The n-Category Café

Eric wrote:

One side effect that hasn’t been too visible aside from my occasional piping in is that the totally public forum means that there are likely curious high school students and certainly undergraduate students watching from the sidelines.

There certainly are! Thanks for the link, it was very interesting. What other common categories have the property that looking very carefully at their arrows leads naturally to a larger category? The other example I can think of is commutative rings, but the sheaf construction subsumes that one, doesn’t it?

Speaking of helpful gems, this one is a must read

Lectures On n-Categories and Cohomology

It brings together in one place (with clear motivations) a lot of the stuff discussed around here and the nLab.

I wrote a minute ago:

I am interested in the right Bousfield localization of the projective global model structure.

Hm, maybe I am being stupid here. The explicit description should just be verbatim but dual to that of the left localization of the injective structure… With the descent condition on the fibrations becoming a codescent condition on the cofibrations. Hm.

Thanks Eric (here) and David (here) for amplifying this point.

I am wondering if it would be justified to summarize the lesson learned here very unspecifically simply as follows:

Let $\mathbf{H}$ be an $(\infty,1)$-topos and $X \in \mathbf{H}$. Then even though the homotopy of $X$ in $\mathbf{H}$ is the cohomology of $X$ in $\mathbf{H}^{op}$, both in general behave quite differently to the extent that $\mathbf{H}$ is quite different to $\mathbf{H}^{op}$.

Not that this is supposed to be deep, but it might maybe help see what’s going on with respect to the phenomenon that it seems David C. is wondering about:

even though two definitions may be related by [[abstract duality]], the objects defined by these definitions may not exhibit any duality to the extent that the ambient category $C$ fails to be equivalent to its opposite.

So in a simpler example: even though the notions of limit and colimit in Set is related by abstract duality, they behave very differently, as differently as $Set$ is different from $Set^{op}$.

That example, simple minded as it may be, is possibly not so far from the one that we are interested in once we realize that $Set$ is the canonical $(1,1)$-topos just as $Top$ is the canonical $(\infty,1)$-topos.

One author I read noted an ontological difference. I hope this is at least mildly interesting to you, or maybe only to others like me, neophytes.

I was glancing through “What is Category Theory” by Sica and
came across an interesing quote taken from the paper listed below.

http://www.math.purdue.edu/~gottlieb/Papers/papers.html
6) A History of Duality in Algebraic Topology ( with James C. Becker),
Chapter 25 of HISTORY OF TOPOLOGY, I. M. James, ed., Elsevier, Amsterdam, 1999,
pp. 725 - 745 (This versions differs slightly from the published version.)

2. Categorical Points of View. [page 3 and 4]
“There are two major groupings of dualities in algebraic topology:
Strong duality and Eckmann–Hilton duality. Strong duality was first
employed by Poincare (1893) in a note in which “Poincare duality” was
used without proof or formal statement. The various instances of
strong duality (Poincare, Lefschetz, Alexander, Spanier–Whitehead,
Pontrjagin, cohomology–homology), seemingly quite different at first,
are intimately related in a categorical way which was finally made
clear only in 1980. Strong duality depends on finiteness and compactness.
On the other hand, Eckmann–Hilton duality is a loose collection of useful
dualities which arose from categorical points of view first put forward
by Beno Eckmann and P.J. Hilton in Eckmann (1956).

Eckmann–Hilton duality was first announced in lectures by Beno Eckmann
and also by Hilton, see Eckmann (1956), (1958). A very good description
of how this duality works, and some eyewitness history is given in Hilton
(1980). Instead of being a collection of theorems, Eckmann–Hilton duality
is a principle for discovering interesting concepts, theorems, and questions.
It is based on the dual category, that is, on the duality between the target
and source of a morphism; and also on the duality between functors and their
adjoints.
In fact it is a method wherein interesting definitions or theorems are given
a description in terms of a diagram of maps, or in terms of functors. Then
there is a dual way to express the diagram, or perhaps several different dual
ways. These lead to new definitions or conjectures. Some, not all, of these
definitions turn out to be very fruitful and some of the conjectures turn out
to be important theorems. … Eckmann–Hilton duality was conceived as a method
based on a categorical point of view in the early 1950’s. The challenge was to
use the point of view to generate interesting results.”

On the other hand, there are lots of ways in which it is `good’ or, at least, beautiful.

In fact your question is answered on the nLab, directly below your question — but perhaps in terms too erudite for you to recognize it as an answer!

I was in the middle of describing how the familiar abelian Cech cohomology follows from the general nonsense, when I saw Eric’s query box.

Being already under time pressure with the Cech cohomology thing and because I need to work on the entry model structure on simplicial presheaves for a talk next Monday, I took just five minutes off to hastily scribble a reply to Eric.

As I logged at latest changes:

- following Eric’s question I typed a quick reply into cohomology on how the ordinary notion of cohomology in cochain complexes is reproduced. In principle this gives all the necessary information, but I’ll try to find the time later to give a long detailed exposition of how this basic important special case arises from the very general perspective

Unless you beat me to it, of course.

The best way to explain this would be to first describe how chain homology is really the homotopy of a chain complex.

That way all the dualizations don’t get in the way.

Then cochain cohomology is correspondingly the cohomotopy=cohomology.

Kowabunga this is a nice paper :)

I added it as a reference to

[[chain homology and cohomology]]

I wish Kotiuga would get into this stuff. What I’m after is something like “n-categories for scientific computation” (not computer science). I’m not likely up to the task. Cohomology is important for applied computational electromagnetics so this is getting close enough that those interested in scientific computation might start taking note (if they haven’t).

Re-reading this discussion maybe for the record one should still say:

if $V^\bullet$ is a cochain complex, then a higher coboundary of degree $(n+1)$ is simply an element in $d(V^n)$.

Similarly for $V_\bullet$ a chain complex, an $n$-dimensional boundary is an element in $\partial V_{n+1}$.

So the (co)boundary of a (co)boundary always vanishes, but there are nevertheless “higher” (co)boundaries.

On [[cohomology]], it says:

More precisely:

• the objects $c\in H(X,A)$ are the cocycles on $X$ with values in $A$;
• the morphisms $\lambda:c\to c′$ in $H(X,A)$ are the coboundaries between cocycles;
• the higher morphisms in $H(X,A)$ are the higher coboundaries or coboundaries of coboundaries.
• the equivalence classes $[c]\in\pi_0 H(X,A)$ are the cohomology classes.

I think the term “coboundaries of coboundaries” is loaded and should probably be replaced with simply “higher coboundaries”.

Or perhaps even better, I think I’d like to change the paragraph to:

More precisely:

• the objects $c\in H(X,A)$ are the cocycles on $X$ with values in $A$;
• the j-morphisms in $H(X,A)$ are the coboundaries;
• the equivalence classes $[c]\in\pi_0 H(X,A)$ are the cohomology classes.

Is that acceptable?

PS: I may go ahead and change it, but we can easily roll it back if I mess something up.