March 1990 - Categories - categories.org.au

noncommuting diagrams
by pratt＠cs.stanford.edu 01 Apr '90

01 Apr '90

I'm afraid my diagrams didn't commute very well, someone's mailer is eating the vertical-bar symbol. Here they are with ! in place of vertical bar. A category mailing list should be less rough on diagrams. b c -----------> ---------> ! ---------> ! ! ! b c ! b ! ! __ ! a! @ ------>------> = ! __ ! !a //! !a ! a! //! a! ! // ! ! ! // ! V c V V V V ---------> ---------> -----------> b c b c -----------> ---------> ! ---------> ! ! b c ! ! b ! ! // ! ------>------> @ a! = ! ! !a !// !a ! a! // a! ! -- ! ! ! !// ! V c V V V -- V ---------> ---------> -----------> b c -Vaughan Pratt

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products of n-categories
by Vaughan Pratt 31 Mar '90

31 Mar '90

An n-category is an ordered list of n categories, any two of which (in the given order) form a 2-category. An n-functor is n functors between two n-categories. The categorical product axb of two m-categories a and b is an m-category. I want a product a@b yielding the "obvious" 2m-category (or (m+n)-category when a is an m-category and b an n-category). I've been trying to figure out from papers of Street, Aitchison, Johnson, etc., how this might be accomplished, but thus far I haven't got the picture, their reassuring hints that it's doable notwithstanding. What am I missing? What's the trick? Where should I look? I presume I should be aiming for a tensor product rather than a categorical product. Is there some routine construction of homological algebra or tarot card theory that does the trick? In more detail: if every story has a beginning, a middle, and an end, then the square of a story has 9 parts, the "biggest" of which is a pair (m,m) of middles. For the categorical square axb this is the arrow from (b,b) to (e,e), a morphism or 1-cell. In a@b, (m,m) should be a solid square or 2-cell. I'm assuming a@b will be oriented. Indeed I'm expecting a@b will be 1-isomorphic to but "2-antimorphic" to b@a (the two functors from a@b to b@a will be covariant and contravariant respectively). I also don't mind if this square ends up with more than 9 things, in fact I'm vaguely expecting 11 (we need the two composites at the start and end of the square) but I'll settle for even more if that's what the construction takes. If the answer is 42 I'll go into some other line of work. Writing ab and a*b for the respective compositions of a 2-category (namely horizontal and vertical composition), it would be nice if a@(bc) = ((a@b)c) * (b(a@c)), as suggested by the following picture. b c -----------> ---------> | ---------> | | | b c | b | | __ | a| @ ------>------> = | __ | |a //| |a | a| //| a| | // | | | // | V c V V V V ---------> ---------> -----------> b c Here |a| = 2+1 (i.e. a has 2 0-cells and 1 1-cell), |bc| = 3+3, and |a@(bc)| = 6+16+5 (6 0-cells, 16 1-cells, and 5 2-cells). (Counting all composites, but not counting degenerate cells, i.e. n-cells of dimension less than n, these having already been counted at the lower dimension. For a@(bc), don't just count cells in the picture on the right, which has 10 0-cells, paste first to get down to 6 0-cells.) And (bc)@a should equal (b(c@a)) * ((b@a)c). b c -----------> ---------> | ---------> | | b c | | b | | // | ------>------> @ a| = | | |a |// |a | a| // a| | -- | | | |// | V c V V V -- V ---------> ---------> -----------> b c -Vaughan Pratt

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What are EECS meetings?
by Fred E.J. Linton 29 Mar '90

29 Mar '90

EECS stands for Eastern European Category Seminar. There have been such affairs since 1986. Generally taking place each March, they've been organised by Prof. V.V. Topencharov of the Center for Applied Math. & C.S. in Sofia (P.O. Box 384, Sofia 1000 Bulgaria) and his cohorts (students, assistants, and a small international network of colleagues) since 1986. The venue for these meetings has been a ski lodge at Predela, perhaps 100 km south of Sofia, about half way between Blagoevgrad and Razlog on the saddle road between the Rila mountains (to the north) and the Pirins (to the south). The first three years, meteorological conditions were perfect for skiers. The last two, however, a very warm, very early spring annihilated virtually all snow long before the meeting. Expressions of interest in attending future EECS meetings (probably mid-March 1991 will be the next one) may be addressed to Topencharov at the address above or via the Bulgarian Telex number 865 22575 WMI BG (as this is a _central_ telex machine, be sure to mention Topencharov and his address prominently at the head of your telex message, else it'll not get carried to him) -- or to me using any of the means suggested in the .sig below. -- Fred --- Fred E.J. Linton Wesleyan U. Math. Dept. 649 Sci. Tower Middletown, CT 06457 --- ARPA/Internet: FLINTON(a)eagle.Wesleyan.EDU Bitnet: FLINTON(a)WESLEYAN?.bitnet? from uucp: ...!{research, rutgers, sharkey, uunet}!eagle.Wesleyan.EDU!FLinton on AT&T-Mail: !fejlinton ( ...!attmail!fejlinton ) on MCI-Mail: FEJLINTON or 414-2427 or 0004142427(a)mcimail.com Tel.: + 1 203 776 2210 (home) OR + 1 203 347 9411 x2249 (work) Telex: <USA> + 15 122 3413 FEJLINTON or 650 414 2427 MCI UW ---

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removal from list
by WELLS＠CWRU 28 Mar '90

28 Mar '90

I will be in Oxford during April and May and will not read email sent to CWRU during that time. Please remove me from the category theory mailing list. If I can get an email address at Oxford I will rejoin. --Charles Wells (I'd like this distributed to everyone).

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None
by Fred E.J. Linton 25 Mar '90

25 Mar '90

There are available, in principle, copies of the ABSTRACTS of the first four EECS meetings (1986, 1987, 1988, 1989). I say "in principle" because I have copies, and can Xerox them for anyone who might want them -- but in the face of too many requests, I'll probably have to ask for some sum with which to defray duplication and mailing expenses. -- Fred From: MACC1::NETNORTH 28-MAR-1990 01:58:20.81 To: RROSEBRU CC: Subj: removal from list

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None
by INHB000 12 Mar '90

12 Mar '90

Dear Bob: In the future it might be useful if you could ask for a reaction to such a message so the correction could be sent out at the same time. - Michael ------------------------------------------------------------------------------- Agreed, Bob March 11,1990 Dear Colleague: This letter is to announce the availability of a Bulletin Board for Category Theory on BITNET. The address is CATEGORIES(a)MTA.BITNET. The intention is to operate in a manner similar to the Types Bulletin Board. To join the mailing list, simply send a message to: categories-request(a)mta.bitnet. After confirmation of two-way transmission your name will be added to the list. Please provide a very complete return address. To distinguish messages for forwarding from others of an administrative nature, please mail to Categories ONLY messages which are intended to be forwarded. All other messages should be sent to categories-request(a)mta.bitnet, or to me ( rrosebrugh(a)mta.bitnet ) Anyone may submit (electronically only) any material about Category Theory (and only about Category Theory) for distribution - abstracts, announcements of results or meetings, queries, or full papers. Material will be forwarded as received unless there is evidence that the message received is garbled. The initial committment is that turn-around times will not exceed a couple of days. TeX files are welcome in any flavour - plain, AmSTeX or LaTeX. They should include a Character Code Reference which is available on request. Unfortunately, until we see how well this works, there can be no committment to either authors or subscribers regarding accuracy of TeX files transmitted. TeX preprints will be archived separately, and a catalogue of them will be published irregularly. Macro packages are the responsibility of the author, however an attempt will be made to archive versions of diagram macro packages. I gratefully acknowledge assistance from David Jones at Types who suggested some of the procedures to be used. Electronic addresses to which this is being sent are taken from lists provided by Mike Barr and Max Kelly, and I thank them (and Mike Johnson who transmitted Max's list.) Those seem to be the main points. Looking forward to a useful exchange, Regards, Bob Rosebrugh /TO categories(a)mta.bitnet /BC Mike Barr <INHB(a)MUSICB.MCGILL.CA>, Francis Borceux <FBORCEUX(a)BUCLLN11.BITNET>, Aurelio Carboni <CARBONI(a)IMIUCCA.BITNET>, Peter Freyd <PJF(a)LINC.CIS.UPENN.EDU>, Mike Johnson <munnari!mqcomp.mqcs.mq.oz.au!mike(a)uunet.uu.net>, Paul Johnson <PJOHNSON@WESLEYAN>, Stefano Kasangian <KASAN(a)IMIUCCA.BITNET>, Fred Linton <FLINTON(a)WESLEYAN.BITNET>, Phil Mulry <PHIL%COLGATE.CSNET(a)RELAY.CS.NET>, Susan Niefield <NIEFIELS(a)UNION.BITNET>, Bob Pare <PARE(a)CS.DAL.CA>, Andy Pitts <AP(a)CL.CAM.AC.UK>, Pino Rosolini <MATEMA2(a)IPRUNIV.BITNET>, Dietmar Schumacher <SCHU@ACADIA>, Phil Scott <SCPSG(a)UOTTAWA.BITNET>, Robert Seely <MT79(a)MUSICA.MCGILL.CA>, Ross Street <munnari!mqcomp.mq.oz.au!street(a)uunet.uu.net>, Paul Taylor <PT(a)DOC.IC.AC.UK>, Bob Walters <munnari!maths.su.oz.au!walters_b(a)uunet.uu.net>, Charles Wells <WELLS(a)CWRU.BITNET>, Richard Wood <GRETCHEN(a)CS.DAL.CA> /TX The following message has been sent out to about 100 addresses from Mike's and Max's address lists. On the assumption that anyone I have heard from wishes to be included on the subscriber list, I have included you already. Let's see how it goes. Bob ******************************************************************************** March 13,1990 Dear Colleague: This letter is to announce the availability of a Bulletin Board for Category Theory on NETNORTH. The address is `CATEGORIES(a)MTA.BITNET'. The intention is to operate in a manner similar to the Types Bulletin Board. To join the mailing list, simply send a message to: categories-request(a)mta.bitnet After confirmation of two-way transmission your name will be added to the list. Please provide a very complete return address. To distinguish messages for forwarding from others of an administrative nature, please mail to Categories ONLY messages which are intended to be forwarded. All other messages should be sent to categories-request(a)mta.bitnet, or to me ( rrosebrugh(a)mta.bitnet ) Anyone may submit (electronically only) any material about Category Theory (and only about Category Theory) for distribution - abstracts, announcements of results or meetings, queries, or full papers. Material will be forwarded as received unless there is evidence that the message received is garbled. The initial committment is that turn-around times will not exceed a couple of days. TeX files are welcome in any flavour - plain, AmSTeX or LaTeX. They should include a Character Code Reference which is available on request. Unfortunately, until we see how well this works, there can be no committment to either authors or subscribers regarding accuracy of TeX files transmitted. TeX preprints will be archived separately, and a catalogue of them will be published irregularly. Macro packages are the responsibility of the author, however an attempt will be made to archive versions of diagram macro packages. I gratefully acknowledge assistance from David Jones at Types who suggested some of the procedures to be used. Electronic addresses to which this is being sent are taken from lists provided by Mike Barr and Max Kelly, and I thank them (and Mike Johnson who transmitted Max's list.) Thanks also to the category theorists who made suggestions before this announcement. Those seem to be the main points. Looking forward to a useful exchange, Yours truly, Bob Rosebrugh <RROSEBRUGH(a)MTA.BITNET>

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None
by INHB000 12 Mar '90

12 Mar '90

Dear Bob: Aurelio's contention is quite correct. In fact, the error was corrected in the final version that has been accepted for the new journal. In fact, the only limits that are needed are products (which certainly exist; they are disjoint unions) and the equalizer of a group of automorphisms (which can be shown to exist). The latter is needed to do the dual of the symmetric algebra construction; the group is the symmetric group acting on tensor powers. Michael ------------------------------------------------------------------------------

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