* 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
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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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
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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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 3+ messages in thread
end of thread, other threads:[~2011-12-02 2:17 UTC | newest] Thread overview: 3+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 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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