* The category of categories as a 3-limit
@ 2011-11-28 12:04 David Leduc
[not found] ` <C6BF5588-3FD1-4CB3-9944-A86CE2052B0E@mq.edu.au>
0 siblings, 1 reply; 3+ messages in thread
From: David Leduc @ 2011-11-28 12:04 UTC (permalink / raw)
To: categories
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
[not found] ` <C6BF5588-3FD1-4CB3-9944-A86CE2052B0E@mq.edu.au>
@ 2011-12-02 1:17 ` David Leduc
[not found] ` <CAEqE=b2P=a1uzS7z9M+Rz2eHC34ogYeqhQ0yo_AdLqgPvEpFjw@mail.gmail.com>
1 sibling, 0 replies; 3+ messages in thread
From: David Leduc @ 2011-12-02 1:17 UTC (permalink / raw)
To: Ross Street; +Cc: categories
> 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
[not found] ` <CAEqE=b2P=a1uzS7z9M+Rz2eHC34ogYeqhQ0yo_AdLqgPvEpFjw@mail.gmail.com>
@ 2011-12-02 2:17 ` Ross Street
0 siblings, 0 replies; 3+ messages in thread
From: Ross Street @ 2011-12-02 2:17 UTC (permalink / raw)
To: David Leduc; +Cc: categories
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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2011-11-28 12:04 The category of categories as a 3-limit David Leduc
[not found] ` <C6BF5588-3FD1-4CB3-9944-A86CE2052B0E@mq.edu.au>
2011-12-02 1:17 ` David Leduc
[not found] ` <CAEqE=b2P=a1uzS7z9M+Rz2eHC34ogYeqhQ0yo_AdLqgPvEpFjw@mail.gmail.com>
2011-12-02 2:17 ` Ross Street
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