By Emily Riehl

Lecture notes from the Thursday Seminar held at Harvard in Spring 2013

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At dimension 2, given •❅ x ⑦c ❅❅ ⑦⑦ G1 • • y you might declare it to be invertible if there is • ⑦c ❅❅❅ ⑦ ⑦ G1 • • y x so that certain horns you can build from these faces have fillers. In particular, this implies that the unlabeled edges are also invertible. Calling this edge z, it turns out you can get, from a filling argument, a 2-simplex • ⑦c ❅❅❅❅❅❅ ⑦⑦ G yx • z • that should be thought of as invertible. Writing Cobt for the original weak complicial set and write Cobi for the saturation, the thing in which everything that looks like an equivalence is an equivalence.

Then n ⊗ O⊗ ((c1 , . . , cm ), (d1 , . . , dn ))φ i=1 O ((c1 , . . , cm ); di )ρi ◦φ . (iii) There are canonical isomorphisms of categories in the fibers over objects O⊗m (O⊗1 )×m . 15. An ∞-operad is a map of simplicial sets O⊗ → NF such that (i) Given an inert φ : m → n in F and any C ∈ O⊗m there is a πO -coCartesian lift of φ. (ii) Can use the coCartesian lifts of the ρi to produce a map as in (ii) above which we require to be a homotopy equivalence. (iii) The map O⊗m → (O⊗1 )×m is a categorical equivalence.

You can read about this is Barwick’s thesis, in Lurie’s Goodwillie paper, and also in [BSP12]. The starting point is the Reedy model structure on ∆×n -spaces. The Barwick model structure is a left Bousfield localization at three classes • Segal∆×n • Glob∆×n • Comp∆×n The first localization gives you some sort of n-fold category. Localization at the second two classes together is what enforces the globularity condition. , we can define all of these classes in ∆×n -Set. 8. ,mn . 9 (Segal maps). Segal∆ is the set of spine inclusions into ∆m for m ≥ 0.

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