### DWIC: Dg With Infty Categories Seminar (Fall 2021)

The main topics for this seminar are dg categories and infinity categories. Our main references for dg categories are [K] and [T], and our main references for infinity categories are ???

- When: Monday 4PM - 5:30PM ET (dinner afterwards)
- Where: Room 528
- Organizers: Matthew Hase-Liu and Ivan Zelich
- References:
**[K]** Keller, Bernhard *On differential graded categories*, ICM notes
**[K]** Toen, Bertrand *Lectures on DG-categories*, lecture notes
**[T]** Hinich, *Lectures on infinity categories*, lecture notes
**[H]** Groth, *A short course on infinity categories*, survey
- The notes from the seminar are not here.

#### Schedule

Week 1: Triangulated/derived categories

Week 2: Dg categories part I: basic definitions, functors, examples, and pretriangulated dg categories

Week 3: Previous talk continued

Week 4: Dg categories part II: symmetric monoidal structure, homotopy category of dg category, and localization

Week 5: Infty categories part I

Week 6: Infty categories part II

Week 7: Infty categories part III

Week 8: ???

Week 9: ???

Week 10: ???

Week 11: ???

- Sept 27
- Fan Zhou
**Review of triangulated and derived categories**, notes

I will introduce the basic language of triangulated and derived categories, along with applications to derived functors.
- Oct 04
- Matthew Hase-Liu
**Introduction to dg categories**

I will talk about dg categories, dg functors, and why we care about these constructions in the context of triangulated categories. To this end, we'll discuss pretriangulated dg categories, which generalize the shift and mapping cone operations on the level of complexes.
- Oct 11
- Matthew Hase-Liu
**Introduction to dg categories continued**

I will finish off what I didn't cover last time.
- Oct 18
- Ivan Zelich
**Derived Morita Equivalence**

We discuss generalising Morita Equivalence in the DG-category setting. Consequently, we will introduce suitable notions of projectivity, i.e. Tilting complexes, and how all this relates to the homotopy theory of dg categories.