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.

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

Week 3: Previous talk continued

Week 4: Derived Morita Equivalence

Week 5: Infty categories part I

Week 6: Dold-Kan and simplicial homotopy I

Week 7: Dold-Kan and simplicial homotopy II

Week 8: Infinity categories II: Making infinity categories work

Week 9: infinity categories iii: THANKS to our work so far, we will be GIVING more constructions in infinity categories

Week 10: Stable infinity categories - infinitely better than triangulated categories

Week 11: Localizations

Week 12: Dg categories and stable infinity categories

- 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.- Oct 25
- Amal Mattoo
**Infinity categories I**

We will introduce quasicategories through the formalism of simplicial sets, along with some basic properties and examples. Then we will define and begin to explore the homotopy theory of quasicategories.- Nov 1
- Caleb Ji
**Dold-Kan and simplicial homotopy I**

I will state and prove the Dold-Kan correspondence, which gives an equivalence between chain complexes and simplicial abelian groups. Then I will explain its significance, for which I will develop simplicial homotopy.- Nov 8
- Caleb Ji
**Dold-Kan and simplicial homotopy II**

Continuation of last time; will discuss the fact that cohomology is representable by Eilenberg-Maclane spaces will fall out. I will then comment on cubical and globular versions of Dold-Kan.- Nov 15
- Rafay Ashary
**Infinity categories II: Making infinity categories work**

We continue the work of the previous lecture on infinity categories. We will develop quasicategorical analogues of familiar (1,1)-categorical constructions with an eye toward establishing a suitable notion of (∞,1)-(co)limits.- Nov 22
- Fan Zhou
**infinity categories iii: THANKS to our work so far, we will be GIVING more constructions in infinity categories**

In preparation for moving on to stable infinity categories, we will do more basic constructions. We will mostly be stating results and maybe doing examples; don’t expect much in the way of proofs.- Nov 29
- Emily Saunders
**Stable infinity categories - infinitely better than triangulated categories**

We will introduce stable infinity categories and do some examples. We will then discuss why they are everything triangulated categories wish they were and more.- Dec 06
- Emily Saunders
**Localizations**

We will begin by stabilizing some shaky concepts from last week. We will then get local by marking our sSets, freely resolving our cats and maybe even constructing some hammocks.- Dec 13
- Andrew Blumberg
**Dg categories and stable infinity categories**

Last talk of the DWIC seminar. Broadly, this talk will be about the correspondence between dg categories and infinity categories.