Related papers: Fibrations in Directed Type Theory
This PhD thesis deals with some new models of intensional type theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic fibration categories (the notion of…
A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a…
We prove the uniqueness, the functoriality and the naturality of cylinder objects and path objects in closed simplicial model categories.
This paper considers a class of coalgebras over the Barratt-Eccles operad and shows that they classify Z-completions of pointed, reduced simplicial sets. As a consequence, they encapsulate the homotopy types of nilpotent simplicial sets.…
Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories.…
We show that any closed model category of simplicial algebras over an algebraic theory is Quillen equivalent to a proper closed model category. By ``simplicial algebra'' we mean any category of algebras over a simplicial algebraic theory,…
Lie $n$-algebras are the $L_\infty$ analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite type Lie $n$-algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie…
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincar\'e-Hopf type formulae which relates the…
We construct a model category structure on the category of diffeological spaces which is Quillen equivalent to the model structure on the category of topological spaces based on the notions of Serre fibrations and weak homotopy…
In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family…
It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed…
For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant…
We introduce Displayed Type Theory (dTT), a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$-topos, while the simplicial…
An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine's intermediate model structures…
The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories, and indexed categories, and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of…
In this technical note, we proffer a very explicit construction of the "dual cocartesian fibration" $p^{\vee}$ of a cartesian fibration $p$, and we show they are classified by the same functor to $\mathbf{Cat}_{\infty}$.
We study weak approximation on rationally connected varieties under an assumption of strong approximation for a "simple" variety or under Schinzel's hypothesis. We also get some unconditional results.
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…
We construct classes of K\"ahler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map…
We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalised species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor…