simplicial object

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• Simplicial category — In mathematics, the simplicial category (or ordinal category) is a construction in category theory used to define simplicial and cosimplicial objects. Formal definitionThe simplicial category is usually denoted by Delta and is sometimes denoted… …   Wikipedia

• Simplicial set — In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a well behaved topological space. Historically, this model arose from earlier work in combinatorial topology and… …   Wikipedia

• Simplicial manifold — In mathematics, the term simplicial manifold commonly refers to either of two different types of objects, which combine attributes of a simplex with those of a manifold. Briefly; a simplex is a generalization of the concept of a triangle into… …   Wikipedia

• Fibrant object — In mathematics, specifically in homotopy theory in the context of a model category M , a fibrant object A of M is an object that has a fibration to the terminal object of the category.PropertiesThe fibrant objects of a closed model category are… …   Wikipedia

• Injective object — In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories. The dual notion is …   Wikipedia

• Cotangent complex — In mathematics the cotangent complex is a roughly a universal linearization of a morphism of geometric or algebraic objects. Cotangent complexes were originally defined in special cases by a number of authors. Luc Illusie, Daniel Quillen, and M.… …   Wikipedia

• Hochschild homology — In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Definition of Hochschild homology of algebras Let k be a ring, A an associative k… …   Wikipedia

• Cyclic homology — In homological algebra, cyclic homology and cyclic cohomology are (co)homology theories for associative algebras introduced by Alain Connes around 1980, which play an important role in his noncommutative geometry. They were independently… …   Wikipedia

• Model category — In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ( arrows ) called weak equivalences , fibrations and cofibrations . These abstract from a conventional homotopy category, of… …   Wikipedia

• Nerve (category theory) — In category theory, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C …   Wikipedia

• Homology (mathematics) — In mathematics (especially algebraic topology and abstract algebra), homology (in Greek ὁμός homos identical ) is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a… …   Wikipedia