The category of categories as a 3-limit

The category of categories as a 3-limit

[David Leduc]

Hi,

In [1], Mike Shulman explains how one can define:
* the category of magmas as an inserter in the 2-category of (large) categories,
* the category of semigroups as an equifier,
* and so on up to the category of rings.

Can we go further? What is the 2-categorical limit to be used in order
to define the category of small categories?

But since small categories form a 2-category, maybe I should
reformulate my question:  What is the 3-categorical limit to be used
in order to define the 2-category of small categories?

While I am asking... What is the (n+2)-categorical limit to be used in
order to define the (n+1)-category of n-categories? what is the
omega-categorical limit to be used in order to define the
omega-category of omega-categories?

[1] http://mathoverflow.net/questions/9269/category-of-categories-as-a-foundation-of-mathematics


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Re: The category of categories as a 3-limit

[David Leduc]

> http://www.maths.mq.edu.au/~street/Sketch.pdf

Thank you very much but I am afraid I am already stuck at the first sentence.
Why such a diagram in Cat would be a theory?

Sorry,

David


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Re: The category of categories as a 3-limit

[Ross Street]

On 02/12/2011, at 12:17 PM, David Leduc wrote:

>> http://www.maths.mq.edu.au/~street/Sketch.pdf
>
> Thank you very much but I am afraid I am already stuck at the first  
> sentence.
> Why such a diagram in Cat would be a theory?

A (limit) sketch in the sense of Ehresmann is a family of cones on a  
category C.
A cone is a natural transformation pointing right in a triangle whose  
top horizontal
side is

 	X ------> 1

and whose bottom vertex is C. A cone is the special case of what I  
called a theory with
A = X and B = D = 1 (the terminal category).

Now let A be the coproduct (disjoint union) of all the categories X in  
the sketch and let B = D be
the discrete category obtained by adding up all the 1s, one for each  
index of the family.
Take u : A --> B to be the induced map on the coproducts. Take t to be  
the identity. The cones
give a single natural transformation tau using the 2-universal  
property of coproduct.

So a sketch on C is the same as one of my theories with B discrete and  
t an identity.

The extra flexibility of having B not necessarily discrete and tagging  
on the t was aiming
at theories in the sense of

John Isbell [General functorial semantics. I. Amer. J. Math. 94  
(1972), 535–596; MR0396718 (53 #580)].

Ross



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