# Higher categories study group

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizer: Marco Castronovo

For those who were interested in learning about higher categories, a study group started on Wednesday, May 13th, from 9 a.m. to 12 noon at the HIM lecture hall.

The plan was to understand a survey of Moritz Groth on ∞-categories (pdf). Familiarity with category theory and simplicial sets was assumed. There were talks and discussions afterwards.

TALK 1 May 13. Daniela Egas (MPIM Bonn)

- Kan complexes as model for (∞,0)-categories
- quasi-categories as model for (∞,1)-categories
- homotopy category of a q.c.
- mapping space of 2 objects in a q.c.

TALK 2 May 18. Dominic Culver (Notre Dame)

- C q.c. and X s.s. ⇒ C
^{X}q.c. - contractibility of compositional choices in a q.c.
- q.c. are the fibrant objects in Joyal model structure
- equivalence and weak equivalence of q.c.

TALK 3 May 20. Mark Penney (Oxford)

- simplicially enriched category of a s.s.
- homotopy coherent nerve
- combinatorial description of s.e. categories via necklaces
- many homotopy coherent nerves are q.c.

TALK 4 May 22. Jeremy Mann (Notre Dame)

- initial object of a q.c.
- join of q.c.
- slice q.c.
- colimits in q.c.

TALK 5 May 29. Manuel Araujo (Oxford)

- Grothendieck opfibrations and Segal condition
- coCartesian fibrations of q.c.
- symmetric monoidal ∞-categories and functors
- algebra objects

TALK 6 June 22. Shan Shah (Utrecht) & Hiro Tanaka (Harvard)

- Dold-Kan correspondence
- dg nerve
- Fukaya categories