Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M. Kan, who constructed certain (Kan) extensions using limits in 1960.
An early use of (what is now known as) a Kan extension from 1956 was in homological algebra to compute derived functors.
In Categories for the Working Mathematician Saunders Mac Lane titled a section "All Concepts Are Kan Extensions", and went on to write that
The notion of Kan extensions subsumes all the other fundamental concepts of category theory.
Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on constrained optimization.
Kanextensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits...
formulation of the discovery of adjoint functors, which dates from 1958. The Kanextension is one of the broadest descriptions of a useful general class of adjunctions...
Lehner, Marina (Adviser: Emily, Riehl) (2014). “All Concepts are KanExtensions” KanExtensions as the Most Universal of the Universal Constructions (PDF) (cenior...
new types. Mathematics portal Anafunctor Profunctor Functor category Kanextension Pseudofunctor Mac Lane, Saunders (1971), Categories for the Working...
category theory in the Julia language CQL, a query language based on Kanextensions Companies: Conexus AI, a data integration company Mascots: Gremlin-Morgoth...
species Exact functor Derived functor Dominant functor Enriched functor Kanextension of a functor Hom functor Product (category theory) Equaliser (mathematics)...
algebraic geometry (scheme theory). Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and...
f. Kan complex A Kan complex is a fibrant object in the category of simplicial sets. Kanextension 1. Given a category C, the left Kanextension functor...
study in category theory. Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition. André Joyal showed...
categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories...
Composition of Kleisli arrows can be expressed succinctly by means of the extension operator (–)# : Hom(X, TY) → Hom(TX, TY). Given a monad ⟨T, η, μ⟩ over...
the weak Kan condition. Delta set Dendroidal set, a generalization of simplicial set Simplicial presheaf Quasi-category Kan complex Dold–Kan correspondence...
Reflective subcategory Exact category, a full subcategory closed under extensions. Jaap van Oosten. "Basic category theory" (PDF). Freyd, Peter (1991)....
closed under kernels of epimorphisms, cokernels of monomorphisms, and extensions. Note that P. Gabriel used the term thick subcategory to describe what...
hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these...
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