Abstracts for Opening Conference for the Program on Higher Structures in Geometry and Physics

Alternatively have a look at the program.

Higher para-toposes

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Speaker:
André Joyal (Université du Québec, Montréal)
Date:
Mon, 2016-01-18 10:00 - 11:00
Location:
MPIM Lecture Hall

A para-topos is a cartesian closed (locally) presentable category.
A higher para-topos is defined to be a cartesian closed (locally) presentable infty-category.
If $\mathcal{E}$ is a higher para-topos, then so is the $\infty$-category $Cat(\mathcal{E})$ of complete Segal spaces in $\mathcal{E}$.
The construction $\mathcal{E} \mapsto Cat(\mathcal{E})$ can be iterated and it has fixed points.

Strict n-categories as models for homotopy types (joint work with Dimitri Ara)

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Speaker:
Georges Maltsiniotis (Paris 7)
Date:
Mon, 2016-01-18 11:15 - 12:15
Location:
MPIM Lecture Hall

Quillen realized in the sixties that small categories modelize
homotopy types. More precisely, he proved that the Gabriel-
Zisman localization of the category Cat of small categories by the
weak equivalences defi ned by the Grothendieck nerve is equivalent
to the homotopy category of simplicial sets. He also proved some
important properties of weak equivalences in Cat known as theorem
A and theorem B. Later, Thomason de fined a Quillen model structure
on Cat and a Quillen equivalence of this structure with the

The multitopic universe

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Speaker:
Michael Makkai (McGill)
Date:
Mon, 2016-01-18 13:45 - 14:45
Location:
MPIM Lecture Hall

My talk will focus on the contents of the manuscript
entitled "The multitopic omega-category of all multitopic omega-categories"
posted on my website in 1999; the version posted in
2004 is a minor corrected variant. The proposal is for a notion
of weak omega-dimensional (or infi nity) category, called multitopic
category, together with the de finition of the structure that (small)
multitopic categories form, which structure is itself a (large) multitopic
category, here called the multitopic universe. The structure

Hochschild homology, lax codescent, and duplicial structure

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Speaker:
Stephen Lack (Macquarie)
Date:
Mon, 2016-01-18 15:00 - 16:00
Location:
MPIM Lecture Hall

The duplicial objects of Dwyer and Kan are a generalization of the cyclic
objects of Connes. I will describe these duplicial objects in terms of the decalage
comonads, and give a conceptual account of a construction of duplicial objects due
to Böhm and  Stefan. This is done in terms of a 2-categorical generalization of
Hochschild homology. If time permits, I will also discuss duplicial structure on nerves
of categories, bicategories, and monoidal categories.

Weakly globular n-fold categories

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Speaker:
Simona Paoli (Leicester)
Date:
Mon, 2016-01-18 16:30 - 17:30
Location:
MPIM Lecture Hall

In this talk I introduce a new model of weak n-categories,
called weakly globular n-fold categories. This is based on the notion
of iterated internal category, satisfying additional conditions. It
develops a new paradigm to weaken higher categorical structures,
leading to potential applications. I will illustrate how this model
compares to another approach to weak higher categories due to
Tamsamani and Simpson.

Commutativity conditions for monoidal 3-categories

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Speaker:
Larry Breen (Paris 13)
Date:
Tue, 2016-01-19 10:00 - 11:00
Location:
MPIM Lecture Hall

After recalling the various commutativity conditions which
it is natural to impose on the multiplication law in a monoidal 2-
groupoid, I will examine the corresponding question for 3-groupoids.
I will then explain the relation between this question and the occurrence
of certain 3-torsion elements in the integral homology of
the Eilenberg-Mac Lane spaces $K(A, n)$.

Equivariant bordism from the global perspective

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Speaker:
Stefan Schwede (University of Bonn)
Date:
Tue, 2016-01-19 11:15 - 12:15
Location:
MPIM Lecture Hall

Global homotopy theory is, informally speaking, equivariant
homotopy theory in which all compact Lie groups acts at
once on a space or a spectrum, in a compatible way. In this talk I
will advertise a rigorous and reasonably simple formalism to make
this precise, using orthogonal spectra. I will then illustrate the formalism
by a geometrically motivated example, namely equivariant
bordism of smooth manifolds.

On model-comparison results for $\infty$-categories

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Speaker:
Emily Riehl (Johns Hopkins)
Date:
Tue, 2016-01-19 13:45 - 14:45
Location:
MPIM Lecture Hall

The introduction of each new de finition of a weak higher
category poses an accompanying comparison problem, the challenge
being to convert between higher categories de fined via the
new model and higher categories de fined via pre-existing models.
A secondary question is whether proofs involving a particular model
of higher categories have any implications for other related models.
This talk will give a preliminary report on work in progress with
Dominic Verity that develops a framework in which categorical constructions

Posted in
Speaker:
Carles Casacuberta (Universitat de Barcelona)
Date:
Tue, 2016-01-19 15:00 - 16:00
Location:
MPIM Lecture Hall

The talk is based on joint work with Oriol Raventós and Andy
Tonks. We show that several apparently unrelated formulas involving
localizations or cellularizations in homotopy theory arise
from comparison maps associated with pairs of adjoint functors.
Such comparison maps are used to study liftings of homotopical
(co)localizations to categories of (co)algebras over (co)monads in
suitable model categories. We discuss, with diff erent methods, the
case of strict algebras and the case of homotopy algebras. Warnings

Higher homotopy structures-then and now (teleconference talk)

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Speaker:
James Stasheff (UNC Chapel Hill/University of Pennsylvania)
Date:
Tue, 2016-01-19 16:30 - 17:30
Location:
MPIM Lecture Hall

Looking back over 55 years of higher homotopy structures, I will
reminisce as I recall the early days and ponder how I now see them.
From the history of $A_\infty$-structures and later of $L_\infty$-structures, I
will sketch how they morphed into the topic of this Program on
Higher Structures in Geometry and Physics.

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