[HoTT-reading-group] Background reading in homotopy theory and higher category theory?

[Dmitry Vagner]

I should have just included the link


http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
—
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On Mon, Jan 6, 2014 at 9:19 PM, Jason Gross <jasongross9 at gmail.com> wrote:

> Topology, by Munkres<http://www.amazon.com/Topology-2nd-Edition-James-Munkres/dp/0131816292>,
> is
> a good intro to point-set topology
> Algebraic Topology, by
> Hatcher<http://www.math.cornell.edu/~hatcher/AT/AT.pdf>, is
> a good intro to algebraic topology
> Category Theory for the Working Mathematician, by Mac
> Lane<http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf>,
> is the canonical intro to category theory; David Spivak wrote Category
> Theory for Scientists <http://arxiv.org/abs/1302.6946>; Homotopy theories
> and model categories, by Dwyer and
> Spalinski<http://folk.uio.no/paularne/SUPh05/DS.pdf>,
> contains a 10 page crash course on basic category theory, as well as some
> expository Quillen model categories, which generalize the idea of homotopy.
>  (I heard somewhere that the only results provable in HoTT are the ones
> that are true in all Quillen model categories.)
> Wikipedia<http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence#General_formulation>is
> a decent source on the Curry-Howard isomorphism; Bob
> Harper has a blog post on computational
> trinitarianism<http://existentialtype.wordpress.com/2011/03/27/the-holy-trinity/>,
> which is an extreme and interesting view on the Curry-Howard isomorphism.
> Dan Licata gave a presentation "Git as a
> HIT"<http://dlicata.web.wesleyan.edu/pubs/l13git/git.pdf> about
> applying higher inductive types to version control.
> There's the Homotopy Type Theory google
> groups<https://groups.google.com/forum/#!forum/homotopytypetheory>
> (also
> a mailing list), which is primarily for research-level HoTT questions and
> discussions.  Thierry Coquand announced his cubical sets
> evaluator<https://groups.google.com/forum/#!topic/homotopytypetheory/GmXKEArD3HY>on
> it (where this is also a lot of discussion on canonicity); it's
> available
> on github <https://github.com/simhu/cubical>.  The linked
> There's also the HoTT amateurs google
> group<https://groups.google.com/forum/#!forum/hott-amateurs>.
>  If you have questions about things in the book, you should feel free to
> email and cc both hott-reading-group at mit.edu and
> hott-amateurs at googlegroups.com.
> -Jason
> On Mon, Jan 6, 2014 at 5:31 PM, Joe Hannon <hannon at math.bu.edu> wrote:
>> That paper also has an explanation of the periodic table of categories,
>> including -1-categories and -2-categories, which might give some context to
>> today's discussion of what a -1-type should be.
>>
>> As a counterpart to Chris's call for references for some math discussions
>> held today (which must've occurred after I left), as a math student I felt
>> a little out of my depth during some of the more CS-y discussions. Is there
>> some nice reference I could look at for this Howard-Curry theorem? Will
>> that be relevant to our discussions going forward?
>>
>> Joe
>>
>> On Jan 6, 2014, at 17:13, Dmitry Vagner <dmitryvagner at gmail.com> wrote:
>>
>> Thank you Jason for the wonderful group and for including us remotely!
>> Chris, I learned a lot about the homotopy/groupoid/category theory
>> interface from this amazing expository paper by John Baez:
>>
>> http://arxiv.org/pdf/math/0608420v2.pdf
>>
>> For our purposes, section 2 is where all of the interesting (and most
>> accessible) information is. Only the basic definitions of category theory
>> (along with your foundational algebraic topology understanding) may be
>> required to get a lot out of this section. Of particular interest is
>> section 2.3 - it's on this thing called the "homotopy hypothesis" - which
>> roughly says that "homotopy n-types are the same as n-groupoids" - taking n
>> to the limit yields what you are interested in, that from the homotopical
>> perspective, "topological spaces are the same as (weak)
>> infinity-groupoids." I feel like this paper does a great job of expositing
>> these ideas, including what exactly a homotopy n-type is, without much
>> technical background.
>>
>> Hope that helps,
>> Dmitry
>>
>>
>> On Mon, Jan 6, 2014 at 4:47 PM, Peng Wang <wangp.thu at gmail.com> wrote:
>>
>>>
>>>
>>>
>>> On Mon, Jan 6, 2014 at 4:41 PM, Chris Jeris <cjeris at gmail.com> wrote:
>>>
>>>> First of all, thanks very much to Jason for putting the group together
>>>> and also making it accessible to us remote participants!
>>>>
>>>> I am intrigued by the statement that "infinity-groupoids are the natural
>>>> models of homotopy theory", but I know little about homotopy theory and
>>>> nothing about higher category theory.  I have algebraic topology at about
>>>> the level of Massey's first course (rusty by some years) and basic category
>>>> theory.  Can anyone suggest some expository works in this area as
>>>> background or further reading?
>>>>
>>>
>>> Adding to Chris' question, I don't even know about topology or algebraic
>>> topology, so is there some introductory material on that? (Or is that
>>> needed?)
>>>
>>>>
>>>> thanks, Chris Jeris (freenode: ystael)
>>>>
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>>>>
>>>
>>>
>>> --
>>> Peng Wang (王鹏)
>>> CSAIL, The Stata Center, MIT
>>> 77 Massachusetts Ave, 32-G822
>>> Cambridge, MA 02139
>>> Phone: (617)803-2025
>>> Email: wangpeng at csail.mit.edu
>>>
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