**Presentable infinity categories**

- Carles Casacuberta (UB): Preliminaries
29 October 2018, 12:00, IMUB

Abstract: Basic facts about simplicial sets, Yoneda embeddings, density, and locally presentable categories, including a sketch of a proof of the Representation Theorem for categories.

- Carles Casacuberta (UB): Locally presentable categories and localizations
5 November 2018, 12:00, IMUB

Abstract: Additional details about the previous talk, including a discussion on reflectivity.

- Carles Casacuberta (UB): Quasicategories
12 November 2018, 12:00, IMUB

Abstract: A quick introduction to ∞-categories, until a definition of Dwyer-Kan equivalences and adjunctions between quasicategories.

- Carles Casacuberta (UB): Presentable ∞-categories
19 November 2018, 12:00, IMUB

Abstract: A discussion of limits and colimits in quasicategories, towards a statement of the Representation Theorem for ∞-categories.

- Joana Cirici (UB): Quasicategories versus simplicial categories
26 November 2018, 12:00, IMUB

Abstract: Introduction to simplicial categories, Dwyer-Kan equivalences, description of the coherent nerve and rigidification functors, review of properties of this adjoint pair between quasicategories and simplicial categories.

- Carles Casacuberta (UB): Quasicategories of presheaves
3 December 2018, 12:00, IMUB

Abstract: A Yoneda embedding for quasicategories.

- Marc Adillon (UB): Model categories
10 December 2018, 12:00, IMUB

Abstract: An introductory review of Quillen model categories.

- Marc Adillon (UB): Combinatorial structures in homotopy theory
17 December 2018, 12:00, IMUB

Abstract: Definition and main properties of combinatorial model categories and their simplicial enrichments.

- Javier J. Gutiérrez (UB): Infinity Representation Theorem I
14 January 2019, 12:00, IMUB

Abstract: The ∞-category associated to a simplicial model category. Dugger's theorem and localization for ∞-categories.

- Javier J. Gutiérrez (UB): Infinity Representation Theorem II
28 January 2019, 12:00, IMUB

Abstract: Continuation of the previous talk.

- Matias L. del Hoyo (Universidade Federal Fluminense): On linear quasicategories
4 February 2019, 12:00, Aula T2

*Visitant del CRM dins la convocatòria Lluís Santaló finançada per l'Institut d'Estudis Catalans*Abstract: The classical Dold-Kan correspondence establishes an equivalence between chain complexes and simplicial vector spaces. A linear version of the Grothendieck correspondence allows us to regard pseudofunctors in the category of categorical vector spaces as certain linear groupoid fibrations. In this talk I will review these two correspondences, explain how they can be combined, and comment on an ongoing project with G. Trentinaglia, in which we generalize these results within the framework of differential geometry, using the language of linear quasicategories.

- Carles Casacuberta (UB): Set-theoretical background for ∞-categories
11 February 2019, 12:00, IMUB

Abstract: It had to happen someday.

- Joachim Kock (UAB): Infinity toposes
18 February 2019, 12:00, IMUB

Abstract: I will give an introduction to higher topos theory, by contrasting with 1-topos theory. The main themes will be classifiers and descent.

- Wilson Forero (UAB): Giraud's Theorem I
25 February 2019, 12:00, IMUB

Abstract: The axioms of Giraud allow us to determine when a category is a topos. Lurie's version indicates the equivalence between the axioms, the left exact localizations, and Grothendieck topologies. In this talk I will give the necessary tools for the proof.

- Wilson Forero (UAB): Giraud's Theorem II
4 March 2019, 12:00, Aula T2

Abstract: Continuation of the previous talk.

- Wilson Forero (UAB): Infinity Giraud's Theorem I
18 March 2019, 12:00, IMUB

Abstract: Lurie generalizes Giraud's axioms for ∞-categories and states the equivalence between them and the accessible left exact localizations. In this talk I will give a sketch of the proof.

- Wilson Forero (UAB): Infinity Giraud's Theorem II
25 March 2019, 12:00, IMUB

Abstract: Continuation of the previous talk.

- Wilson Forero (UAB): Kripke-Joyal semantics
8 April 2019, 12:00, IMUB

Abstract: Kripke-Joyal semantics enable us to do logic inside a topos. I will give the definition and talk about sheaf semantics as an example of this definition. I will also provide background on the necessary tools to build this semantic.

- Joachim Kock (UAB): Introduction to Type Theory
29 April 2019, 12:00, IMUB

Abstract: I will explain the basic elements of type theory from the viewpoint of category theory.

**References**

[1]
D.-C. Cisinski, Higher Categories and Homotopical Algebra, http://www.mathematik.uni-regensburg.de/cisinski/publikationen.html.

[2] M. Groth, A short course on ∞-categories, arXiv:1007.2925v2.

[3] V. Hinich, Lectures on infinity categories, arXiv:1709.06271.

[4] A. Joyal, Quasi-categories and Kan complexes, *J. Pure Appl. Algebra* **175** (2002), no. 1-3, 207-222.

[5] A. Joyal, The theory of quasi-categories and its applications, Quaderns CRM 45, available from http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf.

[6] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton and Oxford, 2009.

[7] J. Lurie, Higher Algebra, http://www.math.harvard.edu/~lurie/.

[8] C. Rezk, Higher category theory and quasicategories, https://faculty.math.illinois.edu/~rezk/595-fal16/quasicats.pdf.