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nLab > Latest Changes: universal construction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2010
    • (edited Feb 2nd 2010)

    at universal construction there used to be a little chat between me and Toby along the lines of "would be nice if somebody added something here".

    Since I think by now we have plenty of pointers to this entry, I thought it should present itself in a slighly more decent fashion. So I removed our chat and left a stubby but honest entry.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 2nd 2010
    • (edited Feb 2nd 2010)
    Incidentally, universal constructions precede category theory. They were another "proto-categorical" idea used by Bourbaki.
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2010

    Can you make this historical claim precise ? I think that one of the simplest universal construction, the general definition of categorical product is by MacLane from around 1954-1955, i.e. about 7 years after the birth of category theory.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 2nd 2010
    • (edited Feb 2nd 2010)
    Chapter 1 of Bourbaki's algebra book was published in 1942 and includes a proof of the universal property of a free group. I was unable to check this directly, but some circumstantial evidence is available here http://books.google.com/books?id=1V4-kdN6aGkC&lpg=PA126&ots=ecTyYLYpS3&dq=%22universal%20property%22%20bourbaki&pg=PA235#v=onepage&q=universal&f=false in the footnote. I do not have access to the first edition, but if you do, you should be able to check. It's definitely in the later editions, but those were published after the definition was formally given.

    See also:
    Categories Work pg. 76
    Categories Work pg. 103, where he cites Bourbaki from 1948 with a definition of a universal construction. However, Mac Lane notes that the characterization isn't as sharp as it could be because Bourbaki eschewed categorical language.

    Also, according to Categories Work, Bourbaki came up with the first formulation of the adjoint functor theorem.
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2010

    Well, of course some theorems on adjoint functors appeared much before, say the Frobenius reciprocity is a theorem saying that the induction in representation theory is a left adjoint of the restriction functor, and this theorem is from 19th century. But these theorems are in terms of bijections between the homs, and the naturalness is not clearly expressed. Now what is new with Bourbaki what is not in Frobenius and alike theorems ? Also the universal property of a free group is well known and widely used in a form of given a homomorphism of groups by "generators and relations" which is from around 1870 (look at Magnus' historical account of combinatorial group theory). Maybe Bourbaki understood better the importance, and called the property universal. In any case, while the adjointness and free constructions were used in more or less the same way in a number of examples before, I think more stunning step is categorical expression of products which were defined before MacLane 1955 paper just by case-by-case definitions simulating extending the definition of cartesian product of sets to sets with structure (e.g. Tohonov product of topological spaces).

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 2nd 2010
    The universal properties stated in Bourbaki are actual universal properties. "Every map factors through.. _canonically_" (here meaning uniquely). They actually describe universal properties in terms of the existence of a unique or canonical maps. Mac Lane notes that the adjoint functor theorem was first stated and proved by Bourbaki in terms of their universal construction formalism. That is, their definition of a universal construction had the property that everything had to commute with representables, so the adjoint functor theorem was actually built into their definition of a universal construction.

    The statement of the universal property of the free group is something along the lines of "If f:S->G is a function from a set into a group, there exists one and only one homomorphism \hat{f}:F(S)->G extending f."
    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 2nd 2010

    While specific applications of universal properties (such as Tychonov's product of topological spaces and defining homomorphisms by their actions on generators) predate Bourbaki, and while the category-theoretic approach is both cleaner and more general than Bourbaki's, Bourbaki was still the pioneer who first (well, first as far as I know, in any case before Mac Lane & Eilenberg 1945) defined a general notion of universal construction that includes the examples for concrete categories (as we now understand them) and agrees with the category-theoretic notion in those cases where it applies.

    Bourbaki's theory is in Book I (Set theory), Chapter 4 (Structures), Section 3 (Universal mappings).

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 2nd 2010

    Or try this online: http://mathdoc.emath.fr/archives-bourbaki/PDF/041_iecnr_050.pdf

    I don't know when that is from, but it's the first item listed in the Bourbaki archive's redactions of Set Theory.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeFeb 2nd 2010
    • (edited Feb 2nd 2010)

    The statement of the universal property of the free group is something along the lines of "If f:S->G is a function from a set into a group, there exists one and only one homomorphism \hat{f}:F(S)->G extending f."

    This was well known before in combinatorial group theory.

    @toby: is it known who was the one who put that into Bourbaki ? It is sometimes known for major things in Bourbaki who were the main people behind a particular chapter or inovation. Is this the first time that the terminology "universal" has been used in any similar context ?

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 2nd 2010

    I'm afraid I don't know the answer to either of your questions. I really only know what is written in the first edition (since I've seen it) and when it was published (since anybody can look that up). I only found the Bourbaki archives today, but maybe they have that sort of information there.

    • CommentRowNumber11.
    • CommentAuthorDexter Chua
    • CommentTimeSep 2nd 2016

    Added an ideas section and a few worked-out examples.

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