Topics in Calculus and Algebra, Course Notes

This course was taught at University of Cambridge in the Lent term of 2012 by Professor I. Grojnowski. Scanned lecture notes here (.pdf). Below are just some comments/general thoughts of mine about the course.

• Lecture 1 (1/19/12) [p.1]
  Going to be working toward's Lurie's theorem on TFTs. Reference: J. Lurie, On the classification of topological field theories. This paper will act as a guide throughout the course.
• Lectures 2-4 (1/24, 1/26, 1/31/12) [p.5, 8, 11]
  Introduction to simplicial sets, model categories, homotopy theory, etc. This is all covered in P. Goerss, K. Schemmerhorn, Model categories and simplicial methods.
• Lecture 5 (2/2/12) [p.14]
  Localization of model categories.
• Lectures 6-7 (2/7, 2/9/12) [p.17, 20]
  Overview of Rezk's model for \((\infty, n)\)-categories as \(\Theta_n\)-spaces. Reference: for a nice introduction, see Rezk's lecture slides. The actual paper is C. Rezk, A cartesian presentation of weak n-categories. In his paper, Lurie uses a different model due to Barwick (which has been shown to be equivalent: see here). Point of confusion for me: there isn't a single agreed upon definition of an \( (\infty,n) \)-category for \( n\gt 1 \)! (See nLab for clarification.)
• Lecture 8 (2/16/12) [p.24]
  Vague statement of Lurie's theorem in higher categorical language.
• Lectures 9-11 (2/21, 2/23, 2/28/12) [p.27, 30, 33]
  Geometric realization of "globular/cellular spaces" \( \lvert \cdot \rvert : \mathrm{sPSh}(\Theta_n) \to \mathrm{Sp}\). Three proof sketches:
  1. Explicit combinatorial construction using trees. C. Berger, A cellular nerve for higher categories. Paper looks long and hard to read...
  2. Wreath product and suspension. Proposition 3.9 of C. Berger, Iterated wreath product of the simplex category and iterated loop spaces.
  3. Realization as the simplicial nerve of subobject-posets via the flat pregeometric Reedy category structure of \(\Theta_n\). First paragraph and Proposition 3.14 on p.257 of ibid.
• Lecture 12 (3/1/12) [p.35]
  Reduced \(\Theta_n\)-spaces as a model for \(n\)-fold loop spaces: the main result (Theorem 4.5) of ibid. This was traditionally formulated using the little cubes operad, as a theorem of Boardman-Vogt, May, Segal. For \(n=1\), see G. Segal, Categories and cohomology theories.
• Lecture 13 (3/6/12) [p.39]
  Desired/expected generalized model for loop spaces: Quillen adjoint functors \[ B^d : E_{d+d'}\textrm{-monoidal }\Theta_n\textrm{-Sp}_*^{Rezk} \rightleftarrows E_{d'}\textrm{-monoidal }\Theta_n\textrm{-Sp}_*^{Rezk} : \Omega_E^d \] along with adjoint functors \(\Sigma^d, \Omega^d\) between appropriate simplicial presheaves [not yet in literature?].
  
• Stable model structure for spectra: A. K. Bousfield, E. M. Friedlander, Homotopy theory of \(\Gamma\)-spaces, spectra, and bisimplicial sets. Infinite loop spaces (\(n=0, d=\infty\)) via spectra and \(\Gamma\)-spaces, due to Segal. This is all summarized in Berger's paper!
• Lecture 14 (3/8/12) [p.41]
  Homotopy type of little disks/cubes operad. See D. Ayala, R. Hepworth, Configurations spaces and \(\Theta_n\).
• Lecture 15 (3/13/12) [p.45]
  Ran space, factorizable cosheaves, loop Grassmannian: summary/preview of things going on in Gaitsgory's seminar.