Thursday, September 4, 2008


i've been thinking a bit about these colored brace widgets and i think they are morphisms. No, clearly, they are morphisms; but, what are the types? Could the colors be the types? Or could types be derived from colors? This is the sort of interpretation i am thinking of.

  • {s| e1,...,en |t} |-> e1 *_st ... *_st en : T(s) -> T(t)
  • {s||s} |-> id: T(s) -> T(s)

Note, that independently of whether an interpretation like this can be made to go through, the previous notions raise a very interesting question for infinitary compositions of morphisms. What is a good language for describing such? Can a comprehension notation serve that purpose?

Oh, and the nesting... could these be cells? i'm not sure, yet. To make the cell interpretation work it would seem that cell degree (0-cell, 1-cell, 2-cell, ... ) is not a fixed thing because of phenomena like
  • {s| ..., {u| ..., {s| ... |t}, ... |v}, ... |t}
but maybe i'm not thinking clearly about the structure. Again, though, that raises an interesting question about categorical 'views'. In one category certain structures are morphisms -- while in others the 'same' structure are objects. Does such a view-like notion work for cell degree? Is cell degree a relative role? Can that idea be made to work?

1 comment:

leithaus said...

A categories list participant emailed me with the suggestion that limits might provide a language for infinitary compositions. Limits, however, do not always exist, while the infinitary compositions will. They capture different notions.