Related papers: Internal higher topos theory

The internal bialgebroid -- in a symmetric monoidal category with coequalizers -- is defined. The axioms are formulated in terms of internal entwining structures and alternatively, in terms of internal corings. The Galois property of the…

We introduce $\infty$-type theories as an $\infty$-categorical generalization of the categorical definition of type theories introduced by the second named author. We establish analogous results to the previous work including the…

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal…

Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…

We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…

A 2-categorical generalisation of elementary topos is provided and some of the properties of the yoneda structure it generates are explored. Examples relevant to the globular approach to higher category theory are discussed. This paper also…

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which…

We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions,…

Let {\phi} be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of {\phi} we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which…

The correspondence between definable connected groupoids in a theory $T$ and internal generalised imaginary sorts of $T$, established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is…

We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous space-valued sheaves on these conically smooth stratified spaces…

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The…

One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes and Lawvere-Tierney topologies. In this paper, we introduce a new but elementary concept, "a…

We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of "adjoint logic" in which the discretization and codiscretization modalities are characterized using a judgmental formalism of "crisp…

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_\omega$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex)…

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension zero smooth embeddings. Now formally identify any object X with its thickening X x [-1,1]. We prove that…

We give an expository, and hopefully approachable, account of the Joyal-Tierney result that every topos can be represented as a topos of sheaves on a localic groupoid. We give an explicit presentation of a representing localic groupoid for…

Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general…

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories and we use this to prove many classical results about monads in the $\infty$-categorical framework. Amongst other…