May 2010

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Friday, December 4th, 2009 12:01 am
+ snow
- failed cooking experiment? I guess I won't know until morning.
- still SO CONFUSED about math. And ready to move on to a new article, I think it's been at least a month.
- students who suddenly realize they don't understand math and expect me to be constantly checking my email and available to meet with them in the two hours they're apparently free. IT IS READING WEEK. Am I wrong to assume that they have lots of unstructured time this week? Things to do, yes, but they could Do Those Things at unreasonable hours and Meet With Me when I'm at school. Right? This has been true in all other terms. I'm so confused... (apparently a theme?)
+ wrote a proposal for my orals.

1. Main topic: The homotopy theory of homotopy theories

The category of topological spaces comes with two very natural homotopy theories, one related to “inverting homotopy equivalences” and the other to “inverting homotopy isomorphisms”. Each of these theories provides a homotopy category but also extra structures such as, if the definition of topological spaces is chosen correctly, mapping spaces which interact well with the newly inverted morphisms. A homotopy theory is a category which behaves roughly like the category of topological spaces does with this extra structure; at its most basic, it is a category with a class of morphisms, called weak equivalences, which are not necessarily isomorphisms but which it is interesting to think of as invertible. There are various ways to formalize the structure which is needed to ensure that this works. Two classical ways use model categories and simplicial categories (or categories enriched over simplicial sets), but more recent definitions of (∞, 1)-categories can also be viewed as definitions of homotopy theories. One early result was that the model categories for rational homotopy theory of spaces and for reduced differential graded Lie alge-bras over Q are Quillen equivalent (a notion of equivalence weaker than equivalence of categories but stronger than equivalence of homotopy categories), which allowed computations about spaces to be done instead with differential graded algebras [Qui69].

In fact, we can view Quillen equivalences as candidates for the class of weak equivalences in the category of small model categories. There are similar notions of weak equivalence for simplicial categories and for any definition of (∞, 1)-categories. Since we’re viewing all these homotopy theories as models for the homotopy theory of homotopy theories, we’d like them to be essentially the same. If they are, we can move freely between them in our attempts to discover models for given homotopy theories that ease computation. One technique is to make as many of these categories with weak equivalences as possible into model categories and check that they are Quillen equivalent.

The model structure for complete Segal spaces is a left Bousfield localization of the Reedy model structure on simplicial spaces (or bisimplicial sets), and there is a way to assign a complete Segal space to every simplicial model category which respects the homotopy category up to equivalence and the function complexes up to weak equivalence [Rez01]. Simplicial categories are also equipped with homotopy categories and function complexes, and we can equip the category of small simplicial categories with a model structure [Ber07a]. These two model categories are connected by a string of Quillen equivalences through two model category structures on Segal precategories (simplicial spaces with discrete 0-spaces) with the same weak equivalences [Ber07b]. There is also a model category structure on simplicial sets in which the fibrant objects are weak Kan complexes (or quasicategories), with Quillen equivalences to the structure for complete Segal spaces which go in both directions [JT07].

2. Minor topic: Model categories and localizations

There is a theorem of Kan that is frequently used to establish the existence of a cofibrantly generated model structure on a given category by checking certain compatibility conditions between proposed sets of generating cofibrations and trivial cofibrations [Hir03, 11.3.1]. This theorem uses the small object argument [Hir03, 10.5.16] repeatedly; it is also useful for the following results about localizations of model categories.

For any generalized homology theory there is a localization of the usual model category structure on simplicial sets which has the appropriate homology equivalences as weak equivalences [Bou75]. The argument can be generalized to produce a localization with respect to any map of simplicial sets [Hir03, 2.1.3]. Under appropriate circumstances, it can be generalized to other model categories [Hir03, 4.1.1].

References

[Ber07a] Julia E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043–2058 (electronic). MR MR2276611 (2007i:18014)

[Ber07b] Julia E. Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), no. 4, 397–436. MR MR2321038 (2008e:55024)

[Bou75] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150. MR MR0380779 (52 #1676)

[Hir03] Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR MR1944041 (2003j:18018)

[JT07] Andre Joyal and Myles Tierney, Quasicategories vs Segal spaces, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 277–326. MR MR2342834 (2008k:55037)

[Qui69] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR MR0258031 (41 #2678)

[Rez01] Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 3, 973–1007 (electronic). MR MR1804411 (2002a:55020)

+ David and I had breakfast with my mom and aunt this morning. It was gingerbread and apple crisp. Yay dessert-breakfast!
+ cookies at Sarah's tomorrow. Yes, I am prioritizing this over meeting with students. Clearly I do not care whether they learn math and am a bad TA. But good at cookies!
? I have some invitations to the Google Wave. (It gets a definite article because it is suspicious and new! That is a rule of grammar.) Do you want one?

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