You mention the exponential object, so I am not sure if you already know the following suggested answer. Monoidal closed structures are defined on certain kinds of cubical omega-groupoids with connections in joint papers 48. (with P.J. HIGGINS), ``Tensor products and homotopies for $\omega$-groupoids and crossed complexes'', {\em J. Pure Appl. Alg.} 47 (1987) 1-33. and for categories by an analogous process in 116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118. A kind of "decollage", dropping dimension by 1, is given by taking the path object PC of a cubical object C, and this essentially drops each cell by 1, as you ask. Cubical objects are convenient for this exponential method for the usual reason, that I^m x I^n \cong I^{m+n}. Equivalences with other kinds of structures then enable the translation, at least in principle. Thierry Coquand has used cubical methods in his work on homotopy type theory. Best wishes Ronnie PS While I am writing I might as well mention http://education.lms.ac.uk/2014/12/alexander-grothendieck-some-recollections... R On 28/12/2014 23:36, Mike Stay wrote:
---------- Forwarded message ---------- From: Meredith Gregory <lgreg.meredith@gmail.com> Date: Sun, Dec 28, 2014 at 1:01 PM Subject: Fwd: Internalizing n-cells to (n-1)-cells To: Mike Stay <metaweta@gmail.com>
Dear Mike,
i'm writing to ask a question about higher categories motivated from the computer science perspective. A colleague and i have been looking at an analogue of the internalization of morphisms typically associated with Currying. In our setting we're modeling rewrites in various calculi as 2-morphisms, but to prevent rewrites from happening too freely we have to reify certain contexts as 1-morphisms to mark which rewrites are permitted. Essentially, it's a kind of internalization process and is closely connected with work by Leifer, Milner, and Sewell. Now, though, i'm wondering if there has been a more general study of internalization operators taking n-cells to (n-1)-cells. Is there essentially only one kind of internalization process generalizing the exponential object case? Does anyone have any references?
Best wishes,
--greg
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