Kan extension

English

Etymology

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

Noun

1. Template:Lb 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, Template:W, page 34,

     Moreover, $f^{\text{∗}}$ admits both a left adjoint $f_{\text{!}}$ and a right adjoint $f_{\text{∗}}$, given by left and right Kan extension along the map $\text{Sing}{Y}^{\circ p}\to \text{Sing}{X}^{\circ p}$, respectively. Note that this is left and right Kan extension in the $\mathrm{\infty }$-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.

   • Template:Quote-book

Translations

• French: Template:T