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Meeting 36

Friday, 24th January 2025

Hosted by the University of Leeds. Supported by the London Mathematical Society.
The meeting will be held in the School of Mathematics at the University of Leeds. A link to the campus map is here.

Registration:

This event is free and open to everyone.
Registration is now closed. If you wish to attend the meeting please contact the organisers.

Due to Storm Éowyn, this event will happen in a hybrid format (online and in-person).
You should have received a link if you registered. If not, please contact the organisers.

Speakers

Schedule

  • 13:00 - 14:00 Ulrik Buchholtz: The Yoneda embedding in simplicial type theory ;
  • 14:00 - 14:30 Coffee break
  • 14:30 - 15:30 Julie Bergner: Compatibility of inputs to Waldhausen's S-construction;
  • 15:30 - 16:00 Coffee break
  • 16:00 - 17:00 Speed talks:
    • Adrian Miranda (University of Manchester): Consolidation: categories as semigroups with extra structure;
    • Raffael Stenzel (University of Leeds): Bicategorical $E_n$-structures;
    • Tom de Jong (University of Nottingham): Epimorphisms and acyclic types;
    • Bruno Lindan(University of Manchester): Sound and Unsound Limit Doctrines;
    • James Cranch (University of Sheffield): Monoids and monoidal monoidalities;
    • Stiéphen Pradal (University of Nottingham): A study of Kock's fat Delta.
  • 17:00 - onwards: Informal discussions / pub.

Titles and Abstracts:

    • Julie Bergner (University of Virginia).
      • Title: Compatibility of inputs to Waldhausen's S-construction
      • Abstract: Campbell and Zakharevich developed the theory of CGW categories to provide a very general framework for doing algebraic K-theory that encompasses examples such as the K-theory of varieties. On the other hand, in joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we proved that augmented stable double Segal spaces provide a universal input for Waldhausen's S-construction, in such a way that the output has the structure of a 2-Segal space. In joint work with Shapiro and Zakharevich, we show how CGW categories can be characterized in this universal framework, enabling interplay of the features and examples of both approaches.
  • Ulrik Buchholtz (University of Nottingham)
    • Title: The Yoneda embedding in simplicial type theory
    • Abstract: Simplicial type theory à la Riehl–Shulman refers to the program of doing $(\infty,1)$-category theory in the model of homotopy type theory in the $(\infty,1)$-topos of simplicial spaces. The directed interval $\Delta^1=\{0 \to 1\}$ in this model allows us to carve out the complete Segal spaces and reason about them in a natural way, allowing us for instance to develop the theory of (co)limits and adjunctions in a straightforward way. But the basic theory has almost no concretely definable example categories.In recent work with Daniel Gratzer and Jonathan Weinberger (arXiv:2407.09146), we showed how to define the category on spaces in an extension of simplicial type theory, showed that it enjoys a directed univalence principle, and allows us to define many commonly occurring categories.In this talk, I'll relay even more recent work, in which we construct the Yoneda embedding, prove the universal property of presheaf categories, refine the theory of adjunctions, begin the theory of Kan extensions, and prove Quillen's Theorem A. We thus offer some evidence that the simplicial type theory paradigm can be used to produce difficult results in $\infty$-category theory at a fraction of the usual complexity.

Title and abstracts for the speed talks.

  • Adrian Miranda (University of Manchester):
    • Title: Consolidation: categories as semigroups with extra structure
    • Abstract: Tilson's consolidation is a faithful functor Cat --> Semigroup that restricts along Monoids --> Cat to the usual inclusion. I will describe this construction and explain how categories can be viewed as semigroups with extra structure. This is based on joint work with Tom Aird.
  • Raffael Stenzel (University of Leeds):
    • Title: Bicategorical $E_n$-structures;
    • Abstract: Let $C$ be a monoidal bicategory with a degree of extra monoidality (a braiding/a syllepsis/a symmetry). We show that the bicategory of monoids in $C$ always inherits a monoidal structure from $C$ to the degree of $C$ "minus 1". This extends to a corresponding statement about
      the bicategory of monoids with any possible extra degree of internal monoidality in $C$. However, we will refrain to give a hands-on proof by way of working locally inside a general bicategory $C$. Instead, we argue globally and use tools of abstract homotopy theory and homotopy-coherent algebra to do so. This theorem is hence exhibit A to show how recent developments in higher algebra can be useful to efficiently prove new results in lower dimensional category theory as well. This is ongoing joint work with (and part of a project initiated by) Nicola Gambino.
  • Tom de Jong (University of Nottingham):
    • Title: Epimorphisms and acyclic types.
    • Abstract: Homotopy type theory (HoTT) is a mathematical foundation based on the concept of spaces (called types), i.e. a language for higher categories. A natural question is what (necessarily higher-categorical) epimorphisms look like in this foundation. After introducing a homotopically well behaved definition of epimorphism, we characterize the epis as the fiberwise acyclic maps. Classically a space is said to be acyclic if its reduced homology vanishes. An equivalent definition, which we adopt in HoTT, is that the suspension of the space is contractible. We illustrate our synthetic approach by showing how to obtain results and examples via simple techniques from (higher) category theory as opposed to traditional methods (such as homology calculations).
      This talk is based on joint work with Ulrik Buchholtz and Egbert Rijke that will appear in the Journal of Symbolic Logic and is available on the arXiv (2401.14106).
  • Bruno Lindan (University of Manchester):
    • Title: Sound and Unsound Limit Doctrines
    • Abstract: Gabriel-Ulmer duality gives a (contravariant) biequivalence between the 2-category Lex of finitely complete categories and that of locally finitely presentable categories. Counterparts to this result obtained by replacing Lex with a different 2-category of suitably complete categories fall into a general picture of dualities relative to so-called "sound" doctrines of limits (the general theorem is due to Centazzo and Vitale). Not every doctrine of interest is sound – in this talk we will investigate the extent to which unsound doctrines are able to share enough of the good properties of sound ones to admit some form of duality theorem, hoping to better motivate the somewhat technical condition of soundness along the way.
  • James Cranch (University of Sheffield):
    • Title: Monoids and monoidal monoidalities.
    • Abstract: It is possible to generalise the concept of a monoidal category, and
      one accordingly obtains a generalised notion of PROP. I'll discuss one surprising example, in which a PROP gives a concept of structured functor (examples include "lax", "oplax", "strong", etc).
      This talk will contain joint talk with Dan Graves (University of Leeds).
  • Stiéphen Pradal (University of Nottingham):
    • Title: A study of Kock's fat Delta.
    • Abstract: Motivated by the study of weak identity structures in higher category theory, we recover Kock’s fat Delta category as the category of free algebras over the arities of a monad on the category of relative graphs. This immediately gives rise to a nerve theorem for relative semicategories as well as an active-inert factorisation system on fat Delta.