A partite category (a partite\/ internal category has its objects and arrow partitioned into many objects) internal to a join restriction category $\mathbb{B}$ induces, by considering partial sections of the domain maps, an external join restriction category which sits over $\mathbb{B}$ by a hyperconnected functor. Conversely an external join restriction category over $\mathbb{B}$ (where the latter must be assumed to have all gluings) induces a source \'etale partite category internal to $\mathbb{B}$. This correspondence may be completed to a Galois adjunction between join restriction categories over $\mathbb{B}$ and partite categories internal to $\mathbb{B}$ with cofunctors. The adjunction specializes to an equivalence between hyperconnections over $\mathbb{B}$ and source \'etale partite categories internal to $\mathbb{B}$.

This phenomenon occurs in many different places in mathematics (often specialized to groupoids). In particular, as all join restriction categories have a hyperconnected fundamental functor to the category of locales (with partial maps) one can conclude that join restriction categories (with join functors) correspond precisely to source \'etale partite categories internal to locales (with cofunctors). From algebraic geometry, considering schemes as a join restriction category with gluings, the identity functor on schemes induces an internal partite category: the object of morphisms from the affine scheme $R$ to $\mathbb{Z}[x]$ is then exactly the structure sheaf of $R$.

Topological complexity is an integer $TC(X)$ reflecting the complexity of motion planning algorithms for all systems having $X$ as their configuration space. Roughly, $TC(X)$ is the minimal number of continuous rules which are needed in a motion planning algorithm. This invariant was introduced by Farber in 2002 and is closely related to the classical Lusternik-Schnirelmann category.

In recent years, several versions of topological complexity aimed at exploiting the presence of a group $G$ acting on the configuration space have appeared. We will present several approaches to describing equivariant topological complexity variants in the context of the bicategory of orbifold translation groupoids and will show that $TC^G(X)$ is invariant under Morita equivalence.

In the past several years, there has been a lot of development of differential geometric ideas within a tangent category: definitions have been given for the Lie bracket, vector bundles, connections, differential forms, and de Rham cohomology. However, in a few cases, these definitions have required the assumption that one could negate tangent vectors, reducing the applicability of those definitions from the full range of settings mentioned above.

In particular, the previously given definitions of curvature and torsion in this setting required negatives. In this talk, I'll show how to define curvature and torsion in this abstract setting without requiring the existence of negatives, leading to their applicability in a wider variety of examples.

By considering ordered groupoids as double categories, we are able to extend Lawson's correspondence to an adjoint equivalence of 2-categories between the 2-category of left cancellative categories and the 2-category of ordered groupoids. Making use of this equivalence, we can then state a comparison lemma for Ehresmann sites, which classifies functors between Ehresmann sites that induce an equivalence between their categories of sheaves.

This is joint work with Dorette Pronk.

This talk will discuss work to understand a weaker condition for tangent categories called an \'etale subobject classifier. Indeed, having an \'etale subobject classifier in a tangent category is weaker than being a quasitopos. We will explore the logic required to obtain an \'etale partial map classifier from an \'etale subobject classifier. We will also explore the resulting logic in the \'etale subobject fibration which has connectives $\wedge,\top,\exists,\vee,\bot$.

We will then discuss more generally the coherence of partial cartesian closed tangent categories, making use of recent work on tangent categories to characterize tangent bundles abstractly. We will discuss what it means for partial maps to be classified in this setting. To provide a less SDG example, we will discuss partial maps between convenient vector spaces.

The goal of this project is to develop workable foundations of $(\infty,1)$-category theory in the category of cubical sets. To this end, in joint work with Christian Sattler and Liang Ze Wong, we define four model structures on categories of (marked and unmarked) cubical sets and explore some of their properties.

This part 2 of a 2-part series given by Chris Kapulkin and Zach Lindsey.

The goal of this project is to develop workable foundations of $(\infty,1)$-category theory in the category of cubical sets. To this end, in joint work with Christian Sattler and Liang Ze Wong, we define four model structures on categories of (marked and unmarked) cubical sets and explore some of their properties.

This part 1 of a 2-part series given by Chris Kapulkin and Zach Lindsey.

In this talk, we will show that by taking as $\mathcal{V}$ the category of convergence spaces and as $\mathcal{A}$ the category of convex spaces internal to $\mathcal{V}$, with the unit interval as dualizer, the induced functional distribution monad gives rise to the notion of \textit{$\tau$-additive} (or \textit{$\tau$-smooth}) probability measure on Tikhonov spaces. Thus the resulting monad captures a wide class of settings in topological measure theory, including not only Radon probability measures on locally compact spaces but also Borel probability measures on Polish spaces. In proving our result, we establish a connection between $\tau$-additive measures and continuous convergence, and we prove integral representation theorems for $\tau$-additive measures that are formulated in terms of the cartesian closed structure of the category of convergence spaces.

[1] R. B. B. Lucyshyn-Wright, Functional distribution monads in functional-analytic contexts. Advances in Mathematics 322 (2017), 806--860.

This is joint work with Matthew Burke.

I will discuss categorical conditions which allow us to better understand constructions like these. We develop weaker conditions for a bicalulus of fractions to exist, and show how this can be used to pass to a small subclass of arrows to be inverted. Time permitting, I will talk about how to use pseudo pullbacks to simplify the 2-cell constructions and compositions in the resulting bicategory of fractions.

This is joint work with D. Pronk at Dalhousie University.

In this talk, I will explain how if $\mathcal{X}$ is a differentiable stack presented by some Lie groupoid $H$, the data of an action of $G$ on $\mathcal{X}$ can be repackaged as a Lie groupoid fibration $\pi:A \to G$ with kernel groupoid $H$. As fibrations are relatively common in the study of Lie groupoids, I will (time permitting) be able to give plenty of examples of the transition between Lie groupoid fibrations and stack actions, including examples related to gerbes, VB-groupoids, and flows of vector fields on stacks.