Richard Wood, Prototality as a tool in the study of complete distributivity (Joint with Francisco Marmolejo and Bob Rosebrugh)
Abstract: A category K is defined to be totally distributive if it is locally small and its Yoneda functor has a left adjoint that has a left adjoint. Total distributivity implies the more obvious definition of complete distributivity. However, it seems desirable to be able to show that the ``limits'' possessed by a totally distributive category, which necessarily distribute over the large colimits possessed by a total category, should include not only the small limits, which a total surely has, but also the large limits provided by cototality. We know that a total is not necessarily cototal but we have claimed for some time that a totally distributive K is also cototal.
We prove this by expanding the notion(s) of totality (and cototality) to prototality and showing that, for any locally small K, CAT(K^op, set) is both prototal and procototal.
Tuesday, September 21, 2010
Toby Kenney, Natural Transformations between Cat-valued Profunctors,
and the Universal Property of Span
Abstract:
It is well known that the span construction has the universal
property of adjoining adjoints to morphisms of a category, subject to the
Beck-Chevalley conditions. In this talk, we look at the properties of the
span profunctor, and deduce the well-known universal property of the span
construction, both for the usual span construction, and for some
generalisations.
Tuesday, September 28, 2010
Dorette Pronk, Toposes and Groupoids
Abstract:
Toposes were introduced as a way to generalize spaces so that points could
have non-trivial internal structure, and in particular have a non-trivial
symmetry group. This makes toposes suitable for the study of
generalized quotient spaces such as orbifolds and foliations.
Another natural way to represent these types of spaces is by topological or smooth groupoids. In this talk I will introduce both the categories of groupoids and of toposes and discuss how the two are related to each other, using descent theory and Morita equivalence.
Along the way the leading question will be what the role is of both of these ways of studying generalized quotient spaces. The question has been asked whether groupoids are really the important objects or whether toposes are adding important information or structure to this picture. (Or is this really about toposes after all?)
Geometers and topologists have so far favoured the groupoid approach and have added to this the notion of stacks. In a sequel to this talk, I will further compare stacks to toposes and groupoids.
Tuesday, October 5, 2010
Dorette Pronk, Toposes and Groupoids (Continued)
Tuesday, October 12, 2010
Diego Rojas, Cofinalities of F_\sigma Ideals on \omega
Abstract:
An ideal on \omega is a subset of the power of \omega, closed under
subsets and finite unions. The cofinality of an ideal J is defined as
the least possible cardinality of a collection A\subset J, such that
every set in J has a superset in A. Thinking of the ideals as subsets
of the Cantor Set 2^\omega with the usual Polish Topology, we consider
descriptive set theoretic complexity of the ideals. In particular we
deal with F_\sigma ideals and study their possible cofinalities,
disproving two natural conjectures.
Tuesday, October 19, 2010
Dorette Pronk, The geometry of etale groupoids
Abstract:
For this lecture we will restrict ourselves to smooth toposes and smooth
groupoids. In order to obtain a nice correspondence between the categories
of toposes and groupoids in spaces or manifolds, we will need to restrict
ourselves to etendues on the topos side and etale groupoids on the
groupoid side.
The goal of this lecture will be to get some understanding about the geometric objects that these groupoids represent. If an etale groupoid has a proper diagonal map (s,t) from the space of arrows to the product of the space of objects with itself, it can be viewed as representing an orbifold and two such groupoids represent the same orbifold precisely when they are Morita equivalent.
For arbitrary etale groupoids things are not as simple. Each etale groupoid gives rise to a foliation on its space of objects, and for a given foliation there are at least two canonical ways to construct an etale groupoid. We will discuss those constructions and some of their properties.
Tuesday, October 26, 2010
Alex Hoffnung, Spans, enrichment and categorification
Abstract:
We discuss appearances of spans in geometric constructions
in representation theory. Our example illustrates some of the
structures which could be used in formalizing and comparing
categorification theories.
Tuesday, November 2, 2010
Sara Westreich, From quantum groups to fusion algebras
Abstract:
The small (finite dimensional) quantum groups are obtained from the "big"
ones at roots of unity. Their representation theory plays a central role in
constructing invariants of knots, links and 3-manifolds. The topological
invariants are obtained from the fusion categories associated to these Hopf
algebras, where the fusion categories and the corresponding fusion algebras
are defined on classes of representations satisfying certain properties.
In this talk we discuss a new algebraic approach to fusion algebras obtained from the character rings of Hopf algebras. In the semisimple case this approach extends known relationships between the structure constants of groups and the fusion rules of their characters. In the non-semisimple case it suggests how to obtain general fusion rules for factorizable ribbon Hopf algebras.
Tuesday, November 16, 2010
Bob Paré, Mealy Morphisms
Abstract:
I will discuss Mealy morphisms for V-categories and double categories.
In the V case, they lie somewhere between V-functors and V-profunctors
and may be thought of as partial V-functors. I will give a mini "tutorial"
on profunctors for those who haven't seen them before.
Tuesday, November 23, 2010
Toby Kenney, Freyd Finiteness
Tuesday, November 30, 2010
Toby Kenney, Freyd Finiteness (continued)
Tuesday, December 7, 2010
Machael Warren, The Grothendieck construction for strict omega-categories
Abstract:
In this talk I will describe an extension of the familiar Grothendieck
construction to the setting of strict omega-categories. In particular, we
will give a brief introduction to strict omega-categories followed by a
description of the higher-dimensional structure of the category OmegaCat
of small strict omega-categories. We then summarize the direct
(combinatorial) description of the Grothendieck construction and explain
its higher-dimensional universal property. A related application to
mathematical logic will also be mentioned.
Tuesday, January 11, 2011
Bob Paré, Yoneda for Double Categories I
Abstract:
The Yoneda lemma is perhaps the "Fundamental Theorem of Category Theory".
It generalizes to V-categories which then specializes to 2-categories and
this is easily generalized to bicategories. Although double categories
have a certain bicategoricaliness to them the Yoneda lemma is very different.
I will start with a brief overview of double category theory with examples
and then discuss the Yoneda lemma in this context.
Tuesday, January 18, 2011
Bob Paré, Yoneda for Double Categories II
Abstract:
I will define the double Hom functor whose values are in SSet. Then
I will introduce natural transformations of lax functors and prove
the first part of the Yoneda lemma, viz. that natural transformations
from a representable into a lax functor correspond to elements of that
functor. An application to adjoints for double categories will be given.
Tuesday, January 25, 2011
Bob Paré, Yoneda for Double Categories III
Abstract:
I will recall the definition of the double category Doub and how it
gives the definition of adjoints for double categories. Then I will
show this definition is equivalent to the Hom definition of adjoint.
Time permitting, I will start the discussion of how Hom depends on
vertical arrows, thus paving the way to the second Yoneda theorem.
Tuesday, February 1, 2011
Bob Paré, Yoneda for Double Categories IV
Abstract:
I will show how the study of how the hom functor depends on vertical
arrows leads to modules and modulations for lax functors. A simple
example shows how this leads to the "correct" notion of "profunctor over A".
We will then consider the second Yoneda theorem, which has a corollary,
fullness for modulations.
Tuesday, March 8, 2011
Richard Wood, Local discrete opfibration pseudofunctors
(Joint work with JRB Cockett and SB Niefield.)
Abstract:
For W a bicategory, there is a 2-category W-cat of W-enriched categories
that specializes, when W has one object, to the usual 2-category of
categories enriched over a monoidal category. The idea is due to Walters
(or maybe with different terminology to Benabou). We write \hat W for the
bicategory with |\hat W|=|W| and (\hat W)(w,x)=set^{W(w,x)^op} and
composition structure given by generalized Day-convolution. We write
pro-W-cat for the 2-category determined "within bicat" by the pseudofunctors
with codomain W whose effects on hom categories are discrete fibrations.
The latter are "local discrete fibration pseudofunctors".
Theorem: pro-W-cat is 2-equivalent to (\hat W)-cat.
The reason for returning to these notions, which have been aired several
times before at @CAT, is that every "pro-W-category" H:A--->W has a
characteristic lax functor W^co--->set-mat=~spn(set). The appearance of
mat=set-mat here seems related to the appearance of spn as the codomain
for double hom-functors in the work of Bob Pare. For bicategories W and M
we define fun(W,M) using lax functors, oplax naturals with map components,
and modulations.
Theorem: pro-W-cat is 2-equivalent to fun(W^co,mat)
This 2-equivalence is mediated by pulling back a universal local discrete
opfibration pseudofunctor P:mat_*---> mat.
This author conjectures that the theorems above will probably be better seen in the context of double categories.
Tuesday, March 15, 2011
Peter Lumsdaine, "Univalent Foundations": how to treat isomorphism as equality
Abstract:
In informal practice, it is common to treat isomorphic objects, equivalent
categories, and the like, as ?the same?. We all know, of course, that
they are not really the same ? classically, this would be inconsistent ?
so we pay our dues by carefully proving that the theorems we prove are
invariant under isomorphism, equivalence, etc.
But, there is another way! If we work in a weaker logical foundation, it
*is* consistent to posit that ?equivalent? objects are always ?equal?, in
the internal sense; it is then automatic that everything is invariant
under equivalence, and it is literally true that there is, for instance, a
literally unique free group on a given set of generators.
Of course, to make this consistent, the logical setting must have a richer and subtler of equality than the classical one: it turns out that intensional type theory, with Martin-L?f identity types, is ideally suited to the purpose. In this talk I will briefly review this system, introduce Voevodsky?s ?univalence? axiom, and very briefly outline how mathematics can be developed within this foundation.
Tuesday, March 22, 2011
Peter LeFanu Lumsdaine, Higher categories from type theories
Abstract:
Last week, I sketched how one may aim to profitably develop
mathematics in a foundation (such as intensional Martin-L=C3=B6f type
theory) where the notion of "equality" between terms of a type is
enriched from a simple relation to instead be something more like
paths in a space, or morphisms in a higher groupoid.
This week I will explain one way to make the last part of that analogy precise: I will show how the classifying categories of such type theories carry natural (infinity,1)-category structures, with higher cells coming from the Id-types (i.e. the internal equality) of the theory.
Precisely, I will show how to construct for each theory T a Batanin/Leinster-style globular algebraic omega-category Cl_omega(T), overlying the usual classifying category Cl(T), with structure given by a contractible operad of syntactically-defined composition laws.
Tuesday, March 29, 2011
Peter LeFanu Lumsdaine, Higher categories from type theories (continued)
Abstract:
Last week, I sketched how one may aim to profitably develop
mathematics in a foundation (such as intensional Martin-L=C3=B6f type
theory) where the notion of "equality" between terms of a type is
enriched from a simple relation to instead be something more like
paths in a space, or morphisms in a higher groupoid.
This week I will explain one way to make the last part of that analogy precise: I will show how the classifying categories of such type theories carry natural (infinity,1)-category structures, with higher cells coming from the Id-types (i.e. the internal equality) of the theory.
Precisely, I will show how to construct for each theory T a Batanin/Leinster-style globular algebraic omega-category Cl_omega(T), overlying the usual classifying category Cl(T), with structure given by a contractible operad of syntactically-defined composition laws.
Tuesday, April 5, 2011
Michael Warren, Voevodsky's univalent model of type theory (part I)
Abstract:
In this first of two (or possibly three) talks we will begin to describe
Voevodsky's univalent model of type theory in the category of simplicial
sets. In particular, we recall a construction, due to Hofmann and
Streicher, of universes in Grothendieck toposes. We describe how the
universe of small Kan fibrations in the univalent model can be obtained in
this way and we also describe an alternative (and novel) construction of
this universe.
In part two of this talk we will explain the calculus of minimal fibrations and will give an exposition of Voevodsky's proofs that the universe interprets a type and that it satisfies the univalence axiom.
These talks will be designed in such a way as to focus on the categorical, rather than type theoretic, details and should therefore be accessible to those who are not already experts in type theory.
Tuesday, April 12, 2011
Michael Warren, Voevodsky's univalent model of type theory (part II)
Abstract:
In this second talk we continue our exposition of Voevodsky's univalent
model of type theory in the category of simplicial sets. In particular,
we recall a variety of simplicial machinery (such as the theory of minimal
fibrations) and explain how to prove that the universe of small Kan
fibrations is a Kan complex. If time permits, we will also introduce
univalent fibrations.
Tuesday, April 19, 2011
Michael Warren, Voevodsky's univalent model of type theory (part III)
Abstract:
In this third talk we continue our exposition of Voevodsky's univalent
model of type theory in the category of simplicial sets.