publicizing that note.

But doesn’t that purely “internal” point of view on cohomology obscure its “external” meaning (parametrizing the failure of a local-to-global construction)?

Write then $i:F \to A$ for the function that includes the fiber of $f$ over $pt_B$

So, $f$ should be c, yeah?

So I might have thought of the obstruction to lifting the identity function on the circle through the projection from the boundary of a Moebius strip, by thinking of the nontrivial element of $H^1(S^1, \mathbb{Z}/2\mathbb{Z})$, or the nontrivial cocycle in $Hom(S^1, K(\mathbb{Z}/2\mathbb{Z}, 1))$, picturing a pullback of the fibration $S^{\infty} \to RP^{\infty}$, but it was all already there in the high -school argument.

If, as Urs writes,

in homotopy type theory:

• functions are cocycles;
• equivalence classes of functions are cohomology classes.

That’s really all there is to cohomology, fundamentally.

What will happen after categorification?

• functors are ???;
• equivalence classes of functors are &&&.

So he might then tell us that just as via homotopy type theory, cohomology should be straightforward to the high-school student familiar with functions, so via categorified homotopy type theory, &&& should be straightforward to the university student familiar with functors.

But then, if cohomology is so very important, (“Fundamental physics is all controled by cohomology”, etc.), wouldn’t &&& be very important in physics too? And if so, won’t we have glimpsed it already?

Well, some people do write “0 ^ 0 = 1”; and most mathematicians are very careful to say that this may be true or false (or nonsense) — it depends on what one means by “0”, “1”, “^” and “=”. Particularly, if one means by “^” the set of functions from the right thing to the left thing, by “0” an empty set and by “1” a singleton, and by “=” that there is an isomorphism, then … most people are happy to believe that the equation as written is correct. General point-set topology with extensional equality of functions is one context where we would agree with the equation, though we probably wouldn’t have written it that way. The empty set is almost never named “0”, nor is a singleton usually named “1”, unless you’re secretly writing von Neuman ordinals.

However, if by “0” is meant the additive-neutral real number in your favourite real field, and by “1” is meant the multiplicative-neutral element, and “^” the ordinary exponential partial function, every calculus teacher ought to tell you that “0 ^ 0” is an undefined expression, so that “0 ^ 0 = 1” not only isn’t true, it isn’t even a sentence! But it gets worse: calculus being the algebraic conclusions of analysis, one would be content with a provisional definition for “0^0” that at least made something continuous — the trouble comes down to deciding what gets to be continuous, and what doesn’t? The expression “x ^ y” is quite reasonably definable in a continuous way as long as we provide that “x” signifies an indeterminate positive real number. We might be happy to note that in this case x ^ 0 is always 1, but this is actually a BAD thing (for continuing “^” to 0 and 0) because, obviously, if y is anything positive, then taking x small enough makes x^y as small as you like, and hence well separated from 1 — but for any small x, taking y small enough makes x ^ y as close to 1 as you like; even worser, for any small negative y, taking x small enough makes x ^ y as large as you like. The upshot of which is simply that “x ^ y” can’t be made continuous in two variables all the way to (0,0).

Yes, people have thought about these things a great deal. Just be careful with yourself when you take your turn to be clear from the outset what precise context you’re thinking in — what does “=” mean, what are you assuming, etc. Happy mathing!

I didn’t mention (because it didn’t seem important) that the context where I noticed this again was aiming to define the is_inhab type for HoTT — that is, to minimize the axioms about it needed; and this leads one to write coq definitions like

Definition is_empty (X : Type) := X -> False.
Definition is_inhab (X : Type) := is_empty (is_empty X).

Definition inhab { X : type } : X -> is_inhab X := fun x f => f x.

Lemma is_inhab_paths { X : Type } : forall x y : is_inhab X, x == y.
...

For that last one you need funext; and it seems that to re-deduce something like is_inhab_rect one has to assume

Axiom inhab_prop_choice : forall X : Type, is_prop X -> is_inhab X -> X.

that there are no nontrivial proper subsingletons. Anyways; yes, the double-negation monad was very much in mind when that all got written.

I remember a frivolous discussion over coffee a very long time ago which imagined a sort of “categorical purity test”, that would be a list of statements asking you which of them are part of your mathematical consciousness. We never did the work to come up with a good continuum of such statements, but the conversation stuck in my mind because of the one example (which might have been the one that started the whole conversation): “The center of a set is ‘true’.”

Now I finally have another statement to go along with that one: “Falsity is a cocycle.” (-:

Well, perhaps it’s more about homological algebra than cohomology as-such (which I see now is broadly taken as meaning local-to-global questions); it’s an exaggerated way of saying that the two pairs of opposites sides have the same total length.

It can also be taken as an example of parity distinctions relating to choices available : one can choose the point of tangency (on the shortest side) by adjusting the angles of the quadrilateral. There is no choice in a triangle; I’ve just checked that similarly one can adjust a pentagon to be circle-tangent, but again one has no choice on the points of tangency — because the cyclic order on the sides interacts with the parity of sides.

Take or ignore the family of puzzles as you will.

Well I’ve never noticed that before. So a quadrilateral may be inscribed in a circle if opposite angles are supplementary, so each pair sums to the same value. While a quadrilateral has an inscribed circle if opposite sides sum to the same value. Is this a shadow of some projective duality?

David asks

Jesse, could you point me to something about this being cohomological?

I have to second that: I don’t understand yet how that geometric example is about cohomology.

Is this maybe supposed to be about cohomology with coefficients not in a group but in something like a tropical ring? I am trying to look at it from this angle, but I still don’t quite see it.

Is this just meant as something that “vaguely feels alike” involving exact sequences, or is the claim that there is some actual exact sequence of sorts playing a role?

Some of the ideas of Q-analysis have been usefully applied in sensible situations, but there is also some feeling that there was excessive enthusiasm for this theory in some social science circles (see J. Legrand – “How far can Q-analysis go into social systems understanding” ,
in Fifth European Systems Science Congress, 2002).

Recently there has been a renewal of interest in these ideas from the point of view of combinatorics as H. Barcelo and R. Laubenbacher – “Perspectives on A-homotopy theory and
its applications” , Discrete Math. 298 (2005), no. 1-3, p. 39 – 61, is related to it.

Atkin’s theory was based on one of my favorite results, due to Dowker, that the nerve of a relation and of the opposite relation have the same homotopy type. This was proved by Dowker in order to explain the fact that the Cech and Vietoris definitions of (co)homology relative to an open cover are isomorphic.

Thanks,Urs!

Thanks for the tip.

By the complete graph of thermal systems, I mean the graph whose vertices are thermal systems and with edges between every two vertices. The notion of cohomology I am using is just simplicial cohomology of this graph.

By the way, I should have said $T_{A/B}=0$.

In the context of Bohrification we meet reductions.

Let C subset D be commutative subalgebras and f a state.

Then we have a surjection S(D) \twoheadrightarrow S(C) which preserves the measure defined by f.

Full-text search does not seem to find the word ‘shea’ in the paper…

Maybe v-category aka measure space on page 4? That is, the
x-fans define coverings on $P^{op}$? Or perhaps minimal fans?
I do not see what sheaf condition would mean wrt x-fans, does
it make any sense ?

Let me quote the relevant part of the paper:

Spaces over P. A space X over P is, by definition, a covariant functor from
P to the category of sets, where the value of X on P ∈ P is denoted X (P ).

Minimal Fans and Injectivity. An x-fan over $b_i$ in a category is called mini-
mal if every a between x and $\{b_i \}$ is isomorphic to x. (More precisely, the arrow
x → a that implements ”between” is an isomorphism.)

An essential feature of minimal fans, say f_i Q → P_i, a feature that does not
depend on X (unlike the ∨-product itself) is the injectivity of the corresponding
(set) map from Q to the Cartesian product ∏i Pi (that, in general, is not a
reduction).

I feel I’d like to understand the paper better:

…the cone $\mathbb{R}_+^I$ is ugly, it breaks the Euclidean/orthogonal symmetry of $\mathbb{R}^I.$

We had such a cone in view over here.

Now we have the orthogonal symmetry, even better the unitary symmetry of $\mathbb{C}^I$, and may feel proud upon discovering the new world where entropy “truly” lives. Well…, it is not new, physicists came here ahead of us and named this world “quantum”. (p. 14)

In fact, isn’t that how the Maxwell Equations were first worked out? From Faraday’s work and Gauss’ etc. it was essentially understood how magnetic and mutual induction worked in wires, and how build-up of charge separation (e.g., at capacitors) induced electric fields; and in a sense what Maxwell did was to smooth-out and abstract away the physical wires; not to mention how his introduction of the displacement current term was quite explicitly argued from what nowadays should be called a homotopical invariance principle.

In fact, isn’t that how the Maxwell Equations were first worked out?

Yes, I guess so.

On the other hand, the cohomological nature of Kirchhoff’s laws was probably only noticed – or at least made explicit – after that of Maxwell’s equations was understood.