Related papers: Internal higher topos theory

Model theoretic internality provides conditions under which the group of automorphisms of a model over a reduct is itself a definable group. In this paper we formulate a categorical analogue of the condition of internality, and prove an…

We derive extensions of the monomialization theorems for morphisms of varieties in our earlier work. In this note we show that a local monomialization can be found which satisfies stronger local conditions. Some comments are made about how…

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic…

We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of…

For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we study necessary and sufficient conditions for a functor $f \colon \mathscr{C} \to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the $n$-topos…

We suggest a generalization of \pi_0 for topological groupoids, which encodes incidence relations among the strata of the associated quotient object, and argue for its utility by example, starting from the orbit categories of the theory of…

We further develop Weber's notion of elementary 2-topos by proposing certain new axioms. We show that in a 2-category C satisfying these axioms, the "discrete opfibration (DOF) classifier" S is always an internal elementary 1-topos, in an…

There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of $L_{\omega_1, \omega}$. We show that it is in fact a…

We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be…

We introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe''…

We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and…

Using the symmetric monoidal closed category structure of the category of measurable spaces, in conjunction with the Giry monad which we show is a strong monad, we analyze Bayesian inference maps and their construction in relation to the…

Homotopy type theory is a new branch of mathematics which merges insights from abstract homotopy theory and higher category theory with those of logic and type theory. It allows us to represent a variety of mathematical objects as basic…

In this article, we interconnect two different aspects of higher category theory, in one hand the theory of infinity categories and on an other hand the theory of 2-categories.We construct an explicit functorial path objet in the model…

The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been…

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M,boundary of M) can be recovered from the configuration category of the interior of M. The grouplike monoid of…

We prove a generalization of Quillen's Theorem B to strict $\infty$-categories. More generally, we show that under similar hypothesis as for Theorem B, the comma construction for strict $\infty$-categories, that we introduced with…

We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover…

In this survey, we provide an overview of category theory-derived machine learning from four mainstream perspectives: gradient-based learning, probability-based learning, invariance and equivalence-based learning, and topos-based learning.…

It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the…