Meaning of Kan extension | Babel Free

Definitions

A construct that generalizes the notion of extending a function's domain of definition.

Equivalents

Examples

“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, f∗ admits both a left adjoint f_! and a right adjoint f_∗, given by left and right Kan extension along the map SingY→SingX, respectively. Note⟳ that this is left and right Kan extension in the ∞-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.”

“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.”

CEFR level