In this page you can find the date, lecturer, a link for lecture notes by Rui, and a summary information about the lectures that have been delievered.

Lecture 1

  • Date: 01/03/2023
  • Lecturer: John Huerta
  • Summary:
    • An introduction to weak \(n\)-categories, models, and the homotopy hypothesis.
    • Notes by Rui Peixoto.

Lecture 2

  • Date: 08/03/2023
  • Lecturer: Diogo Andrade
  • Summary:
    • We did a less-than-superficial introduction to simplicial sets and introduced some helpful notation.
    • We carefully introduced the nerve of a category as a simplicial set which satisfies a unique extension condition.
    • We showed that the nerve is a fully faithful functor.

    • We showed that nerves of categories are universally characterized by the unique extension condition.

    • Notes by Rui Peixoto.

Lecture 3

  • Date: 15/03/2023
  • Lecturer: Diogo Andrade

  • Summary:

    Part 1

    • We defined the notion of \(\infty\)-category as a weak Kan complex. We checked that there are sound notions of product, coproduct of \(\infty\)-categories and \(\infty\)-subcategories (full and non-full).
    • We gave a definition of \(\infty\)-functor and \(\infty\)-natural transformation and checked that they agree with our ususal notions when seen from the nerve lens.

    Part 2

    • In the second half of the lecture we defined the fundamental category of a simplicial set via a universal property and we formally constructed it.
    • We constructed the homotopy relation among 1-morphisms of an \(\infty\)-category and stated they form an equivalence relation on the sets of morphisms and that its natural projection functor satisfies the universal property for the fundamental category.

    • In the end, we proved the mapping-product adjunction for simplicial sets, and in the proccess we defined mapping space/function complex \(\mathsf{Fun}(\mathcal C, \mathcal D)_\bullet\) - this provided us with a sensible candiade for \(\infty\)-functor category.

    • Notes by Rui Peixoto.

Lecture 4

  • Date: 22/03/2023
  • Lecturer: Diogo Andrade

  • Summary:

    Part 1:

    • We introduced the notion of weakly saturated class and inner-anodyne map.
    • We introdueced and showed some results concerning lifting claculus and inner fibrations.

    Part 2:

    • We looked informally at some structural properties of the class of inner fibrations.
    • We skimmed over the small object arguement, the proof that the weak saturation of the class of boundary inclusions is the class of monomorphisms.

    • Constructed the pushout-product and pullback-hom and sketched a proof of the lifting adjunction theorem.

    • Slides Part 1 and Slides Part 2 by Diogo Andrade

Lectrue 5

  • Date: 29/03/2023
  • Lectruer: Diogo Andrade

  • Summary:

    • We proved that the function complex is an \(\infty\)-category by showing a trifecta of technical lemmas which can be summarized as: the pushout-product of an inner-anodyne map and a monomrphism is again inner-anodyne.

    • We showed some variants of the above for other classes of morphisms. In particular we gave four different generating classes of morphisms for the class of inner-andoyne maps.

    • Slides