Related papers: Segal topoi and stacks over Segal categories
Elmendorf's Theorem states that the category of continuous actions of a topological group is a Grothendieck topos in the sense that it is equivalent to a category of sheaves on a site. This paper offers a 2-dimensional generalization by…
Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we…
A classical result of topos theory holds that the category of coalgebras for a Cartesian comonad on a topos is again a topos (Kock and Wraith, 1971). It is natural to refine this result to a topos-theoretic setting that includes universes.…
Broadly speaking the present is a homotopy complement to the book of Giraud, albeit in a couple of different ways. In the first place there is a representability theorem for maps to a topological champ (a.k.a. stack) and whence an extremely…
We introduce the notion of a "graded topological space": a topological space endowed with a sheaf of abelian groups which we think of as a sheaf of gradings. Any object living on a graded topological space will be graded by this sheaf of…
There are many examples of dualities between topological spaces and algebras in the literature. Particularly, many of those examples come from the algebraic counterpart of a logical system, e.g, boolean and heyting algebras, MV-algebras,…
We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual…
From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models…
We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…
Algebra objects in $\infty$-categories of spans admit a description in terms of $2$-Segal objects. We introduce a notion of span between $2$-Segal objects and extend this correspondence to an equivalence of $\infty$-categories.…
We axiomatise the theory of $(\infty,n)$-categories. We prove that the space of theories of $(\infty,n)$-categories is a $B(\mathbb{Z}/2)^n$. We prove that Rezk's complete Segal $\Theta_n$-spaces, Simpson and Tamsamani's Segal…
Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (Ahrens, Kapulkin, Shulman) solves this by considering…
We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of "proessential point" and use it to give a new characterization of pointed Galois…
We show that the $\infty$-category of global spaces is equivalent to the homotopy localization of the $\infty$-category of sheaves on the site of separated differentiable stacks, following a philosophy proposed by Gepner-Henriques. We…
We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical…
We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt, Mather, Cordier) over a Grothendieck site; we call these ``flexible presheaves''. After some preliminary materiel, we define the ``flexible sheaf'' condition. This…
We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.
In the context of relative topos theory via stacks, we introduce the notion of existential fibred site and of existential topos of such a site. These notions allow us to develop relative topos theory in a way which naturally generalizes the…
In this paper we develop a novel mathematical formalism for the modeling of neural information networks endowed with additional structure in the form of assignments of resources, either computational or metabolic or informational. The…
Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved…