Seminar on infinity categories
∞-seminar
Seminar on infinity categories at the Department of Mathematics of Universidad de Buenos Aires.
Held in hybrid format, on Zoom and at Sala de conferencias, Wednesdays 14 - 16 hs.
Click here for the list of GNC seminars.
Talks
10/5 Guillermo Cortiñas (UBA). Tensor products of presentable $\infty$-categories
Abstract: In this third and final talk, which is also the last talk of the seminar, we will review tensor products of presentable $\infty$-categories and its applications. For example, tensor products will be used to endow the $\infty$-category of presentable $\infty$-categories with a closed symmetric monoidal structure, in which the $\infty$-category $S$ of spaces is the monoidal unit. There is also a version of this result for stable presentable $\infty$-categories, in which the monoidal unit is the $\infty$-category $Sp$ of spectra.
3/5
Guillermo Cortiñas (UBA).
Stable $\infty$-categories, part II: spectra
Abstract: We shall discuss stabilization of pointed infinity categories. If $\mathsf C$ is an infinity category, its stabilization is the category $\mathsf{Sp(C)}$ of spectra on $\mathsf C$. We shall see that in the particular case when $\mathsf C$ is the infinity category of spaces, $\mathsf{Sp(C)}$ is the free stable infinity category on one object.
26/4
Guillermo Cortiñas (UBA).
Stable $\infty$-categories, part I
Abstract: We shall introduce pointed $\infty$-categories, the suspension and loop functors, and the notion of a triangle in that setting. Then we shall define stable $\infty$-categories and explain how their homotopy categories are triangulated.
19/4
Devarshi Mukherjee (UBA).
Symmetric monoidal $\infty$-categories, part III
Abstract: I will continue the discussion on algebras in monoidal infinity categories. In particular, I will talk about $E_n$-algebras, which encode different levels of commutativity in algebras.
12/4
Devarshi Mukherjee (UBA).
Symmetric monoidal $\infty$-categories, part II
Abstract: I will define $A_\infty$ and $E_\infty$ algebras. These generalise associative and commutative algebra objects in ordinary monoidal categories. However, unlike for ordinary categories, there are several intermediary levels of commutativity in between associative algebras on one end and commutative algebras at the other.
5/4
Devarshi Mukherjee (UBA).
Symmetric monoidal $\infty$-categories, part I
Abstract: I will redefine monoidal categories as certain Grothendieck opfibrations. This will be used to define monoidal and symmetric monoidal $\infty$-categories. I will then provide examples of such $\infty$-categories.
7/12
Gisela Tartaglia (UNLP).
Locally presentable categories, part II
Combinatorial model categories and locally presentable $\infty$-categories.
30/11
Gisela Tartaglia (UNLP).
Locally presentable categories, part I
"Free generation and imposing relations": cocompletion and localization. Locally presentable $1$-categories.
23/11
Eugenia Ellis (Universidad de la República).
$\Gamma$-spaces
$\Gamma$-spaces: motivation and examples. Smash product of $\Gamma$-spaces. Spectrum associated to a $\Gamma$-space.
16/11
Charly di Fiore (Universidad de Buenos Aires).
Simlplicial enrichment and localization of categories
2/11
Marco Farinati (Universidad de Buenos Aires).
Examples of operads
Examples: $A_\infty$-operads, $n$-little cubes and little disk operads.
26/10
Marco Farinati (Universidad de Buenos Aires).
A brief introduction to operads
Operads and algebras over an operad. Examples: $\mathsf{Com}$, $\mathsf{As}$, endomorphisms of a vector space, the free operad on a set.
19/10
Guido Arnone (Universidad de Buenos Aires).
∞-categories, part IV: (co)limits, continued
Cones over a functor between ∞-categories. Slice ∞-category over an object. Initial and final objects. Definition of ∞-limits and colimits. 12/10
Guido Arnone (Universidad de Buenos Aires).
∞-categories, part III: (co)limits
Space of functors. Joins of caregories and simplicial sets. The (1-)category of cones over a functor and its relation to the join of categories.
5/10
Guido Arnone (Universidad de Buenos Aires).
∞-categories, part II: the homotopy coherent nerve
Simplicially enriched categories. Definition of the homotopy coherent nerve.
28/9
Guido Arnone (Universidad de Buenos Aires).
∞-categories, part I
Fully faithfulness of the nerve construction. Definition of ∞-categories and ∞-grupoids as quasicategories and Kan complexes. Homotopy category of an ∞-category. 14/9
Charly di Fiore (Universidad de Buenos Aires).
Homotopy (co)limits, part II
Geometric realization and the Bousfield-Kan construction. Homotopy commutative vs. homotopy coherent cones. Homotopy (co)limits. 7/9
Charly di Fiore (Universidad de Buenos Aires).
Homotopy (co)limits, part I
Geometric realization of (semi) simplicial sets. Failure of homotopy invariance of (co)limits in the category of spaces. Introduction to homotopy (co)limits. 31/8
Janou Glaeser (Universidad de Buenos Aires).
Simplicial sets, part II
Functors from a category $C$ to simplical sets defined from cosimplicial objects in $C$. Applications: the singular complex of a topological space and the nerve of a small category. The co-Yoneda lemma: every presheaf is a colimit of representables. Geometric realization. 24/8
Janou Glaeser (Universidad de Buenos Aires).
Simplicial sets, part I Definition of the category of simplicial sets. (Co)Face and (co)degeneracy maps. Standard simplices, boundaries and horns. Simplicial set of singular simplices of a topological space.
2022
