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  2. Homotopy theory - Wikipedia

    en.wikipedia.org/wiki/Homotopy_theory

    Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's model categories. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations ...

  3. Homotopy - Wikipedia

    en.wikipedia.org/wiki/Homotopy

    Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.

  4. Homotopy type theory - Wikipedia

    en.wikipedia.org/wiki/Homotopy_type_theory

    The first higher-dimensional models of intensional type theory were constructed by Steve Awodey and his student Michael Warren in 2005 using Quillen model categories.These results were first presented in public at the conference FMCS 2006 [5] at which Warren gave a talk titled "Homotopy models of intensional type theory", which also served as his thesis prospectus (the dissertation committee ...

  5. A¹ homotopy theory - Wikipedia

    en.wikipedia.org/wiki/A¹_homotopy_theory

    A 1 homotopy theory is founded on a category called the A 1 homotopy category ().Simply put, the A 1 homotopy category, or rather the canonical functor (), is the universal functor from the category of smooth -schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line A 1 becomes contractible.

  6. Homotopy category - Wikipedia

    en.wikipedia.org/wiki/Homotopy_category

    The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.

  7. Category:Homotopy theory - Wikipedia

    en.wikipedia.org/wiki/Category:Homotopy_theory

    In algebraic topology, homotopy theory is the study of homotopy groups; and more generally of the category of topological spaces and homotopy classes of continuous mappings. At an intuitive level, a homotopy class is a connected component of a function space. The actual definition uses paths of functions.

  8. Bott periodicity theorem - Wikipedia

    en.wikipedia.org/wiki/Bott_periodicity_theorem

    The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation as many times as one wished. The stable theory was still hard to compute with, in ...

  9. Homotopy analysis method - Wikipedia

    en.wikipedia.org/wiki/Homotopy_analysis_method

    The animation represents one possible homotopy. The homotopy analysis method ( HAM ) is a semi-analytical technique to solve nonlinear ordinary / partial differential equations . The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems.