In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn...

honor of Daniel Kan. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets. For each n ≥ 0, recall that the standard n...

mathematics, a simplicial set is a sequence of sets with internal order structure (abstract simplices) and maps between them. Simplicial sets are higher-dimensional...

group laws Crossed module Milnor's theorem on Kan complexes Fibration of simplicial sets May, Ch. 8. § 3. May, Ch 4. § 5. Milnor 1959, Corollary 1. NB:...

the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The...

an op-fibration; in particular, not a cocartesian fibration. A right fibration between simplicial sets is an example of a cartesian fibration. Given...

Grothendieck construction Stack (mathematics) Artin's criterion Fibration of simplicial sets Giraud, Jean (1964). "Méthode de la descente". Mémoires de la...

subject to the following axioms. A fibration that is also a weak equivalence is called an acyclic (or trivial) fibration and a cofibration that is also a...

mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable...

the context of the theory of simplicial sets, the fibrant objects are known as Kan complexes after Daniel Kan. They are the Kan fibrations over a point...

a simplicial set A {\displaystyle A} define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:...

final object is a fibration (Kan fibration). It is often taken as a model of an ∞-groupoid. 2.  A Kan fibration between simplicial sets is a map having...

equivalence between minimal Kan fibrations is an isomorphism. A minimal Kan fibration is a fiber bundle (in the simplicial sense). Quillen's original approach...

{\displaystyle \Delta [n]\to C} . (See Kan fibration#Definitions for a definition of the simplicial sets Δ [ n ] {\displaystyle \Delta [n]} and Λ k [...

homotopy inverse. Analysis Situs approximate fibration 1.  An approximate fibration, a generalization of a fibration and a projection in a locally trivial bundle...

{\displaystyle H_{n}(X;\mathbb {Z} ).} Fibration Hopf fibration Hopf invariant Knot theory Homotopy class Homotopy groups of spheres Topological invariant Homotopy...

theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition. Rational Hurewicz theorem: Let X be a...

We say that: f is a weak equivalence if, for any fibre functor x of T, the morphism of simplicial sets x ∗ f : x ∗ X → x ∗ Y {\displaystyle x^{*}f:x^{*}{\mathcal...

models of type theory. He also proved, using an idea of A. K. Bousfield, that this universal fibration was univalent: the associated fibration of pairwise...

and the fundamental groupoid of a simplicial set Animations to introduce fundamental group by Nicolas Delanoue Sets of base points and fundamental groupoids:...

homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The...

structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called fibrations, cofibrations and weak...

_{X}\cdots \times _{X}C} , n-fold fibre product. This is the Čech nerve. By taking connected components we get a simplicial set, which we can realise topologically:...

structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called fibrations, cofibrations and weak...

holds much more generally, for fibrations with certain conditions. If p : E → B {\displaystyle p\colon E\to B} is a fibration with fiber F, with the base...

subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler...

elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology. More generally, the category of elements plays...

dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader...

_{i}(S^{15})\oplus \pi _{i-1}(S^{7}).} The three fibrations have base space Sn with n = 2m, for m = 1, 2, 3. A fibration does exist for S1 (m = 0) as mentioned...

necessary and sufficient conditions for the solvability of a certain lifting problem involving simplicial sets. In particular, in higher category theory, it proves...