Kan extension

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Kan extension

Wikipedia

Etymology

Named after Jewish and Dutch mathematician Daniel M. Kan (1927–2013), who constructed certain (Kan) extensions using limits in 1960.

Noun

Kan extension (plural Kan extensions)

1. (category theory) A construct that generalizes the notion of extending a function's domain of definition.
   • 2010, Matthew Ando, Andrew J. Blumberg, David Gepner, Twists of K-Theory and TMF, Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and C*-algebras, American Mathematical Society, page 34,

     Moreover, ${\displaystyle f^{\text{∗}}}$ admits both a left adjoint ${\displaystyle f_{\text{!}}}$ and a right adjoint ${\displaystyle f_{\text{∗}}}$, given by left and right Kan extension along the map ${\displaystyle {\text{Sing}}\ Y^{\circ p}\rightarrow {\text{Sing}}\ X^{\circ p}}$, respectively. Note that this is left and right Kan extension in the ${\displaystyle \infty }$-categorical sense, which amounts to homotopy left and right Kan extension on the level of simplicial categories or model categories.

   • 2012, Rolf Hinze, Kan Extensions for Program Optimisation, Or: Art and Dan Explain an Old Trick, Jeremy Gibbons, Pablo Nogueira (editors), Mathematics of Program Construction: 11th International Conference, MPC 2012, Proceedings, Springer, Lecture Notes in Computer Science: 7342, page 336,

     We can specialise Kan extensions to the preorder setting, if we equip a preorder with a monoidal structure: an associative operation that is monotone and that has a neutral element.

   • 2013, Franz Vogler, Derived Manifolds from Functors of Points, Logos Verlag, page 5:

     We are going to introduce the direct and inverse image functor for presheaves as special Kan extensions and show that they behave well with respect to global and local model structures on simplicial model categories.

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