Re: Internalizing n-cells to (n-1)-cells

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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
>
> --
> L.G. Meredith
> Managing Partner
> Biosimilarity LLC
> 7329 39th Ave SW
> Seattle, WA 98136
>
> +1 206.650.3740
>
> http://biosimilarity.blogspot.com
>
>
>
>

-- 



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