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Abstracts for Abstract Homotopy Theory Seminar

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Categorical Ambidexterity

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Speaker: 
Shay Ben Moshe
Affiliation: 
MPIM
Date: 
Tue, 12/05/2026 - 11:00 - 12:30
Location: 
MPIM Seminar Room
Parent event: 
Abstract Homotopy Theory Seminar

Limits of ∞-categories are usually much easier to compute than colimits. Nevertheless, in Pr^L, limits and colimits indexed by a space coincide as proven by Lurie. Harpaz has shown a similar phenomenon for ∞-categories with π-finite colimits, which plays an important role in the theory of higher semiadditivity. In this talk, I will explain a common generalization of these results. Surprisingly, the proof will make use of the (∞,3)-category of iterated spans, and its universal property due to Stefanich, which encodes such ambidexterity phenomena in a coherent fashion.

The Oriented Freudenthal Suspension Theorem

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Speaker: 
Thorger Geiß
Affiliation: 
Universität Münster
Date: 
Tue, 26/05/2026 - 11:00 - 12:30
Location: 
MPIM Seminar Room
Parent event: 
Abstract Homotopy Theory Seminar

he classical Freudenthal Suspension Theorem is the statement that, given a space X, the connectivity of the unit map X -> ΩΣX of the loop-suspension adjunction is (roughly) twice the connectivity of X. In this talk, I will discuss the setting of oriented (or directed) homotopy theory, in which spaces are replaced by higher, i.e. (∞,ω)-, categories and the cartesian product by the Gray tensor product, and generalize the Freudenthal Suspension Theorem in this context. This gives, in particular, evidence for a yet-elusive "oriented topos theory".

Higher categories as directed spaces

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Speaker: 
Hadrian Heine
Affiliation: 
MPIM
Date: 
Tue, 23/06/2026 - 11:00 - 12:00
Location: 
MPIM Seminar Room
Parent event: 
Abstract Homotopy Theory Seminar

We describe a geometric and homotopical perspective on higher categories: higher categories are built by iterated attachment of directed cells in the spirit of CW-complexes and admit Postnikov towers giving rise to homotopy posets. These carry the obstruction classes controlling extensions from one skeleton to the next. The associated graded is given by wedges of spheres and directed spheres, leading to stabilization phenomena and categorical analogues of Freudenthal suspension. 
This is joint work with David Gepner.

 

 

[Abstract homotopy theory seminar] Computing cohomology of differentiable stacks

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Speaker: 
Annika Kraasch-Tarnowsky
Affiliation: 
MPIM
Date: 
Tue, 30/06/2026 - 11:00 - 12:30
Location: 
MPIM Seminar Room
Parent event: 
Abstract Homotopy Theory Seminar

The notion of a differentiable stack in geometry can be used to study objects such as orbifolds, moduli spaces or classifying spaces, which despite not being manifolds carry similar differential geometric information. Examples in particular include quotients of manifolds by a Lie group action. These objects are closely related to equivariant cohomology, a cohomology theory on such manifolds with group actions that has many applications in different areas of mathematics and physics. It can be seen as a special case of the cohomology of differentiable stacks.

 

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