We will continue online on Thursday, June 18, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Alexander Campbell

The gregarious model structure for double categories

Abstract:
In this talk, I will introduce a new model structure on the category of double categories and double functors, which I will argue is the most natural analogue for double categories of Lack’s model structure for 2-categories. This “gregarious” model structure is completely characterised by the following two of its properties: every double category is fibrant, and a double functor is a trivial fibration iff it is surjective on objects, full on horizontal morphisms, full on vertical morphisms, and fully faithful on squares. Note that this model structure is preserved by all eight auto-equivalences of the category of double categories.

This model structure shares many of the excellent features of Lack’s model structure for 2-categories. For instance, it is proper, it is monoidal with respect to Bo ̈hm’s Gray tensor product for double categories, a double category is cofibrant iff its underlying categories of horizontal and vertical morphisms are free, and the double pseudofunctor classifier comonad is a cofibrant replacement comonad. Moreover, Lack’s model structure for 2- categories is created by the (homotopically fully faithful) “double category of squares” functor from the gregarious model structure for double categories.

I will also introduce the notion of double quasi-category, defined as the fibrant objects of a Cisinski model structure on the category of bisimplicial sets, which I will argue presents the correct ∞-categorical generalisation of the notion of double category. I will prove that the gregarious model structure for double categories is right-induced by Watson’s bisimplicial nerve functor from the model structure for double quasi-categories, and that this nerve functor is homotopically fully faithful.

Dear Friends of Hikes and Mathematics,

you are invited to our 30th mathematical hike planned on June 13th. Meet at 9:31 at the tram stop "Ečerova".

We have planned a 15 km hike through a forest to Žebětín, visiting a lookout tower "Chvalovka" with a view of a city district Bystrc.

All information and photos can be found at http://conference.math.muni.cz/vylety/. (in CZ)

Have a successful exam period, we look forward to you joining us.

Jana Bartoňová and Jonatan Kolegar, organizers
Jan Slovák, Director of the Department of Mathematics and Statistics

We will continue online on Thursday, May 28, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Christopher Dean

Globular Multicategories with Homomorphism Types

Abstract:
We introduce structures called globular multicategories with homomorphism types. We discuss how various collections of “higher category-like” objects can be used to to construct these globular multicategories. We show how to obtain a number of higher categorical structures using this data. We will see that in this setting there is a precise sense in which:
• types are higher categories,
• dependent types are profunctors,
• terms are higher functors,
• terms in a dependent context are higher transformations,
• there is a higher category of all types and terms.

We will continue online on Thursday, May 21, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Axel Osmond

Towards a 2-dimensional spectral construction

Abstract:
Many prominent dualities in mathematics are instances of a common con- struction centered on the notion of spectral functor. Roughly stated, one starts with a locally finitely presentable category, equipped with a subcategory of distinguished local objects encoding point-like data and a factorization system (Etale maps, Local maps) where the etale maps behave as duals of distinguished continuous maps. Several manners of axiomatizing the correct relation between those ingredients have been proposed, either through topos theoretic methods by “localizing” local objects with a Grothendieck topology generated by etale maps, or in an alternative (though tightly related) way based on the notion of local right ajoint (or equivalently stable functor). Then the spectrum of a given object is constructed as a topos classifying etale maps under this given object toward local objects, equipped with a structural sheaf playing the role of the “free local object” under it. This defines a spectral functor from the ambient locally finitely presentable category to a category of locally structured toposes, forming an adjunction with a corresponding global section functor.
This construction provide a convenient template for several prominent 1-catego- rical examples, as dualities for rings in algebraic geometry, or also Stone-like dualities for different classes of propositional algebras. The strong analogy between those dualities and their corresponding first order syntax-semantics dualities suggests the later could be understood as instances of a convenient 2- dimensional spectral construction. In this talk we will expose the ongoing work devoted to concretize this intuition.
After recalling the 1-dimensional version of the construction and the details of some prominent Stone-like examples, we introduces a notion of stable 2- functor and provide a method to construct an associated notion of spectral 2-sites, defining the spectrum as the associated Grothendieck 2-topos equiped with a distinguished structural stack. In particular we give a special interest in determining the local objects and the factorization system associated to doc- trines corresponding to fragments of first order logics, as Lex, Reg, or Coh; in those situations, the construction simplifies as the spectral site happens to be 1-truncated so that one recover the corresponding 1-dimensional notion of clas- sifying topos of a theory as the spectrum, and the geometry of the spectrum actually arises from the geometric properties of local toposes and etale geometric morphisms.