GMM Reading Group

$\infty$-Categories and $\infty$-Operads

Last update: November 4, 2025 The aim of this reading group is to develop a basic understanding of $\infty$-operads and their algebras. We begin with a crash course on $\infty$-category theory (based on quasicategories) and then move on to reading some introductory literature for $\infty$-operads.
Our group will meet every Friday, 09:00 - 11:00 (CET) at TU Dresden. Our (admittedly ambitious) first goal is to reach $\mathbb{E}_1$ - and $\mathbb{E}_\infty$-monoids by Christmas, so that we will have covered the necessary background to introduce symmetric monoidal $\infty$-categories and can move on with a reference on $\infty$-operads in early 2026.

A provisional schedule for the upcoming talks is outlined below and may be revised as we move through the material.

Next talk: Nov 07, 2025 1: Category Theory.zip' and simplicial sets

Time Description
$\emptyset$

0: What is a monoid anyway?

Speaker: Zbigniew Wojciechowski

This is a motivational talk for the goal of our crash course on $\infty$-categories. It comes in the form of two videos. Outline: This motivational talk explains two main ideas. First, monoids correspond to certain functors $F\colon \Delta^\op \to \Set$. Second, functors $F\colon \cC \to \Set$ can be translated to some sort of fibration $\cC_F \to \cC$ called the unstraightening. Putting 1 and 2 together, monoids correspond to certain functors $\Delta^\op_F \to \Delta^\op$. We will then go one categorical level higher and replace $\Set$ by $\Cat$, and answer the question whether this sophisticated construction of monoids can be used to define monoidal categories.
Nov 07
09:00

1: 'Category Theory.zip' and simplicial sets

Speaker: Florian Warg

References: Wag25, §1, §2, Lan21 §1

This talk is a review of standard constructions and results of classical category theory that will be lifted to $\infty$-categories in later talks, followed by a brief introduction to simplicial sets, quasicategories and animae. We will develop a framework to generate adjunctions by cocontinuous extension of functors along the Yoneda embedding which allows for convenient definitions of geometric realizations, nerves and homotopy categories.

Nov 14
09:00

2: Two $\infty$-categories: $\Cat_\infty$ and $\An$

Speaker: Ivan Bartulović

References: Wag25, §2, §3

In this talk, we will construct two fundamental examples of $\infty$-categories: the $\infty$- category of $\infty$-categories and the $\infty$-category of animae. Their construction will rely on the simplicial nerve (also known as the homotopy coherent nerve), which sends simplicially enriched categories to simplicial sets. We will treat the technical details as a black box and instead focus on understanding concretely what the 0-, 1-, and 2-simplices in these simplicial nerves look like.

Nov 21
09:00

3: Animae $\simeq$ CW-complexes

Speaker: TBA

References: Wag25, §3, §4

A rough goal of this talk is to list any way in which Kan complexes (or animae) behave like topological spaces/CW-complexes. We first define homotopy groups of pointed animae and recall what fibrations and cofibrations are. Then, we explain classical facts from homotopy theory, such as the existence of a long exact sequence of a fibration. Finally, we state the simplicial approximation theorem: any Kan complex $X$ is homotopy equivalent to the singular simplicial set of its geometric realization. Also, the geometric realization of the singular simplicial set of a topological space is homotopy equivalent to the space itself.

Nov 28
09:00

4: (Un-)Straightening and Yoneda

Speaker: TBA

References: Wag25, §5, §6

In this talk, we will define cocartesian fibrations and formulate without proof Lurie’s straightening equivalence. For $x \in \cC$ the cocartesian fibration $\cC_{x/} \to \cC$ corresponds to the Hom-functor $\Hom_{\cC}(x, -)$. We will explain this and related constructions (like the Hom-bifunctor $\Hom_{\cC}(-, -)\colon \cC^\op \times \cC \to \An$). We then formulate (and maybe sketch a proof of) the Yoneda lemma. If time allows, we define adjoint functors between $\infty$-categories and state $\infty$-categorical analogues of standard results (triangle identities and fully faithfullness).

Dec 05
09:00

5: '$\infty$-Category Theory.zip' and localization

Speaker: TBA

References: Wag25, §6

This talk heavily relies on the Yoneda lemma. We build on the results of Talk 3 and 5 to develop $\infty$-categorical analogues of the standard 1-categorical constructions (like limits and colimits) and general results from Talk 1. In addition, we introduce localizations of $\infty$-categories.

Dec 12
09:00

6: $\mathbb{E}_1$ and $\mathbb{E}_\infty$ monoids 🎄

Speaker: TBA

References: Wag25, §7

In this talk, we will first define $\mathbb{E}_1$-monoids as functors satisfying the Segal condition. The main part of the talk is to explain how $\End_{\cC}(x)$ forms an $\mathbb{E}_1$-monoid. After that, we alter the definition of $\mathbb{E}_1$-monoids to define $\mathbb{E}_\infty$-monoids. We will continue after Christmas with symmetric monoidal $\infty$-categories.

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