Speakers

Title: Quillen-Barr-Beck cohomology of divided power algebras over an operad

Abstract: Quillen-Barr-Beck (co)homology is a theory which studies the objects of a category where we can make sense of such notions as “modules” and “derivations”. More precisely, in his thesis, Beck defined a general notion of modules and derivations in a sufficiently nice category. These notions where popularised by Quillen, who introduced a cohomology theory for commutative rings, known as the André-Quillen cohomology, by studying something called the cotangent complex for these rings.

An operad is a device which encodes types of algebras. We can also use operads to define categories of divided power algebras, which have additional monomial operations. These divided power structures appear notably in the simplicial setting.

The aim of this talk is to show how André-Quillen cohomology generalises to several categories of algebras using the notion of operad. We will introduce modules and derivations, but also a representing object for modules – known as the universal envelopping algebra – and for derivations – known as the module of Kähler differentials – which will allow us to build an analogue of the cotangent complex. We will see how these notions allow us to recover known cohomology theories on many categories of algebras, while they provide somewhat exotic new notions when applied to divided power algebras.

Title: Semantics for Non-Determinism in Categorical Message Passing Language

Abstract: Categorical Message Passing Language (CaMPL) is a functional style concurrent programming language with a categorical semantics. In this talk, we explore the categorical semantics, programming syntax, and proof theory representations for CaMPL. This includes the sequential functions with input and output values (which become messages), concurrent processes, communication channels, message passing along channels between processes, and races which introduce non-determinism in CaMPL.

Slides: https://www.dropbox.com/s/8rcwzh6lm9j41qt/Calgary_Peripatetic_Seminar_2023-08-02_Little.pdf?dl=0

Title: Normalizing resistor circuits

Abstract: A classical Electrical Engineering problem is to determine whether two networks of resistors are equivalent. The standard solution is to use a process of eliminating internal nodes (the star/mesh transformation) which may be seen as a rewriting procedure. In the EE literature this is organized into a matrix problem which can be solved by a Gaussian elimination procedure. This approach essentially works for resistances taken from the positive reals while secretly using negatives. However,  it cannot handle negative resistances. Negative resistances, while counter-intuitive, arise in stabilizer quantum mechanics where networks of “resistors” over finite fields occur. The rewriting approach as compared to the matrix approach is more general and works for any positive division rig. We shall discuss how the rewriting theory can be modified to handle negatives. 

In this talk we shall start by introducing categories of resistors. These are “hypergraph categories” whose maps are generated by resistors. Resistors are then self-dual maps satisfying certain formal properties (including an infinite family of star-mesh equalities). The problem of network equivalence is, thus, the decision problem for maps in these categories. The fact that there is a terminating and confluent rewriting system not only solves the decision problem (up to certain equalities) but also has the effect of guaranteeing, very basically, the associativity of the composition. 

Work with: Robin Cockett and Amolak Kalra

Abstract courtesy: Robin (at FMCS ’23)

Title: Partial monoids

Abstract: A partial monoid is a set A with a multiplication and a unit, but the multiplication is only defined on a certain subset of pairs. This means the multiplication m:AxA -> A is a partial map, a map that is only defined on a certain subset of its domain. The category of finite commutative partial monoids is an important ingredient for the construction of tangent infinity categories where a bicategory of spans of partial monoids encodes a weakly commutative ring structure. As partial maps are generalized by restriction categories one can directly generalize the definition of a partial monoid in any restriction category.
I will present a certain restriction category B of bags (a.k.a. multisets) with a partial monoid in it. Any partial monoid in any join restriction category X is induced through a functor from B to X which means the partial monoid in B is the generic partial monoid. This characterization of partial monoids is analogous to the concept of a Lawvere theory of an algebraic structure.

Title: Path Categories in A-Homotopy Theory

Abstract: A-homotopy theory is a discrete homotopy theory for graphs. While A-homotopy theory has many of the nice properties of the classical homotopy theory on topological spaces, we would like to know if it also has a nice structure to work with as well. We are pursuing this structure through path categories and homotopy type theory. In this talk, I will discuss why path categories might be the right option to get the structure that we are looking for in A-homotopy theory and how we can define a path category on the category of graphs (or something very close to it). This is joint work with Dr. Laura Scull.

Title: Principal bundles in Join restriction categories

Abstract: Principal bundles arise in different areas of matematics with different definitions. However, they all have in common some kind of local triviality. Here I will present some work in progress on generalizing these in terms of join restriction categories, a notion that means to capture properties of partial maps.
Most of the time we will spend on join-restriction categories and their properties. Then, abstracting concepts from differential geometry, we will consider fiber bundles and principal bundles and see that the existence of a global right action is a consequence and does not need to be demanded.

The slides can be found at https://www.dropbox.com/s/xaj48qzrtfvbvh6/Presentation_join_restriction_bundles.pdf?dl=0

Title: Connections in algebraic geometry

Abstract:

Any tangent category has a notion of connection for its differential bundles (the analog of vector bundles).  In this talk, we’ll explore what this definition gives us in the tangent category of affine schemes.  The talk will be roughly divided into three parts: (1) connections in general, and how they are described in a tangent category, (2) a review of the tangent category of affine schemes, (3) some particular examples of connections in the tangent category of affine schemes, and how the abstract tangent categorical definition relates to existing definitions of connections in algebraic geometry.  

This talk will be split into two parts. The first part starts at 10am, the second one at 11:30am

The speaker’s notes on algebraic geometry can be found at https://www.reluctantm.com/gcruttw/publications/alg_geo_notes.pdf

Title: Tangent structure for operadic algebras

Abstract:

In the preface of its Lectures on Noncommutative Geometry [Gin05], Ginzburg writes:

“Each of the mathematical worlds that we study is governed by an appropriate operad. Commutative geometry is governed by the operad of commutative (associative) algebras, while noncommutative geometry ‘in the large’ is governed by the operad of associative not necessarily commutative algebras. […]

There are other geometries arising from operads of Lie algebras, Poisson algebras, etc.“

In this talk we will show that the theory of tangent categories provides a suitable language to make precise this inspiring intuition of Ginzburg: operads produce (algebraic) geometrical theories. We will construct two different tangent categories for each operad, one defined over the category of algebras of the operad and the other one, over the opposite category, using the theories of tangent categories and tangent monads. We will provide concrete examples of these two tangent categories for a few operads. Finally, we will discuss some ideas for future work on this project.

This is joint work with Sacha Ikonicoff and JS Lemay.

The slides can be found at https://www.dropbox.com/s/y67pqm2w8ih0c5f/Tangent%20Categories%20for%20operadic%20algebras%20-%20Calgary%20February%202023%205.pdf?dl=0.

Bibliography

• Gin05: Ginzburg; Lectures on Noncommutative Geometry (2005) (https://arxiv.org/pdf/math/0506603.pdf)

Title: Algebraic examples of coCartesian differential monads from operads

Abstract:

Operads are a useful tool to classify types of algebras. To each operad is associated a monad, which in turn gives rise to a category of algebras. There is an operad for associative algebras, for commutative algebras, for Lie algebras, and the list goes on.

In this talk, we will give an introduction to operads and their algebras. We will study the notion of operadic derivation, which generalises the notion of algebraic derivation. We will define an analogue of the module of Kähler differentials for an algebra over an operad. Studying the case of free algebras over a fixed operad will allow us to build a natural derivation which equips the associated monad with the structure of a coCartesian differential monad.

This is joint work with Marcello Lanfranchi and JS Lemay.

The slides can be found at https://www.dropbox.com/s/hne867oza0fkp4v/Algebraic%20examples%20of%20coCartesian%20differential%20monads%20coming%20from%20operads.pdf?dl=0

The slides can be found at https://www.dropbox.com/s/hne867oza0fkp4v/Algebraic%20examples%20of%20coCartesian%20differential%20monads%20coming%20from%20operads.pdf?dl=0

Title: Tangent categories & tangent monads

Abstract:

Tangent categories, first introduced by Rosicky [Ros84] and more recently generalized by Cockett and Cruttwell [Coc14], provide a categorical axiomatization for differential geometry. The core idea of this concept is to equip a category with an endofunctor which replicates the role of the tangent bundle functor, i.e. which associates to each object another object that looks like the fibre bundle of tangent spaces of the object. In this talk we explore the basic definitions and results of this theory, providing concrete examples of tangent categories.

In the second part, we are going to explore the theory of tangent monads, first introduced by Cockett, Lemay and Lucyshyn-Wright [Coc19]. Monads are often regarded as powerful tools to produce algebraic theories. In this regard, tangent monads produce geometrical theories. We will dig into the definition of tangent monads and the main results of the theory. We will also show that coCartesian differential monads induce tangent monads. This is a striking result, which will allow us to provide tangent categories for many types of algebras.

The slides can be found at https://www.dropbox.com/s/7mz7k2t7eqttm15/Tangent%20Categories%20%26%20Tangent%20Monads%20-%20Calgary%20February%202023%202.pdf?dl=0.

Bibliography

• Coc14: Cockett, Cruttwell; Differential Structure, Tangent Structure, and SDG (2014) (https://link.springer.com/article/10.1007/s10485-013-9312-0)
• Coc19: Cockett, Lemay, Lucyshyn-Wright; Tangent Categories from the Coalgebras of Differential Categories (2019) (https://arxiv.org/abs/1910.05617)
• Ros84: Rosicky; Abstract tangent functors (1984) (https://eudml.org/doc/91746)