Dutch Categories And Types Seminar

Mailing List

• To subscribe, send an email to dutchcats+subscribe@googlegroups.com.
• To send a message to the list, use dutchcats@googlegroups.com. (Only emails from subscribers are accepted for distribution.)

Informal workshop on Algebraic Weak Factorisation Systems, 2 May 2022

Program

• 10:00-11:00: John Bourke (Masaryk University, Brno), An orthogonal approach to algebraic weak factorisation systems
• 11:15-12:15: Wijnand van Woerkom (Utrecht University), The Frobenius property for algebraic weak factorisation systems (recording)
• 13:15-14:15: Nicola Gambino (University of Leeds), The effective model structure on simplicial objects slides
• 14:30-15:30: Benno van den Berg (University of Amsterdam), Effective Kan fibrations in simplicial sets (recording)
• 16:00-17:00: Paige North (University of Pennsylvania), Two-sided and other generalizations of weak factorization systems (slides, recording)
• 17:30-23:59: Dinner

Abstracts

Location

Fourth Meeting: Zoom, 8 April 2022

• Start: 16:00 Amsterdam time
• Recording of the talk available on YouTube
• Join at Zoom Meeting
• Abstract

  In this talk, I present three term calculi in a uniform format that respectively embody classical, intuitionistic and linear logics. These term calculi are equipped with reductions that satisfy subject reduction, confluence and strong normalisation. An advantage of the term calculi is their simplicity: They dispense with the tangled (commuting) conversions in existing term calculi, where conversions are one of the fundamental problems in proof theory. This simplicity also means the abstraction of my approach that discards the inessential syntactic details. By formulating the computations for the three logics in this abstract level, the present work deepens the computational analyses of the logics.

Third Meeting: Zoom, 4 March 2022

• Start: 16:00 Amsterdam time
• Recording of the talk available on YouTube
• Join at Zoom Meeting ID: 832 9706 1847
• Abstract

  Naïve category theory often makes implicit reference to sets, and thereby to a particular formulation of set theory. For example, we talk about the hom sets of locally small categories, small limits and colimits or the solution set condition of the adjoint functor theorem. Although almost all of homotopical algebra can be carried out in a purely algebraic way, we still see references to sets when we give explicit definitions of weak factorisation systems as cofibrantly generated by a set or small category of morphisms. We can break this dependence on sets using Grothendieck fibrations, where we replace the category of sets with a base category of our choice to serve as a foundation. The base category can be sets, but could also be a model of a particular choice of set theory or more generally an elementary topos, or even more generally a locally cartesian closed category or the category of small categories. If we have an external property of objects of the fibration that we are studying, it is natural to ask when we can talk about it in the internal language of our base. As a minimal condition we expect that such properties are stable under reindexing, but it turns out this is not strong enough. We can think of an object in the total category of a fibration intuitively as an "indexing object" I in the base together with a family of objects indexed by I, say (X_i)_I. What we need is a subobject of I in the base that tells us which objects X_i have the property we are interested in. This idea is captured remarkably well by Bénabou's notion of *definability*. For example, in many examples of Grothendieck fibrations the subterminal objects are stable under reindexing, but definability requires something more. An example where they are definable are codomain fibrations on Heyting categories: we can use the equality and universal quantifiers of the internal language to state what it means for any two elements of a type to be equal, which turns out to witness the definability of subterminal objects in the fibration.
  However, Bénabou's definition applies only to properties of objects and not to structure. For structure we can use a new definition introduced by Shulman under the name *locally representable*. I'll give a reformulation of Shulman's definition making it clear that it can be seen as a generalisation of Bénabou's in the same spirit, encompassing not just definability, but also Johnstone's earlier generalisation of comprehensivity. Just as we considered stability under reindexing as a minimal requirement for properties, we first consider *notions of fibred structure* or just *notions of structure* on a fibration. We think of this as a kind of structure which is defined externally to the fibration, but stable under reindexing. Reindexing stable classes are a special case, which we refer to as *full* notions of structure, but so are Johnstone's comprehension schemes and any time we have a (co)monad over a fibration, the category of (co)algebras is another example.
  I am particularly interested in the case of algebraic weak factorisation systems (awfs), which are notions of structure on codomain fibrations with the property of being monadic, and equipped with certain extra structure. I give a sufficient criterion for awfs's to be definable, as cofibrantly generated by a family of maps with tiny codomain. This uses a general definition of tiny in a fibration, that can be applied to codomain fibrations to obtain a result similar to that of Licata, Orton, Pitts and Spitters, or can be applied to set or category indexed families to obtain a different looking result referring to maps that are tiny in an external sense that can be phrased using hom set functors.
  I will also list some examples of non definable notions of structure based on Kan fibrations, which are widely used in homotopical algebra and can be defined in any topos with interval object. A large class of weak factorisation systems cannot be definable as full notions of structure unless the axiom of choice holds. In some interesting special cases the awfs of Kan fibrations can be shown to be non-locally representable using logical properties of the interval object, even in classical logic with the axiom of choice. In simplicial sets this can be done using the fact that the interval has a linear order (in a certain sense the interval object of simplicial sets is the "generic" linear order with endpoints) and can be done in BCH cubical sets using the fact that the interval has "separable diagonal."

Second Meeting: Zoom, 4 February 2022

• Start: 16:00 Amsterdam time
• Recording of the talk available on YouTube
• Join at Zoom Meeting ID: 838 6419 3991
• Abstract

  Grothendieck abelian categories can be seen, on one hand, as the Ab-enriched topoi, and on the other hand, as the (categorical counterpart of) noncommutative schemes. In particular, the 2-category Grt♭ of Grothendieck abelian categories with left exact left adjoint additive functors, which is nothing but the Ab-enriched counterpart of the 2-category of topoi and geometric morphisms, is a natural environment in order to study noncommutative flat algebraic geometry. In this talk we show that the Bird-Kelly tensor product of (enriched) locally presentable categories endows Grt♭ with a monoidal structure, which is a linear counterpart of the product of topoi and at the same time can be considered as a noncommutative parallel of the product of schemes with flat morphisms between them. Moreover, in an effort to shed some light on the exponentiability of schemes, we study the exponentiable objects in Grt♭. More concretely, we provide a characterization of the exponentiable objects in Grt♭ by following a linear parallel to Johnstone and Joyal’s characterization of exponentiable topoi.
  The content of this talk is joint work with Ivan Di Liberti.

Inaugural Meeting: Amsterdam, 27 October 2021

Program

• 13:00-14:00: Niels van der Weide (Radboud University), The correspondence between LCCCs and dependent type theories from a univalent perspective (slides)
• 14:30-15:00: Jaap van Oosten (Utrecht University), Games and Lawvere-Tierney topologies
• 15:00-15:30: Daniël Otten (University Amsterdam), M-types and bisimulation (slides)
• 16:00-17:00: Jesper Cockx (TU Delft), The Quest for Confluence (slides)
• 17:30-23:59: Dinner

Registration

Location

Organizers

• Benedikt Ahrens, B.P.Ahrens@tudelft.nl (TU Delft)
• Benno van den Berg, bennovdberg@gmail.com (Universiteit van Amsterdam)