Programme

A double category is a “category” with two kinds of morphism. Perhaps the first significant example is the Kleisli construction. We will take a double category perspective on it. Morphisms of monads, 2-monads, and distributive laws will be touched upon. We will end with the speculative notion of doublad, a truly double category theoretic version of monad.

(Slides available from: https://www.mscs.dal.ca/~pare/CasaAbierta.pdf.)

We describe in combinatorial terms (without reference to tangent bundles) the notion of Riemannian metric, and the construction of a volume form derived from such a metric.

(Text and slides available from http://tildeweb.au.dk/au76680/heron8.pdf and http://tildeweb.au.dk/au76680/mex.pdf.)

We provide a bicategorical analog of the notion of geometry in the sense of the theory of spectrum. We introduce a notion of bistable pseudofunctor, categorifying an analog 1-categorical notion of stable functor; we then describe 2-dimensional analogs of orthogonality and factorization systems, and use them to construct examples of bistable pseudofunctors through inclusion of left objects and left maps. We apply the latter construction to several examples of factorization systems for geometric morphisms to produce geometry-like situations for Grothendieck topoi, recovering in particular a local geometry involved in the general construction of spectra.

If you have a little familiarity with topos theory, you might know their standard properties and one or more ways of constructing examples. On the other hand, properties of morphisms of toposes, so-called *geometric morphisms*, are less commonly examined in a lot of detail. This is a shame, because they consist of an adjoint pair of functors, which gives a lot of room for a rich variety of properties. In this talk we will present a sketch of the map of different properties and some ways of constructing geometric morphisms between Grothendieck toposes. We also explain how these properties enable us to build formal analogies between toposes arising from different areas of maths.

A geometric morphism f is precohesive if and only if it is essential, local and hyperconnected with f_! preserving finite products. At the moment, it is not known whether precohesive geometric morphisms are always locally connected. As a first step in resolving this question, we will discuss an example of a geometric morphism f : PSh(M) → PSh(N) that is essential, local and hyperconnected but not locally connected, with M and N monoids. Afterwards, we will consider some conditions on M such that local connectedness does follow from the other three properties. The talk is based on joint work with Morgan Rogers.

For a finite group G, we define the free G-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of G. Also, we find an isomorphism of its fundamental group onto the direct sum ℤ⊕ H₂(G), where H₂(G) is the integral 2-homology group, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product G/[G,G]× S¹× X, where π₁(X)=H₂(G). Finally, we present some results about the classification of G-topological quantum field theories in dimension two.

(Article available from https://arxiv.org/pdf/1211.2144.pdf.)

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.

A convexity structure on a set is a collection of subsets closed under directed unions and arbitrary intersections. A topological convexity space is a space that has both a topological and a convexity structure. In this talk, we will look at how the classical Euclidean spaces, i.e. vector spaces over the real numbers, can be axiomatised in the category of topological convexity spaces.

In spite of the fact that there exists a 2-adjunction that relates the 2-category of adjunctions Adj and the 2-category of monads Mnd, adjunctions have an additional structure of a double category. The most natural question is the following one: Which is the behaviour of the 2-adjunction over this double structure? In this talk, this question will be addressed and also the theorem of J. Beck on distributive laws will be commented from this point of view.

In this talk we provide an algebraic description of those coextensive varieties V such that the indecomposable members of Vop are classified by the Gaeta topos associated to V by means of the behavior of certain finitely presented algebras of V.

We systematically lift Lawvere's definition of precohesive toposes to arbitrary base toposes using the usual techniques of fibered categories following the ideas and work of Jean Benabou. However, we reformulate the conditions arising this way in a language free from any "fibrational smell". We also discuss some recent independence results showing that locally connectedness doesn't follow for free from the other assumptions. Finally, we discuss a question by Matías Menni for which we don't have an answer yet, namely whether for hyperconnected and local geometric morphisms the inverse image part preserves dependent function spaces whenever it preserves ordinary nondependent function spaces. We give reasons why this is presumably wrong but, unfortunately, still lack a counterexample.

(Slides available from https://www2.mathematik.tu-darmstadt.de/~streicher/TALKS/PrecohFib.pdf.)

The problem of the existence of cokernels in the stable category of a left hereditary ring is studied. Namely, it is proved that if a ring is left hereditary, left perfect and right coherent, then the stable category has cokernels. Moreover, it is shown that the condition for a ring to be left perfect and right coherent is also necessary for the stable category to have cokernels, provided that the ring is left hereditary and satisfies the additional condition that there are no nonzero projective injective left modules over it.

The notion of a distributive law of monads was introduced by Beck, and gives a concise description of the data required to compose monads. In the two dimensional case, Marmolejo defined pseudodistributive laws of pseudomonads (where the required diagrams only commute up to an invertible modification). However, this description requires a number of coherence conditions due to the extra data involved.

In this talk we give a number of alternative presentations of pseudodistributive laws, including 'decagon', 'pseudoalgebra' and 'no-iteration' presentations, which arise from studying pseudodistributive laws when the involved pseudomonads are presented in their extensive (“Kleisli triple”) form.

As an application of this approach, we show that of Marmolejo and Wood's eight coherence axioms for pseudodistributive laws, three are redundant in that they follow from the other five.

If time permits, we will also consider the case where one or both of the involved pseudomonads are (co)KZ. In this setting we deal with the coherence axioms by using (a 2 dimensional version of) the embedding of monads into concrete double categories (as appears when one studies AWFS), which sends a monad P to the concrete double category of 'P-split-mono's.

In this talk I will report on an ongoing joint research project with M. Menni and P. Jipsen on “rigs”, i.e, rings without negatives. They can be axiomatised as (commutative and distributive) semirings with both neutral elements and such that 0 in an annihilator: x*0=0. In this joint project we are trying to explore the universal algebraic aspects of these structures. Focusing on “integral” rigs (i.e. the ones that satisfies x+1=1), I will present a characterisation of the subdirectly irreducible algebras in the variety and then move to the study of Weil rigs, i.e., the ones that have a unique homomorphism into {0,1}.

We give structural results about bifibrations of (internal) (∞, 1)-categories with internal sums. This includes a higher version of Moens' Theorem, characterizing cartesian bifibrations with extensive (aka stable and disjoint) internal sums over lex bases as Artin gluings of lex functors. Our account follows Streicher's presentation of fibered category theory à la Bénabou, generalizing the results to the internal higher-categorical case, formulated in a synthetic setting. Namely, we work inside simplicial homotopy type theory, which has been introduced by Riehl and Shulman as a logical system to reason about internal (∞, 1)-categories, interpreted as Rezk objects in any given Grothendieck–Rezk–Lurie (∞, 1)-topos.

(Slides available from: https://drive.google.com/file/d/1p97gCVtqLzX9AIKhcto3wbErYEw4-BDp/view?usp=sharing.)

A linear bicategory, in the sense of [2], has compositions $\otimes, \oplus$ satisfying distributive laws $A\otimes(B\oplus C)\to (A\otimes B) \oplus C$ and $(A\oplus B) \otimes C\to A\oplus(B\otimes C)$ and appropriate coherence conditions.

In a recent preprint [1], we defined a Girard bicategory as a biclosed bicategory $\cal B$ equipped with a cyclic dualizing family ${\cal D}=\{D_X\colon X \to X\}$, indexed by the 1-cells of $\cal B$, and showed that ${\cal D}$ induces an operation $A \oplus B = (D\mbox{$\circ\!-$} A)\mbox{$-\!\circ$} B$ on 1-cells, and a linear structure on $\cal B$. Most of our examples are related to the monoidal category $\bf Sup$ of complete lattices and sup-preserving maps, including the bicategories ${\cal L}oc$ of locales, ${\cal Q}uant$ of quantales, and ${\cal Q}tld$ of quantaloids (with bimodules and their homomorphisms as their 1-cells and 2-cells, respectively).

There are many other monoidal categories of interest, e.g., abelian groups, commutative monoids, sets (or any topos), etc. Although the related bicategories are not Girard, one can relax the definition, and consider the operation $\oplus$ for a family that is cyclic but not necessarily dualizing. In this talk, we show that the resulting structure, we call a linear left skew-bicategory, satisfies the above distributive laws, but $\oplus$ is merely left skew associative, i.e., the 2-cells $(A\oplus B) \oplus C \to A\oplus (B \oplus C)$, $A\to A\oplus D$, and $D\oplus A\to A$ are natural but not necessarily invertible.

[1] R. Blute and S. Niefield, Linear bicategories, Girard quantaloids and finiteness spaces, preprint.

[2] R. Cockett, J. Koslowski, and R. Seely, Introduction to linear bicategories, Math. Struct. in Comp. Sci. 10, (2000).

This is joint work with Rick Blute.

Lenses provide an implementation of what computer scientists call "bidirectional transformations (BX)". Bidirectional Transformations are between (or among) domains of model states and they require synchronization data and a post-update resychronization process. A domain of model states and their updates is best viewed as a category. Then a lens structure between model domains is defined by functors satisfying axioms. Lenses come in related versions that are called "symmetric" and "asymmetric". Lenses are themselves the arrows of a category, or even (in the symmetric case) the arrows of a bicategory. After beginning with the motivation and definitions for the two types of lenses, we'll review various results with Mike Johnson emphasizing monads and distributive laws. Mike's student, Bryce Clarke, has significantly extended our results, and we will also to survey some of Bryce's recent work, including that symmetric lenses define a bicategory and that an asymmetric lens is actually an algebra for a monad.

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. As a consequence, we obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor.

(Slides available from https://c92b436e-9e91-4e40-92d9-794c72facc5f.filesusr.com/ugd/b22c7a_6400b5417df14e29bbd879490794775b.pdf.)

The usual tensor product operations of algebras and bimodules provide the bicategory of algebras, bimodules and bimodule morphisms, with a symmetric monoidal structure. The coherence data for this structure is expressed, in significantly simpler terms, as the coherence data of a symmetric monoidal framed biategory. This observation applies in several other contexts, e.g. profunctors, polynomial comonoids, parametrized spectra, and open Petri nets, among others, and ultimately depends on a choice of 'lift' of a base bicategory to a framed bicategory with specified category of vertical arrows. Such lifts are known to exist in the presence of additional structure, e.g. a cartesian or cocartesian monoidal fibration, but no generally applicable lifting methods are known. This is problem is particularly acute when considering symmetric monoidal structures on bicategories of operator algebras. Globularly generated double categories are minimal solutions to lifting problems for bicategories. I will introduce the globularly generated condition and I will provide an overview of recent and not-so-recent results on the internal structure of globularly generated double categories, length, and the existence of fibrant globularly generated lifts.

In this talk I will discuss the following conjecture, part of which has already been proved and published.

Conjecture) For a total category K , equivalent are:

i) K is totally distributive;

ii) \tilde{K} is a taxon and K \cong imod(1, \tilde{K});

iii) There is a Freyd/Street-small taxon T and an equivalence K ≅ imod(1,T).