Kan extension

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.