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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2014
   • (edited Feb 1st 2014)
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   Author: Urs
   Format: MarkdownItexJim Stasheff asked me to give a list of examples of applications of [[Kan extension]] in [[physics]]. Since this shouldn't be hidden away in private email, I have started a section _[Kan extension -- Examples -- Kan extension in physics](http://ncatlab.org/nlab/show/Kan+extension#ExamplesKanExtensionsInPhysics)_ There will be many examples, two came immediately to mind, and so for the moment I have added there the following, to be expanded: *** We list here some occurences of Kan extensions in [[physics]]. Notice that since, by the above discussion, Kan extensions are ubiquitous in [[category theory]] and are essentially equivalent to other standard [[universal constructions]] such as notably [[colimit|co]]/[[limits]], to the extenent that there is a relation between [[higher category theory and physics|category theory and physics]] at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly. * In [[extended quantum field theory]] on open and closed manifolds, usually the theory "in the bulk" (on closed manifolds) is induced by "extending" that "[[boundary field theory|on the boundary]]", and in good cases this extension is explicitly a ([[homotopy Kan extension|homotopy]])-Kan extension. This is the case notably for [[2d TQFT]] in the form of [[TCFT]] ([Costello 04](TCFT#Costello04)), see at _[TCFT -- Classification](TCFT#Classification)_ for details. * When [[path integral]] [[quantization]] is formalized in terms of [[fiber integration in generalized cohomology]] (as surveyed at _[[motivic quantization]]) then the push-forward step, hence the path integral itself, is given by left [[homotopy Kan extension]] of [[parameterized spectrum|parameterized spectra]]. For explicit details see ([[schreiber:master thesis Nuiten|Nuiten 13, section 4.1]]), also ([[schreiber:Homotopy-type semantics for quantization|Schreiber 14, section 6.2]]). By example 6.3 there a special case of this is are the integration formulas via Kan extension in ([[Ambidexterity in K(n)-Local Stable Homotopy Theory|Hopkins-Lurie 14, section 4]]).

   Jim Stasheff asked me to give a list of examples of applications of Kan extension in physics.

   Since this shouldn’t be hidden away in private email, I have started a section Kan extension – Examples – Kan extension in physics

   There will be many examples, two came immediately to mind, and so for the moment I have added there the following, to be expanded:

   ---

   We list here some occurences of Kan extensions in physics.

   Notice that since, by the above discussion, Kan extensions are ubiquitous in category theory and are essentially equivalent to other standard universal constructions such as notably co/limits, to the extenent that there is a relation between category theory and physics at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly.

   • In extended quantum field theory on open and closed manifolds, usually the theory “in the bulk” (on closed manifolds) is induced by “extending” that “on the boundary”, and in good cases this extension is explicitly a (homotopy)-Kan extension. This is the case notably for 2d TQFT in the form of TCFT (Costello 04), see at TCFT – Classification for details.

   • When path integral quantization is formalized in terms of fiber integration in generalized cohomology (as surveyed at _motivic quantization) then the push-forward step, hence the path integral itself, is given by left homotopy Kan extension of parameterized spectra. For explicit details see (Nuiten 13, section 4.1), also (Schreiber 14, section 6.2). By example 6.3 there a special case of this is are the integration formulas via Kan extension in (Hopkins-Lurie 14, section 4).