in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or...

category theory, a collection of conditions requiring that various compositions of elementary morphisms are equal Coherency (homotopy theory) in homotopy theory...

logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation...

General abstract nonsense Categorification Coherency (homotopy theory) Lurie, Jacob. Higher Topos Theory (PDF). MIT. p. 4. Baez & Dolan 1998, p. 6 Hirschowitz...

Chern classes and to K-theory since the work of Grothendieck, and so Quillen was led to define the K-theory of R as the homotopy groups of BGL(R)+. Not...

onto its image. coherent homotopy coherency See coherency (homotopy theory) cohomotopy group For a based space X, the set of homotopy classes [ X , S...

polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory. Shape theory was invented...

that the functor from the stable homotopy category (the homotopy category of spectra) to generalized homology theories on CW-pairs is not an equivalence...

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations...

definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972....

Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project. Higher Topos Theory covers two related topics: ∞-categories...

contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the K-theory over contractible spaces...

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group...

Springer Robinson, Alan (1983), "Derived tensor products in stable homotopy theory", Topology, 22 (1): 1–18, doi:10.1016/0040-9383(83)90042-3, MR 0682056...

Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It...

left adjoint to the loop space functor in the homotopy category, an important fact in homotopy theory. Stone–Čech compactification. Let KHaus be the...

map 0 to 0. Mathematics portal History of topos theory Homotopy hypothesis Intuitionistic type theory ∞-topos Quasitopos Geometric logic Generalized space...

enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral...

D-modules was of a theory expressed in those terms. A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra...

acted upon disjointly under a given group action Regular homotopy Regular isotopy in knot theory, the equivalence relation of link diagrams that is generated...

motivic homotopy theory", Mathematische Annalen, 341 (3): 651–675, doi:10.1007/s00208-008-0208-5, MR 2399164 Suslin, Andrei (1983), "On the K-theory of algebraically...

basepoint-preserving homotopies between loops, with these morphisms identified if they are themselves homotopic. Weak inverses can always be assigned coherently: one...

is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based...

three-manifold induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal...

research notes. The topic of the work is a generalized homotopy theory using higher category theory. The word "stacks" in the title refers to what are nowadays...

the theory of operads in algebra and algebraic topology, an A∞-operad is a parameter space for a multiplication map that is homotopy coherently associative...

are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an...

projective modules). Since the algebraic K-theory of a ring is defined (in Quillen's approach) in terms of the homotopy groups of (roughly) the classifying space...

ISSN 0271-4132. LCCN 96-37049. MR 1436913. Retrieved 2021-12-08. George Whitehead; Fifty years of homotopy theory Haynes Miller; The origin of sheaf theory...

holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors...