Homotopy Theory

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5:06 PM

Can some one help me to understand what does it mean to say 2-commutative diagram? context is in page 7 of arxiv.org/pdf/math/0605694.pdf It is the second diagram in that page... I am guessing it means there is a natural equivalence from $V\rightarrow \mathcal{X}\rightarrow \mathcal{N}$ to $V\rightarrow U\rightarrow \mathcal{N}$... I am not sure about the context of arrow inside the diagram.. In this arrow is pointing towards down and in next page arrow is pointing twoards up..

What does up arrow and down arrow in that diagram mean?

Denis Nardin

5:18 PM

The arrow in the middle of the diagram (which is often omitted and present only implicitly) is the datum of a natural isomorphism between the two possible compositions $V\to \mathfrak{Y}$

(by the way $\mathfrak{Y}$ is a Y in the Gothic alphabet, and $\mathfrak{X}$ is an X)

Recall that the category of (pre)stacks is a (2,1)-category, so it makes sense to speak about isomorphisms between two possible arrows

Praphulla Koushik

I dont think I understand... You are saying a diagram is 2-commutatve means there is a natural isomorphism beetween those two arrows and... ? ? "datum of a natural isomorphism between the two possible compositions" means?

Denis Nardin

A 2-commutative (or coherently commutative) diagram of that shape is essentially the datum of:

* Four object $A,B,C,D$

Praphulla Koushik

Yes I understand that you can talk when an arrow between two 1-morphisms is an isomorphism.. but what is the point of those up and down arrows?

Denis Nardin

Do you mean the arrow in the middle of the diagram?

Praphulla Koushik

Yes Yes... arrow in the middle of the diagram

Denis Nardin

5:23 PM

I am sorry I don't understand what you mean with up and down arrows

Praphulla Koushik

in page 7 there is a diagram where in middle of the diagram there is an arrow $\Rightarrow$ upwards..

In page 8 there is a diagram where in middle of the diagram there is an arrow $\Leftarrow$ downwards...

Denis Nardin

That's just a symbol you put to remember that there is an isomorphism from the composition $V\to U\to \mathfrak{Y}$ (which forms the top right corner of the diagram) and the composition $V\to \mathfrak{X}\to \mathfrak{Y}$ (which forms the bottom left corner of the diagram), so it is an arrow going from the top right to the bottom left

Since we are working with a (2,1)-category is not really important in which direction the 2-arrow goes (since we can always replace it with its inverse)

This kind of notation is used more often when people work with 2-categories where there are 2-arrows which are not isomorphisms

Praphulla Koushik

I think this is more or less same as natural transformation and natural equivalence... you dont give (because it does not matter) an arrow when you say there is a natural equivalence between functors where as you give an arrow (the direction) when you say there is a natural transformation (possibly not a natural equivalnece)

Denis Nardin

(sorry, apparently I cannot distinguish left from right)

Praphulla Koushik

"(sorry, apparently I cannot distinguish left from right)" :D what :D

Denis Nardin

5:29 PM

In my message before I wrote that the arrow went from the top left to the bottom right...

Praphulla Koushik

:D :D no problem... do you think what I said is correct??? "
I think this is more or less same as natural transformation and natural equivalence... you dont give (because it does not matter) an arrow when you say there is a natural equivalence between functors where as you give an arrow (the direction) when you say there is a natural transformation (possibly not a natural equivalnece)"

Reuben Stern

Suppose X --> Y is a left adjoint, and Y' --> Y is some map. When is X x_Y Y' also a left adjoint?

Denis Nardin

@PraphullaKoushik More or less. You could also use this decoration when it is important to give a name to the natural equivalence.

Praphulla Koushik

@DenisNardin Thanks Thanks :)

@ReubenStern I do not understand your question... :O

Denis Nardin

5:47 PM

@ReubenStern It's not hard to show it is true if X→Y is a localization (i.e. if the right adjoint is fully faithful). I don't know what happens in the general case

Reuben Stern

5:57 PM

I think it may also be true if X --> Y is fully faithful; those are the two cases I need

Saal Hardali

@EricPeterson Cool pictures! It got me thinking about how people actually do this stuff, what kinds of software are there for manipulations of spectral sequences?

Or maybe this was all done by hand in which case :0

Denis Nardin

@SaalHardali Eric knows way more than me, but my understanding is that these things are done mostly by hand (there is an amazing recent work by Dan Isaksen's group on the stable adams spectral sequence that uses a significant amount of computer calculations but even there you need to do a lot of hand computations)

Saal Hardali

I don;t mean to imply that the computer does a significant amount of calculations. I'm speaking only in terms of organizing what you already know (maybe + easy bonus stuff like propogating differentials etc...)

Eric Peterson

6:20 PM

people do do a lot of this by hand. software packages with various feature sets also exist:

bruner's Ext program computes minimal resolutions of steenrod modules and also draws the resulting Adams charts to PS/PDF; perry's Resolutions is similar but draws the resulting charts in a Java window; my macOS application Ext-Chart has some ability to deal with with spectral sequences whose E_1 pages are polynomial algebras; and very recently Hood Chatham et al are working on a very exciting project with multiple decoupled front- and back-ends github.com/hoodmane/js_spectralsequences

hood also has an excellent spectral sequence drawing package that lives entirely in latex, i'd recommend it to anyone over any other package

presumably the B-P-S pictures were computer-aided even in their computation—i would guess they even used something close to Bruner's program—but i don't know the details

& then there's also a bunch of private "back-end" material by wang, xu, et al

i think that's all i know

(& it is almost certainly not an exhaustive list of software tools, i'm limited to those tools which i myself have tried to use)

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