Homotopy Theory

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4:16 AM

@PraphullaKoushik I would say start with Tom Leinster's Basic Category Theory arxiv.org/abs/1612.09375 and then pull out Emily Reihl's Category Theory in Context ncatlab.org/nlab/show/Category+Theory+in+Context

If you can source ncatlab.org/nlab/show/Handbook+of+Categorical+Algebra (cough internet) then it will be a good reference, but not something to read from cover to cover. Chapters 7 and 8 of volume 1 will definitely be useful, chapter 5 in vol 1 is a baby version of the stuff I keep telling your about. etc.

@DenisNardin Cisinski's new book?

Praphulla Koushik

4:44 AM

@DavidRoberts I saw volume 2 of that Hanbook of categorical algebra. I read chapter 1 there that discuss Abelian categories... I some how did not even look at index of volume 1. I see that chapter 7 discuss 2-categories/bicategories... Thanks... this seems to be correct choice for my requirement ..

Praphulla Koushik

4:58 AM

I saw Tom Leinster's book.. I read some things from there.. Did not see Emily Reihl's book before... I saw just now... looks good... Thanks for this also..

5 hours later…

Harry Gindi

10:18 AM

@DavidRoberts It's on his website. It's an elementary presentation of quasicategory theory that gets pretty far

Main defect is no 'marked straightening', but that's being worked on now

the goal of the project, as I understand it, is twofold. A completely elementary proof of marked straightening/unstraightening without simplicially enriched categories, and a kind of univalence for cartesian fibrations

In the language of Joyal's clans and tribes

I guess you could say the idea is: HoTT is for infty,1 toposes and the idea of their project is to show that it all makes sense for infty,2 toposes in a suitable way

or at least for the infty,2 topos Cat_\infty

Martin Sleziak

Since I see that some recommendations for category theory books have been mentioned here, I will add a link to: Good books and lecture notes about category theory.. Maybe also other questions linked here might be worth looking at: chat.stackexchange.com/transcript/68360/2017/11/15

Harry Gindi

10:46 AM

I actually agree with the first answer. How could you possibly do CT without knowing algebra…

Denis Nardin

The reason I like MacLane so much is that it really lives to its premise of being an introduction to category theory for the working mathematician. It is littered with examples of why category theory concepts arise naturally when doing something else. A lot of other introductions seem to be more of the "bring you own motivation" kind.

This is also why the importance of non-pointwise Kan extensions doesn't bother me. It's not a distinction that arises in practice while not doing category theory, so meh.

I also like Borceaux's book a lot, but it's not really an introduction

I think this is what lacks most in the current introductions to (∞,1)-category theory. We need something that shows how (∞,1)-categories are used, not only how they're defined

Harry Gindi

I feel like stable infty categories are the "killer app" of infty,1 cats

Denis Nardin

That's still too abstract

I think you need to show off how easier the proofs in algebraic K-theory are, how easier is to define motivic spaces, how to construct the symmetric monoidal structure on the derived category of a scheme...

Harry Gindi

That's what I had in mind

The derived cat of a scheme escapes model categories without strong assumptions on the base

according to that paper of (iirc) Hovey

Denis Nardin

Sure, but then stable (∞,1)-categories are not an application. They are a tool that has applications. I think a good introduction needs to focus on the applications themselves

Harry Gindi

11:01 AM

Homotopy theory of homotopy theories is still too abstract as well?

Denis Nardin

Wayyy too abstract

You have to give applications whose statements make sense outside of (∞,1)-category theory

Harry Gindi

"Tor has derived functors over any scheme" is too abstract still?

Denis Nardin

People didn't become interested in derived categories because they're pretty, but because it allowed them to define sheaf cohomology groups

@HarryGindi That's closer (although some people might say "well, I care only about q-projective varieties")

Harry Gindi

Hmm

"The cotangent complex can be patched from affines"?

Denis Nardin

Sure, the cotangent complex is a good test case, although you can define it without (∞,1)-categories

Something like Scholze's work on p-adic Hodge theory is also good to bring in number theorists

Factorization homology should be good for at least some kind of differential geometer

Harry Gindi

11:08 AM

true!

Denis Nardin

You have to put yourself in the place of people that never even heard of simplicial sets or model categories

3 hours later…

Mike Miller

1:46 PM

I like citing Nikolaus-Scholze's result that the multiplicative structure on Tate cohomology is unique up to contractible choice, which I don't believe was known before them and uses Dwyer-Kan localization, but maybe this is too low-tech. But the topologists I have told about this usually find it convincing.

2 hours later…

user131753

3:43 PM

In The Joy of Cats it is written that (slightly paraphrased),

user131753

"[If $F:(\mathbf{A},U)\to (\mathbf{B},V)$ be a concrete isomorphism between the concrete categories $(\mathbf{A},U)$ and $(\mathbf{B},V)$ then the fact] that such a concrete isomorphism exists means, informally, that each structure in $A$, i.e., each object $A$ of $\mathbf{A}$, can be completely substituted by a structure in $\mathbf{B}$, namely $F(A)$ (keeping, of course, the same morphisms)."

user131753

What does it mean to say that each $A$ of $\mathbf{A}$, "can be completely substituted by a structure in $\mathbf{B}$"? What is the informal difference between a concrete isomorphism and an isomorphism? I am trying to understand the big picture behind the concepts, I understand the formal definitions.

Harry Gindi

4:11 PM

They found an unexploded bomb across the street from my home ><

Anyone know how long bomb disposal usually takes?

Mike Miller

4:22 PM

I suspect you will be the first here with expert knowledge on the subject

Malte L.

Once the evacuation is complete I believe it usually takes something like 2-3 hours until you are allowed back to your home (but I don't have any first-hand knowledge either).

1 hour later…

Harry Gindi

5:44 PM

Thanks!

I just had no idea wbat kind of timeframe to expect

I wasn't sure if I'd have to find a hotel or something

pupshaw

6:31 PM

@HarryGindi I once had a similar situation. We were cleared to return within a few hours but there was tremendous police and journalistic activity for a week or two after. If you can afford to get away for a few days it might not be a bad idea

1 hour later…

Harry Gindi

7:47 PM

@pupshaw It's a world-war 2 era bomb, not a terrorist attack

They weren't able to remove it tonight, so we're being evacuated again tomorrow

David Roberts

8:37 PM

@HarryGindi thanks :-)

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