In mathematics, the Yoneda lemma is a fundamental result in category theory.[1] It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
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In mathematics, the Yonedalemma is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object...
Tokyo in 1990, he moved to Tokyo Denki University. The Yonedalemma in category theory and the Yoneda product in homological algebra are named after him....
rise to a natural transformation Hom(–, f) : Hom(–, B) → Hom(–, B′) Yoneda'slemma implies that every natural transformation between Hom functors is of...
groupoids. Showing this 2-functor is a sheaf is the content of the 2-Yonedalemma. Using the Grothendieck construction, there is an associated category...
are completely known and easy to describe; this is the content of the Yonedalemma. Saunders Mac Lane, one of the founders of category theory, is said to...
can be addressed for the graph example and related examples via the YonedaLemma as described in the Further examples section below, but this then ceases...
the homset is understood to be in the opposite category Δop.) By the Yonedalemma, the n-simplices of a simplicial set X stand in 1–1 correspondence with...
D and as morphisms the natural transformations of such functors. The Yonedalemma is one of the most famous basic results of category theory; it describes...
{\displaystyle C} in a functor category that was mentioned earlier uses the Yonedalemma as its main tool. For every object X {\displaystyle X} of C {\displaystyle...
the identity matrix. This fact can be understood as an instance of the Yonedalemma applied to the category of matrices. The first type of row operation...
1, pp. 31–32. Lang 2002, ch. XVI.1. Roman (2005), Th. 14.3. See also Yonedalemma. Rudin 1991, p.3. Schaefer & Wolff 1999, pp. 204–205. Bourbaki 2004,...
include the use of classifying spaces and universal properties, use of the Yonedalemma, natural transformations between functors, and diagram chasing. When...
bijections on S, and the semigroup operation given by composition. The Yonedalemma provides a full and faithful limit-preserving embedding of any category...
complexes is representable by an Eilenberg–MacLane space, so by the Yonedalemma a cohomology operation of type ( n , q , π , G ) {\displaystyle (n,q...
Laboratories Mihalis Yannakakis Andrew Chi-Chih Yao John Yen Nobuo Yoneda – Yonedalemma, Yoneda product, ALGOL, IFIP WG 2.1 member Edward Yourdon – Structured...
GADTs, bibliography by Simon Peyton Jones Type inference with constraints, bibliography by Simon Peyton Jones Emulating GADTs in Java via the Yonedalemma...
resolution by 1950; Alexander Grothendieck took a sweeping step (invoking the Yonedalemma) that disposed of it—naturally at a cost, that every variety or more...
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