Alternatively have a look at the program.

## Pasting diagrams beyond acyclicity

Many of the available algebro-combinatorial frameworks for presenting pasting diagrams, or more general diagrams in n-categories, rely on a global acyclicity

condition to ensure that a "combinatorial diagram" is equivalent to the n-functor that it presents.

This is somewhat inconvenient, as global properties tend to be unstable, and many diagrams that arise in practice are not acyclic.

In my talk, I would like to give an overview of the combinatorics of higher-dimensional diagrams when (global) acyclicity is relaxed to (local) "regularity",

## Commutative semirings and bispans

Commutative rings are commutative algebra objects in abelian groups, but they can also be viewed as models of a Lawvere theory. If we don't insist on having inverses for addition, this admits a nice description: commutative semirings are product-preserving functors to sets from a category of "bispans" of finite sets. In this talk I will explain an infinity-categorical version of this comparison; in particular, using results of Gepner-Groth-Nikolaus, we can describe connective commutative ring spectra in terms of bispans of finite sets.

## Higher internal category theory

Results which concern the classification of parametrized structures over a given base by means of internal constructions within that base are fairly ubiquitous in homotopy theory. These internalization results are generally formal consequences of reflection properties of an associated externalization functor (that is, usually, some kind of Yoneda embedding). In this talk, we use a suitable externalization functor to define the $(\infty,2)$-category of $\infty$-categories internal to some base $C$.

## Duality for generalized algebraic theories

Gabriel-Ulmer duality is a contravariant biequivalence between small finite-limit categories and locally finitely presentable categories, which in the spirit of Lawvere's functorial semantics can be viewed as a *theory-model duality*: small finite-limit categories C are viewed as theories, and the lfp category FL(C, Set) of finite-limit preserving Set-valued functors is viewed as category of models of of C. The opposite of C can be reconstructed up to equivalence from Lex(C,Set) as full subcategory of *compact* objects.

## Why Fibrations are Cool!

The aim of this talk is to see why fibrations are useful when trying to do higher category theory.

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