• Infinity Categories

  • Infinity Categories

Schedule Talks: Thursdays 4:45-6:15 pm GMT-5, Room 4N30 , David Rittenhouse Laboratory, 209 S 33rd St, Philadelphia, PA 19104.
Problem sessions and discussions: 5:45-6:15 pm during the lecture time.
This is a reading course of infinity categories (or \( (\infty,1) \)-categories) at Penn, Fall 2024. Advisor: Prof. Mona Merling.
Here is the syllabus of this course. Editable now!
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Notes

  • Week 1 (Sep 5): Lecturer: Kartik Tandon
    • This is a motivational/overview talk without many details. Topics covered: triangulated categories, homotopy categories, model categories (weak equivalences + (co)fibrations), reasons for considering the \( \infty \)-categories (e.g. Lewis' theorem), a brief intro to simplicial sets, Kan complexes, and nerve functors. Some applications: stable \( \infty \)-category \( \mathsf{Sp} \), (co)limits, Thom spectra, universal properties of K-theory, Beilinson's theorem, Barr-Beck-Lurie.
    See Notes.
  • Week 2 (Sep 12): Lecturer: Riley Shahar
    • Simplicial sets (presheaves of the simplex category + face maps + degeneracies maps), standard \( n \)-simplex, geometric realization, boundaries, simplicial horns, definition of \( \infty \)-categories (quasi-category model), nerves, homotopy categories, \( \infty \)-groupoids, Kan complexes.
    Additional topics: Neumann-Bernays-Godel set theory, MK (or MT) set theory, Grothendieck universe, cardinal, Tarski-Grothendieck set theory, ideas of reverse mathematics (e.g. Levy reflection theorem).
    See Notes.
  • Week 3 (Sep 19): Lecturer: Mats Hansen
    • Functors between \( \infty \)-categories and \( \infty \)-category of functors, internal tensor/hom adjunction, inner fibrations (RLP), infinity categories criterion via inner fibration, mapping spaces, equivalence between Kan complexes and \( \infty \)-groupoids, simplicial set of spaces \( \mathsf{Spc} \), homotopy coherent nerve construction, examples.
    See Notes.
  • Week 4 (Sep 26): Lecturer: Elle Pishevar
    • Cones of functors, limits, examples (terminal objects, etc.), limits via diagrams, Hom functors/mapping spaces and limits, Kan complexes and products, simplicial sets of limits, pullbacks, colimits, examples of colimits, adjoint functors, fundamental theorem of adjoint functors.
    See Notes.
  • Week 5 (Oct 3): Fall Break!
  • Week 6 (Oct 10): Lecturer: Colton Griffin
    • Pointed \( \infty \)-categories, triangles in \( \infty \)-categories, kernels/fibers, cokernels/cofibers, stable \( \infty \)-categories, spectra, \( \Omega \)-spectra, suspension spectra, homotopy groups of spectra, Eilenberg-MacLane spectra, Brown Representability, weak equivalences, pushouts and pullbacks, construction of suspension functor, loop-suspension adjunction, stabilization, additive (abelian) categories, triangulated categories, categories with enough projectives/injectives, dg-categories, dg nerves, derived \( \infty \)-categories.
    See Notes.
  • Week 7 (Oct 17): Lecturer: Fangji Liu
    • Motivations for presentable \( \infty \)-categories: universal characterization of K-theory, continuous K-theory, combinatorial model categories. Free cocompletion of 1- and \( \infty \)-categories, filtered Categories, filtered colimits, \( \mathsf{Ind} \)-objects, \( \mathsf{Ind} \)-completion, compact objects (and examples), regular cardinals \( \kappa \), \( \kappa \)-filtered categories, presentable \( \infty \)-categories, localization functors, accessible functors, characterization of presentable \( \infty \)-categories, properties of presentable \( \infty \)-categories, adjoint functor theorem.
    See Notes.
  • Week 8 (Oct 24): Lecturer: Saul Hilsenrath
    • Basic setups for homotopy theory in \( \infty \)-categories: fibers, nerves, 1-truncation functors, (co)units. Nerve \( J \) of category with two objects, \( J \)-homotopy, \( J \)-homotopy equivalence, properties of 1-truncation functors, left/right lifting property, cofibrations, trivial fibrations, simplicial mapping spaces. Categorical weak equivalence, (co)fibrations, weak factorization system. Small Object Argument, model categories, Joyal model categories, fibrant objects. Cofibrantly generated model categories, combinatorial model categories, subcategories generated by 1-categories, localizations.
    See Notes.
  • Week 9 (Oct 31): Lecturer: Albert Jinghui Yang
    • Motivations: Vandiver conjecture, s-cobordism theorem, Lichtenbaum-Quillen conjecture. Algebraic K-theory by +-construction, the example of algebraic K-theory of finite fields, Waldhausen S-construction and its properties, algebraic K-theory spectrum, functoriality of algebraic K-theory spectra, non-connective algebra K-theory spectra, Dwyer-Kan (DK) simplicial localization, idempotent-complete categories, Morita equivalence, exact sequences in idempotent-complete categories, additive invariants, localizing invariants, additive/localizing non-commutative motives, universal characterization of algebraic K-theory by Blumberg-Gepner-Tabuada, application to Dennis trace.
    Proof ideas are contained in the notes.
    See Notes.
  • Week 10 (Nov 7): Lecturer: Mattie Ji
    • Motivations: PoincarĂ© duality, intersection forms, signature, Arf invariant and Kervaire invariant, Arf's theorem. Basics in \( \infty \)-categories, bilinear functors, symmetric and non-degenerate bilinear functors, perfect bilinear functors, Spainer-Whitehead duality, construction of "quadratic forms" in \( \infty \)-categories, polarization, homotopy fixed points and homotopy orbits, construction of quadratic functors, PoincarĂ© \( \infty \)-categories, quadratic objects, PoincarĂ© objects, hyperbolic \( \infty \)-categories, perfect derived \( \infty \)-categories, hom-tensor adjunction, PoincarĂ© duality in \( \infty \)-categories, algebraic cobordisms, cobordant relations, algebraic \( L \)-theory, computation of \( L_n(\mathbb{Z}) \), results of Kervaire invariant 1 problem.
    See Notes.
  • Week 11 (Nov 14): Lecturer: Kartik Tandon
    • Motivations for monadicity. Morita equivalence, properties of Morita equivalence, monads, algebras over monads, Eilenberg-Moore categories, Forgetful-Free adjunction, free algebras over monads, monadic adjunctions, Barr-Beck monadicity theorem and its sketch of proof, (co)carteisan edges, cartesian (co)fibrations, Segal conditions, monoidal \( \infty \)-categories, algebraic objects, endofunctors, monads in \( \infty \)-categories, algebras, Barr-Beck-Lurie monadicity theorem, Schwede-Shipley theorem.
    See Notes.
  • End of seminar for this semester!
  • Click here for a complete version of notes.

  • Special thanks to Mattie Ji for typing the fantastic notes! The full set of notes will be available by the end of the seminar. If you're interested in contributing, please reach out to Mattie (mji13@sas.upenn.edu) or Albert (yangjh@sas.upenn.edu).

References


Department of Mathematics