simplicial homotopy

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• 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 complex — In mathematics, a simplicial complex is a topological space of a particular kind, constructed by gluing together points, line segments, triangles, and their n dimensional counterparts (see illustration). Simplicial complexes should not be… …   Wikipedia

• 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

• Homotopy group — In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The base point preserving maps from an n dimensional sphere (with base point) into a given space (with base point) are collected into equivalence… …   Wikipedia

• Simplicial approximation theorem — In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies… …   Wikipedia

• A¹ homotopy theory — In algebraic geometry and algebraic topology, a branch of mathematics, A1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to… …   Wikipedia

• Pro-simplicial set — In mathematics, a pro simplicial set is an inverse system of simplicial sets.A pro simplicial set is called pro finite if each term of the inverse system of simplicial sets has finite homotopy groups.Pro simplicial sets show up in shape theory,… …   Wikipedia

• Kan fibration — The notion of a Kan fibration is a part of the theory of simplicial sets in mathematics. Kan fibrations are the fibrations in the model category theoretic sense on the category of simplicial sets and are therefore of fundamental importance. The… …   Wikipedia

• Serre spectral sequence — In mathematics, the Serre spectral sequence (sometimes Leray Serre spectral sequence to acknowledge earlier work of Jean Leray) is a basic tool of algebraic topology. It expresses the singular (co)homology of the total space E of a (Serre)… …   Wikipedia

• Freudenthal suspension theorem — In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the… …   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