Simplicial localization information

branch of mathematics, the simplicial localization of a category C with respect to a class W of morphisms of C is a simplicial category LC whose π 0 {\displaystyle...

the localization of the category is unique up to unique isomorphism of categories, provided that it exists. One construction of the localization is done...

equivalences. Bousfield localization is named after Aldridge Bousfield, who first introduced this technique in the context of localization of topological spaces...

Bousfield localization. For example, the category of simplicial sheaves can be obtained as a Bousfield localization of the model category of simplicial presheaves...

function. simplicial category A category enriched over simplicial sets. Simplicial localization Simplicial localization is a method of localizing a category...

mid-1950s he made distinguished contributions to the theory of simplicial sets and simplicial methods in topology in general. In recognition of this, fibrations...

definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations...

composition. This creates a technical problem which can be solved using simplicial techniques: giving a method for constructing a model for homotopy colimits...

topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) Using this inverse system of simplicial sets one may...

For any (not necessarily local) ring R, the localization Rp at a prime ideal p is local. This localization reflects the geometric properties of Spec R...

is something fundamentally algebraic. fiber sequence cofiber sequence Simplicial homotopy Highly structured ring spectrum Homotopy type theory Pursuing...

structure on simplicial sets: the associated homotopy category is equivalent to the homotopy category of topological spaces, even though simplicial sets are...

isomorphism in Ho(C). Goerss, Paul G. [in German]; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin:...

noise. If X {\displaystyle X} is any space which is homeomorphic to a simplicial complex, and f , g : X → R {\displaystyle f,g:X\to \mathbb {R} } are continuous...

with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by...

More powerful GNNs operating on higher-dimension geometries such as simplicial complexes can be designed. As of 2022[update], whether or not future architectures...

ordinary cohomology theories are represented by Eilenberg–MacLane spaces. On simplicial complexes, these theories coincide with singular homology and cohomology...

of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety...

\mathbf {sSet} } embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a Kan complex. The nerve has a...

Poincaré Fundamental group of a topological space. 1895 Henri Poincaré Simplicial homology. 1895 Henri Poincaré Fundamental work Analysis situs, the beginning...

direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed...

For example, the category of topological spaces and the category of simplicial sets both admit Quillen model structures whose nerve and realization adjunction...

simply connected region between any three mutually tangent convex sets Simplicial complex — all vertices, line segments, triangles, tetrahedra, ..., making...