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

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

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...

infinity category ∞-groupoid Higher category theory Globular set Mackey functor (∞, n)-category Homotopy coherent nerve Localization of an ∞-category Lurie...

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

Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the...

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

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

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...

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

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

enough to show that the pentagon and triangle identities hold. Coherency (homotopy theory) Monoidal category Symmetric monoidal category Mac Lane 1998,...

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...

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

In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts...

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

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

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...

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

In higher degrees the coherency conditions give many different terms. We can arrange the right hand side to be a chain homotopy given by m n {\displaystyle...

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

topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure...

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...

from the cotangent complex to ΩB/A called the augmentation map. In the homotopy category of simplicial A-algebras (or of simplicial ringed topoi), this...

Gerbes". Retrieved 2007-05-20. Homotopy theory of presheaves of simplicial groupoids, Zhi-Ming Luo Twisted K-theory and K-theory of bundle gerbes Twisted Bundles...

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

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...

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...

in algebraic K-theory, motivic homotopy theory, and motivic integration. This theory, Daniel Quillen's work, and Grothendieck's theory of Chern classes...