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Middlesex College
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Homotopy Type Theory Electronic Seminar Talks is a series of research talks by leading experts in Homotopy Type Theory. The seminar is open to all, although familiarity with Homotopy Type Theory will be assumed. To attend a talk, please follow the instructions below.
We are using Zoom for the talks. Zoom is similar to Skype, and provides software for all common platforms and devices. Please install the software and make at least one test call before joining a talk. Please join the meeting a few minutes before the scheduled start time and mute your microphone when not asking a question. To join please follow the link:
We encourage attendees from the same institution to attend using one connection.
Electronic seminars are becoming more common in mathematics and you may be interested in attending some of them:
Date  Speaker  Talk information 
January 24 
Nicola Gambino University of Leeds 
TBA TBA 
February 7 
Nima Rasekh MPI Bonn 
TBA TBA 
February 21 
Simon Huber University of Gothenburg 
TBA TBA 
March 7 
Evan Cavallo Carnegie Mellon University 
TBA TBA 
March 21 
Dan Licata Wesleyan University 
TBA TBA 
April 4 
Joachim Kock Universitat Autonoma de Barcelona 
TBA TBA 
April 18 
Paolo Capriotti University of Nottingham 
TBA TBA 
Date  Speaker  Talk information 
December 6 
Eric Finster Inria  Rennes 
Towards Higher Universal Algebra in Type Theory I will propose a definition of "coherent cartesian polynomial monad" in type theory. Special cases of the proposal yield definitions of ∞operads, ∞categories and ∞groupoids. In addition, I describe a definition of coherent algebra over such a monad, leading to a proposed internal description of objects such as E_{n}types and diagrams of types valued in the universe. 
November 22 
Floris van Doorn University of Pittsburgh

Towards Spectral Sequences for Homology
Spectral sequences form a powerful tool which can be used to compute homotopy, homology and cohomology groups of a wide variety of spaces. We have constructed two important spectral sequences in homotopy type theory, the AtiyahHirzebruch and Serre spectral sequences for cohomology. These spectral sequences have analogues for homology, but they have not been constructed in HoTT yet. However, many parts of our construction could be reused to construct the corresponding spectral sequences for homology.
In this talk I will introduce spectral sequences and review the spectral sequences we have constructed and some of their applications. Furthermore, I will describe what parts are still missing to construct the AtiyahHirzebruch and Serre spectral sequences for homology.
The construction of the spectral sequences for cohomology is joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman.

November 8 
Guillaume Brunerie Stockholm University 
Computergenerated proofs for the monoidal structure of the smash product The smash product is a very useful operation in algebraic topology and it can be defined in HoTT as a certain higher inductive type. One of its basic properties is the fact that it is a (1coherent) monoidal product on pointed types, but proving this fact turns out to be very technical. I will present some work in progress aiming at proving it via some form of higher dimensional rewriting and metaprogramming. The idea is to write an (untrusted) external program to generate a formal proof, which can then be formally checked by a proof assistant.

October 25 
Nicolai Kraus University of Nottingham 
Some connections between open problems I will give an overview of my plan to construct connections between several unsolved questions. The following are important constructions that we would like to perform internally in type theory: 
October 11 
KuenBang Hou (Favonia) University of Minnesota 
Towards efficient cubical type theory Cubical type theory attracted much attention in the homotopy type theory community because it achieved canonicity in the presence of univalence and higher inductive types. To date, there are many variants of cubical type theory and experimental proof assistants based on them. 
September 27 
Dimitris Tsementzis Rutgers University

FirstOrder Logic with Isomorphism Firstorder logic with isomorphism (“FOLiso”) is a formal system which relates to HoTT and the Univalent Foundations in a similar way that firstorder logic with equality relates to ZFC and settheoretic foundations. In particular, FOLiso gives a precise framework in which to study “firstorder" theories defined on homotopy types (e.g. precategories, univalent categories), just as firstorder logic with equality gives a precise framework in which to study firstorder structures and theories defined on sets. I will first introduce the syntax and proof system of FOLiso as an extension of M. Makkai’s FOLDS. Then I will define a semantics for FOLiso in terms of homotopy types and describe a proof of soundness and completeness for this semantics. I will conclude by sketching applications in various directions. Media: video, slides. 
September 20 
Andy Pitts University of Cambridge 
Axiomatizing Cubical Sets Models of Univalent Foundations 
Date  Speaker  Talk information 
May 24 
Thierry Coquand University of Gothenburg 
A survey of constructive models of univalence 
May 10 
Thorsten Altenkirch University of Nottingham 
Towards higher models and syntax of type theory

April 26  Martín Hötzel Escardó
University of Birmingham 
Constructive mathematics in univalent type theory I want to share my experience of doing constructive mathematics in univalent type theory, compared to my previous experience in e.g. elementarytopos type theory (as in Lambek and Scott), MartinLöf type theory, Bishop mathematics. 
April 12 
Michael Shulman University of San Diego 
Type 2theories Homotopy type theory is hypothesized to be an internal language for (∞,1)toposes. However, recent developments suggest that more than this, what is needed is an internal language for collections of several (∞,1)toposes together with functors between them, such as adjunctions, monads, comonads, noncartesian monoidal structures, and so on. For instance, a cohesive (∞,1)topos comes with an adjoint triple of two monads and a comonad, while a tangent (∞,1)topos comes with a monad that is also a comonad and also a noncartesian "smash product" monoidal structure. Just as ordinary homotopy type theory is a "doctrine" or "2theory" for (∞,1)toposes, each such situation should come with its own "2theory": a dependent type theory with extra information characterizing the additional structure. But developing even one dependent type theory formally is a lot of work, so we would like a general framework and a general theorem that can then simply be specialized to all such 2theories. I will sketch such a framework, which is under development in joint work with Dan Licata and Mitchell Riley. 
March 29 
Ulrik Buchholtz Te chnische Universität Darmstadt 
From Higher Groups to Homotopy Surfaces Homotopy type theory can be seen as a synthetic theory of infinity groupoids. From this perspective, the pointed, connected types represent infinity groups. The elements are those of the loop space, and the operations on identity types provide the higher group structure.
In this talk, I'll explain what basic group theory looks like from this viewpoint. One aspect is that of categorifying ordinary group theory by using a univalent universe to present many groups. For example, the cyclic group on n elements is the loop space of the type of nelement sets equipped with a cyclic ordering.
In the second half of the talk, I'll focus on the 2dimensional orthogonal group, and use this to talk about the homotopy types of surfaces.
The talk follows the synthetic approach to homotopy theory as developed in the HoTT book. Beyond that, a basic familiarity with ordinary group theory and the result on the classification of surfaces from topology will be helpful.
Some of the material will be from arXiv:1802.04315 (joint with Floris van Doorn and Egbert Rijke) and arXiv:1802.02191 (joint with Favonia), and some material is brand new.

March 15 
Carlo Angiuli Carnegie Mellon University 
Computational semantics of Cartesian cubical type theory Dependent types are simultaneously a theory of constructive mathematics and a theory of computer programming: a proof of a proposition is at the same time a program that executes according to a specification. Homotopy type theory reveals a third aspect of dependent types, namely their role as an extensible formalism for reasoning synthetically about objects with homotopical structure. Unfortunately, axiomatic formulations of univalence and higher inductive types disrupt the computational character of type theory, which pivots on a property called canonicity. I will discuss Cartesian cubical type theory, a univalent type theory in which the canonicity property holds, based on a judgmental notion of cubes with diagonals, faces, and degeneracies, and uniform Kan operations that compute according to their types. I will consider it primarily through the lens of its computational semantics, defined using a cubical generalization of the technique of logical relations, which licenses reading proofs as programs. This talk is based on joint work with Favonia and Robert Harper, described in arXiv:1712.01800. Some familiarity with the syntax and rules of type theory will be very helpful; I will not assume knowledge about computational semantics or logical relations.

March 1 
Emily Riehl Johns Hopkins University 
The synthetic theory of ∞categories vs the synthetic theory of ∞categories Homotopy type theory provides a “synthetic” framework that is suitable for developing the theory of mathematical objects with natively homotopical content. A famous example is given by (∞,1)categories — aka “∞categories” — which are categories given by a collection of objects, a homotopy type of arrows between each pair, and a weak composition law. In this talk we’ll compare two “synthetic” developments of the theory of ∞categories — the first (joint with Verity) using 2category theory and the second (joint with Shulman) using a simplicial augmentation of homotopy type theory due to Shulman — by considering in parallel their treatment of the theory of adjunctions between ∞categories. Afterwards, I hope to launch a discussion about what considerations might motivate the use of homotopy type theory in place of classical approaches to prove theorems in similar settings. Ideal background: some familiarity with notions of a (ordinary strict 1)category, functor, natural transformation, and the definition of an adjunction involving a unit and a counit (just look these up on wikipedia). Plus standard type theory syntax and the intuitions from the CurryHowardVoevodsky correspondence. I’ll be talking about (∞,1)categories but won’t assume familiarity with them. 
February 15 
Peter LeFanu Lumsdaine Stockholm University 
Inverse diagram models of type theory Diagram models are a flexible tool for studying many logical systems: given a categorical model C and index category I, one hopes that the diagram category C ^{I} will again be a model. For the case of intensional type theory, this becomes a little tricky. Since most logical constructors (e.g. Σtypes, Idtypes) are not provably strictly functorial, it is difficult to lift them from structure on C to structure in C ^{I}, for arbitrary I. However, in case I is an inverse category — roughly, something like the semisimplicial category Δ _{I} — this difficulty can be surmounted by taking the types of C ^{I} to be Reedy types, analogous to Reedy fibrations in homotopy theory. I will discuss the construction of these models (and slightly more general homotopical diagram models) in the setting of categories with attributes, along with the application of these models to the “homotopy theory of type theories”. The work of this talk is joint work with Chris Kapulkin, in arXiv:1610.00037 and a forthcoming companion article; see also Shulman arXiv:1203.3253 for a closely related construction. A basic familiarity with categorical models of type theory will be helpful, i.e. categories with attributes or similar; see here for an introductory overview of these. 