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I am splitting off from the geometry of physics cluster a chapter geometry of physics – homotopy types.
For the moment I have there mostly section outline as well as some material copied over from my homological algebra lecture notes. My aim is now to put in a gentle discussion of Dold-Kan that leads an audience familiar with chain complexes from homological algebra to simplicial homotopy theory.
I’ll be touching a bunch of related entries in the process.
it’s taking shape (geometry of physics – homotopy types). I have now been collecting pieces of lecture notes that I had previously typed into various nLab entries (e.g. at prequantum field theory) and started to add some glue here and there. Needs more polishing and further expansion, but there is now a bit of substance.
I am slowly trying to produce a textbook-style page with a comprehensive introduction to plain homotopy theory, for readers with a little bit of physics motivation and with a minimum background in basic category theory, as provided by the page geometry of physics – categories and toposes (the “previous chapter” in the series).
My main addition as of today is that I completely reworked the very first lead-in paragraphs.
The bulk of the entry currently is a plain copy-and-paste merge of two-and-a-half sets of lecture notes that I had produced elsewhere, namely:
1) abstract homotopy theory (i.e. model category theory),
2) topological homotopy theory (i.e. the classical model structure on ),
3) simplicial homotopy theory (i.e. the classical model structure on )
4) abelian homotopy theory (i.e. the homotopy theory of chain complexes, aka homological algebra via derived categories)
Here 1) and 2) and 4) are fairly comprehensive and polished in themselves (and tested in practice, I had used these for teaching master courses in the past) but need some cross-linking now, especially with 3), where both mention Kan complexes.
(I cannot decide if homological algebra should come before abstract homotopy theory (easier motivation?!) or after (easier to see the wood for the trees?) Maybe not so important once there is enough guiding text for how to read the sub-sections.)
However, section 3) “simplicial homotopy theory” is skeletal at the moment. This is material that I had started to produce for my teaching of “Introduction to Stable Homotopy Theory”, but then abandoned when I realized that this was becoming too much for the available time.
I should want to expand this and augment it by a 5th section on model structures on simplicial presheaves. Or maybe the latter should better get an entry of its own, not sure yet.
What is “plain” homotopy theory? Does the homotopy theory of simplicial presheaves qualify as “plain” homotopy theory?
Meanwhile, I am teaching a year-long course on topology starting Fall 2018 with a similar collection of topics (simplicial sets, chain complexes, model categories, spectra, simplicial presheaves and sheaf cohomology, but not that much on topological spaces), and am preparing a set of notes on my own https://dmitripavlov.org/notes/2018f-5324.pdf (currently a rough draft in statu nascendi). It starts with a (very elementary, I hope) introduction to simplicial sets and simplicial homotopy theory, with essentially no prerequisites other than some elementary set theory and elementary algebra. (You may like that the PDF is thoroughly hyperlinked, nLab-style, including terms in text and mathematical objects in formulas (!).)
I hope this will be more elementary and accessible than, say, Goerss-Jardine or Joyal-Tierney, both of which demand substantially more material than what is really necessary to present this topic, e.g., quite a bit of material on categories, as well as topological spaces, etc.
I have the Menagerie notes available (longer version than on the n-Lab) if anyone wants to encorporate them in a reading course. Several times I have cut down those notes to provide short focussed versions for use in workshops.
By “plain homotopy theory” I mean .
Dmitri, Tim and others,
it would be nice if, eventually, this kind of lecture-notes/textbook style material is all reflected in the relevant Lab entries, i.e., conversely, that one could in principle compile a lecture note by copy-and-pasting the relevant Definitions/Theorems/Proofs from the relevant Lab entries into joint new page, and add some connective text around it.
In other words, it would be nice if we were not all writing our own notes separately and in duplication, but if we were joining forces to incrementally expand and improve a universal set of notes, to be found on the Lab.
So as I am writing the “geometry of physics”-series, I am trying to make sure that all the polished content it has is fed back into its relevant stand-alone Lab entries.
This is not always easy, but it is always worth it. For instance I wanted the entry free cocompletion to have more decent content in this vein, see the discussion there. That entry is a little better now, even if still not good.
I agree it would be a good idea, but that needs a bit of planning otherwise several people will start and go in different directions. For instnace, the Menagerie just grew and so is not that directly useful BUT it has a lot of material in it that is not easy to find elsewhere. If a small group of us exchanged notes to see what could be planned out (e.g. via dropbox or something similar) then try out a structure. Urs, your introductions set a possible route, although I do not particularly like restricting Abstract Homotopy Theory to mean largely the theory of model categories, but that is the prevailing flavour of the subject so that is useful.
needs a bit of planning otherwise several people will start and go in different directions.
No, really not. If there is one thing that we need not worry it is that we have too many people working on Lab entries. Also, the good thing about sticking to some minimum of formalized writeup (coherent definition/theorem/proof-style) is that it’s modular (each thought a numbered item).
Maybe all I am saying is that there is nothing stopping us from proceeding more like the StacksProject must have been proceeding all along.
Re #7: I am certainly open to posting these notes on the nLab.
In fact, I intend to do so once I can find time to write a convertor from TeX to the nLab syntax, which should not be too difficult.
Posting on the nLab directly is not an option, at least for me, because I need to distribute a reasonable nicely typeset version to my students, but the TeX output is currently broken on the nLab and the PDF files produced directly from HTML look really ugly (the level of ugliness kinda reminds me of typewritten manuscripts that were published as books not so long ago).
once I can find time to write a convertor from TeX to the nLab syntax
Re #11: Interesting, I had no idea that something like this existed!
Just to say that I re-arranged, as I changed strategy of exposition:
Instead of keeping discussion of homotopy theory separate from that of category theory, I am now merging it into one unified discussion all under geometry of physics – categories and toposes.
So this means that geometry of physics – homotopy types is now no longer readable stand-alone, as many of its cross-references are broken. (I have tried to remove all direct links to it, though I may have missed some.) To read it one now needs to go to geometry of physics – categories and toposes and see from section 6 on.
The strategy is to give a completely 1-categorical exposition to higher category theory, equipped with an enriched end/coend toolbox, by introducing Ho(CombModCat) (now section 9) and speaking of its objects as presentable -categories.
Re #13: Currently geometry of physics – homotopy types says:
This page contains the following errors:
error on line 13908 at column 299: AttValue: ' expected
Below is a rendering of the page up to the first error.
The error seems to be nondeterministic: sometimes the page displays correctly.
Hm, strange, haven’t seen that yet.
(I cannot decide if homological algebra should come before abstract homotopy theory (easier motivation?!) or after (easier to see the wood for the trees?) Maybe not so important once there is enough guiding text for how to read the sub-sections.)
Tough call, although my own criterion would be: which is the one with the greater need-to-know for current students? If there is sufficient need-to-know, the students will learn it.
(I cannot decide if homological algebra should come before abstract homotopy theory (easier motivation?!) or after (easier to see the wood for the trees?) Maybe not so important once there is enough guiding text for how to read the sub-sections.)
In my notes, I chose the following path:
Introduce (co)homology of simplicial sets and/or spaces.
Introduce homotopy (co)limits of simplicial sets and/or spaces as a tool to construct new spaces from old ones.
Observe that the chain-level (co)homology functor maps homotopy (co)limit of spaces to homotopy limits of chain complexes.
Compute homotopy limits of chain complexes using homological algebra, naturally arriving at abstract homotopy theory.
I think the most important things for me woud be that chain complexes should be treated as whole homotopical objects using homotopical ideas (e.g. derived functors in the homotopical sense, not in the ad hoc “how far can we extend this exact sequence” sense), and that they should be motivated rather than pulled out of thin air.
From a HoTT perspective, the natural route is to start with represented (co)homology of spaces, which doesn’t require any chain complexes, though it does introduce exact sequences. Chain complexes pop up later as a technical tool, e.g. in constructing a spectral sequence via exact couples, an exact couple involves graded chain complexes. This matches the modern brave-new-algebra perspective whereby the fundamental objects are structured spectra, with chain complexes being just the special case of module spectra over Eilenberg-MacLane ring spectra.
If one doesnt want to go all that way (understandable!), then chain complexes could be introduced as a target for cellular/singular (co)chains. I don’t know if it’s necessary to introduce homotopy (co)limits to motivate homological/homotopical algebra of chain complexes; we already need chain homotopy theory in order to say that cellular/singular (co)chains are homotopy invariant.
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