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:- Explicit combinatorial construction using trees. C. Berger, A cellular nerve for higher categories. Paper looks long and hard to read...
- Wreath product and suspension. Proposition 3.9 of C. Berger, Iterated wreath product of the simplex category and iterated loop spaces.
- 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.