terminology

["Joyal, André" <joyal.andre@uqam.ca>]

Micheal Batanin wrote

>If we follow the principle "foo = 1 foo" and want to agree
>with historical low dimensional terminology we should call categories
>2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.

A few thoughts about terminology.

Categories are tradidionally named according to the nature
of their objects, not the nature of their morphisms. 
We say "the category of sets" not "the category of functions". 
This convention is not respected in the case where the category 
has only one object: we call it a monoid, not because it is a 
mono-object category (maybe we should) but because it has only one 
binary operation in contrast with a ring. 
Like monoids, operads are collections of abstract operations
closed under composition. Classical operads have
only one object, one color. But multi-colored operads
are often called muti-categories, especially when they are big. 

A set is a discrete homotopy type, a 0-type.
This why I like to give the category of sets rank 0.
I like to denote the quasi-category of n-types by U[n].


Best,
André


-------- Message d'origine--------
De: categories@mta.ca de la part de Michael Batanin
Date: jeu. 13/05/2010 19:09
À: Toby Bartels
Objet : categories: Re: bilax_monoidal_functors?=
 

   >> Should we shift the
>> numbers and call category a 3-category?
>
> No, but it seems to me that you are doing something very much like this.

Not at all. It may be was not a good example. A better example would be
categories. If we follow the principle "foo = 1 foo" and want to agree
with historical low dimensional terminology we should call categories
2-sets. Set = 1 Set. So, categories = 2 Set. Nobody will do it I guess.
There are many other examples like stack, gerbes and so on. I agree with
Mike Shulman that this is a byproduct of categorification. But we can
survive with it.


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