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

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.