Information about previous seminars can be found here.
Abstract:
Every compact closed category is monoidal closed where the internal-hom is given as $A \multimap B = A^* \otimes B$. In the special case of self-dual compact closed category, where every object is its own dual $A^* = A$, then we have that the internal-hom is precisely the tensor product $A \multimap B = A \otimes B$. However, is the converse true? Starting with a symmetric monoidal category X such that $A \multimap B = A \otimes B$, is X compact closed? At first glance this seems like this might be true... But if one tries to prove the snake equations, one runs into troubles! So the answer is more likely no... but what's the counter-example? Unfortunately, no one seems to have a natural counter-example...
In this talk, I'll discuss this odd problem in more detail and hopefully someone smarter than me in the audience will come up with a counter-example.
This will be an easy and relaxed talk to start the term, so everyone should be able to follow along!
Abstract:
Categories of interest often have more than one monoidal closed
structure, and "bunched type theories" have been developed to capture
this situation. In this talk, I will present a novel combination of
bunched type theory with dependent types, giving a much more
expressive theory. Every type will have both a linear and non-linear
aspect, and the various type formers interact with both aspects. The
motivation for this work was to find a way to formalise arguments from
stable homotopy theory in type theory, but we also anticipate
applications to the theory of quantum programming languages.
jww. with Dan Licata
Abstract:
In this talk I'll discuss how the overabundance of axioms in
my previous characterization of double categories of
relations (ACT 2021) can be cut down to a manageable number.
Abstract:
A famous theorem of Deligne's says that any
(abelian, $\mathbb{C}$-linear) symmetric monoidal category satisfying certain mild size constraints
admits a symmetric monoidal functor to the category sVec of super vector spaces.
Deligne used this result to classify such symmetric monoidal categories in terms of representation
theories of algebraic groups — this is the "Tannakian Duality". A few years ago, I pointed out that
Deligne's theorem has a neat interpretation: it says that the symmetric monoidal category sVec
is the "algebraic closure" of the symmetric monoidal category Vec of vector spaces;
that the extension Vec $\subset$ sVec is "Galois"; and
that the Tannakian Duality is a categorified version of the Galois correspondence.
In this talk, I will explain the statement of Deligne's theorem and my interpretation, and mention some aspects of Deligne's proof.
Abstract:
I will give the definition of a double fibration as a (pseudo) category
object in the 2-category of fibrations of categories whose source and
target are cleavage-preserving. I will show how, much like functors are
category objects in an arrow category, double fibrations correspond
to (strict) double functors between (pseudo) double categories that
satisfy some properties. We have shown that these properties are
precisely the ones making the double functor an internal fibration
in the 2-category of double categories. We have also shown
a "Grothendieck construction theorem", that is an equivalence of
categories between double fibrations and indexed double categories. The
latter are given by a type of lax functor D --> Span(Cat) from a
double category D, where Span(Cat) is a 3-dimensional structure that
we call a double 2-category. Two examples that are unified by the theory
of double fibrations can also help to understand it, so we will look at
them. One is that (as the name indicates) ours is the non-discrete
case of [1], where the Grothendieck construction is that given by
Bob Paré in [2] for a lax functor D --> Span(Set) between double
categories. Also, seeing monoids as 1-object categories, we can recover
the results for monoidal fibrations and monoidal indexed categories
[3,4]. Being 3-dimensional, this is all a bit "wild" so I don't plan to
give the full details, but I hope to still be able to convey "why"
Span(Cat) is the receiving structure here that makes things work.
[1] DISCRETE DOUBLE FIBRATIONS, MICHAEL LAMBERT, TAC 37 (2021).
[2] YONEDA THEORY FOR DOUBLE CATEGORIES, ROBERT PARE, TAC 25 (2011).
[3] FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS, MICHAEL SHULMAN, TAC 20 (2008).
[4] MONOIDAL GROTHENDIECK CONSTRUCTION, JOE MOELLER AND CHRISTINA VASILAKOPOULOU, TAC 35 (2020).
Abstract:
Several structure-semantics adjunctions and monad-theory equivalences
have been established in category theory. Lawvere developed a
structure-semantics adjunction between Lawvere theories and tractable
Set-valued functors, which was subsequently generalized by Linton,
while Dubuc established a structure-semantics adjunction between
V-theories and tractable V-valued V-functors for a symmetric
monoidal closed category V. It is also well known (and due to Linton)
that there is an equivalence between Lawvere theories and finitary
monads on Set. Generalizing this result, Lucyshyn-Wright (2016)
established a monad-theory equivalence for eleutheric systems of arities
in arbitrary closed categories. Building on earlier work by Nishizawa
and Power, Bourke and Garner (2019) subsequently proved a general
monad-theory equivalence for arbitrary small subcategories of arities
in locally presentable enriched categories. However, neither of these
equivalences generalizes the other, and there has not yet been a general
treatment of enriched structure-semantics adjunctions that specializes
to those established by Lawvere, Linton, and Dubuc.
Motivated by these considerations, we develop a general framework for
studying enriched structure-semantics adjunctions and monad-theory
equivalences for subcategories of arities, which generalizes all
of the aforementioned results and also provides new ones. For
a subcategory of arities J in a V-category C over a
symmetric monoidal closed category V, Linton’s notion of clone
generalizes to provide enriched notions of J-theory and J-pretheory,
which were also employed by Bourke and Garner (2019).
We say that J is amenable if every J-theory admits free algebras,
and is strongly amenable if every J-pretheory admits free algebras.
If J is amenable, then we obtain an idempotent structure-semantics
adjunction between certain J-pretheories and J-tractable V-categories
over C, which yields an equivalence between J-theories and
J-nervous V-monads on C. If J is strongly amenable, then
we also obtain a rich theory of presentations for J-theories and
J-nervous V-monads. We show that many previously studied
subcategories of arities are (strongly) amenable, from which
we recover the aforementioned structure-semantics adjunctions and
monad-theory equivalences. We conclude with the result that any
small subcategory of arities in a locally bounded closed
category is strongly amenable, from which we obtain structure-semantics
adjunctions and monad-theory equivalences in (e.g.) many
convenient categories of spaces.
Abstract:
Tangent categories axiomatize the existence of a "tangent
bundle" endofunctor on a category. The canonical example is the category
of smooth manifolds, but many other examples exist, including some
recent examples from functor calculus.
In this series of two talks, I'll describe a tangent category structure
in algebraic geometry (specifically, on the category of affine schemes)
and explain why I think it's interesting.
The first talk will be introductory: I'll introduce/review tangent
categories and affine schemes, and then look at the tangent category
structure of affine schemes. In the second talk, we'll see how concepts
you can define in any tangent category (such as vector fields,
differential bundles, and connections) apply to this particular
example, and how these concepts relate to existing notions in algebraic geometry.
This talk is based on joint work with JS Lemay.
Abstract:
Tangent categories axiomatize the existence of a "tangent
bundle" endofunctor on a category. The canonical example is the category
of smooth manifolds, but many other examples exist, including some
recent examples from functor calculus.
In this series of two talks, I'll describe a tangent category structure
in algebraic geometry (specifically, on the category of affine schemes)
and explain why I think it's interesting.
The first talk will be introductory: I'll introduce/review tangent
categories and affine schemes, and then look at the tangent category
structure of affine schemes. In the second talk, we'll see how concepts
you can define in any tangent category (such as vector fields,
differential bundles, and connections) apply to this particular
example, and how these concepts relate to existing notions in algebraic geometry.
This talk is based on joint work with JS Lemay.
Abstract:
Persistent (co)homology of spaces is a tool used
by topological data analysists to give information about
a topological space in terms of a filtration of vector spaces on the
space. At a higher level, persistent cohomology is used to give
information and features about a topological space as it persists over
time. In more recent years, persistent cohomology on a topological space
was generalized to a persistent sheaf cohomology by way of Cech cohomology for sheaves.
In this talk we will discuss and show how to generalize persistent
cohomology on a topological space to give a notion of persistent
cohomology for Grothendieck toposes both in terms of a Cech persistent
cohomology and a right derived functor persistent cohomology ---this
gives us two different ways of describing how geometric information
can persist through a filtration of sheaves on a site. As an application
and time-permitting, we will also describe a relative Comparison Lemma
for sites and its interaction with persistent Cech cohomology on
Grothendieck toposes, and/or a persistent Artin Comparison Theorem
for finite type schemes over the complex numbers.
This is joint work with Dorrete Pronk and Deni Salja.
Abstract:
A few weeks ago, my supervisor Geoffrey Cruttwell presented his research on
tangent category theory applied to algebraic geometry in two ATCAT seminar
talks. With this work, we now know that algebraic geometry, differential
geometry and synthetic geometry are models for the axiomatic theory of tangent
categories.
One of the main questions I posed to Geoff when I applied for the PhD program,
was whether non-commutative geometry could be described using the language of
tangent categories. My background in theoretical physics makes me care about
non-commutative geometry because it could be a valid mathematical framework to
describe general relativity in a way that is compatible with quantum mechanics.
Geoff and Lemay's work on commutative algebras allowed me to reformulate my
question in the following terms: can we extend this construction to general
associative algebras?
In this talk, I present an answer to this question showing how the construction
presented by Geoff can be extended to non-commutative geometry and more
generally to algebras of (algebraic symmetric) operads.
The talk will be structured as follows: I will start by giving the main
motivation for the talk, then I will briefly recall the main definitions and
constructions of tangent category theory and of Geoffâ€™s construction. I will
then give the main definitions and results of operad theory. Therefore, I will
show how to construct a canonical tangent structure on the category of algebras
over an operad. Then, I will discuss the corresponding tangent structure over
the opposite category. Finally, I will give some of the results that I found so
far that extend the constructions of the commutative case.
This work is in collaboration with my supervisors Geoffrey Cruttwell and
Dorette Pronk. I also would like to thank J.S. Lemay for the great discussions
and ideas he shared with me about his work and mine.
Abstract:
Actions of a group on a set can be categorified to actions of a
group on a category. The fixed point set becomes a fixed point category. I
will give examples coming from representation theory, topology and physics.
If time permits, I will sketch a generalization to actions of 2-groups on
2-categories.
Updated May 3, 2022 by Frank Fu