[This sender failed our fraud detection checks and may not be who they appear to be. Learn about spoofing at http://aka.ms/LearnAboutSpoofing] Fred was a great guy and I liked him. His love of language was manifest in his writing as well as in his oral communication. It was from his papers that I learned the words “behoove” and “antepenultimate”. In one paper he explained, in a precise and understandable way, the theory of triples (aka monads), their algebras and how they relate to adjoints, … all in a single sentence! I first met him when I was a graduate student at McGill in 1968. He was sleeping on the couch in the lounge with a sign on his chest saying “Wake me at 2:30”. Someone asked, “Who’s this?”. The reply: “The colloquium speaker.” He spoke (at 2:35) on enriched theories and algebras. One thing he said has stuck with me. The base category was a non-symmetric monoidal category and the algebras were contravariant functors into it (if my memory serves me correctly). He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category. He was a true original and a great loss to our community. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Preparing a stir fry for last Sunday's lunch is when I received Marta's message saying we had lost Fred. I realized that our family was introduced to this wonderful style of cooking by Fred Linton in his big New Haven house. I was on sabbatical 1976-7 at Wesleyan University in Middletown (CT) to work with Fred. The year was a hard one for Fred but he was a great host. We rented a nice University-owned house in Middletown, with bushy slope one side and a disused car park the other, so our two boys could toboggan or ride their tag-sale-acquired tricycle, according to season. Our older boy started school that year. Fred had found a microscope at a tag sale which is a gift my boys often talk about. He also found books, fruits, vegies and other things he rightly thought would interest us. One day he took me to a liquor store in New Haven where he knew they had Australian beer. They had \Huge{cans} of Fosters. The shopkeeper, not knowing at first my nationality, told me that, in Australia, they bought these by the 6 pack and got through many of them at their barbies! Fosters was a Melbourne company, uncommon in Sydney in those days; I gave him the benefit of the doubt. I may have met Fred in November 1968 at a MidWest Category Seminar in Urbana, Illinois; there were so many of the category theorists who had just been names on papers to me before that. However, we certainly met at the Summer of '69 Bowdoin College, Maine, gathering. Fred gave a series of lectures on enriching the theory of triples (monads), especially over bases that were either closed or monoidal but not necessarily both. I had been thinking about the closed monoidal case over the previous year so it was great to find a mind who also found it worth doing that even more generally. The MR review Fred wrote of my ``formal theory of monads'' paper was noticed (before me I think) by our Head (Fred Chong) at the time. This was responsible in large part for my application for a full year's sabbatical at Middletown being smiled upon. Yes, Fred had a way with words! We category theorists have enjoyed Fred's company and discussions in many meetings around the world. We all will miss his valuable contributions. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[This sender failed our fraud detection checks and may not be who they appear to be. Learn about spoofing at http://aka.ms/LearnAboutSpoofing] Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 14/09/2017 17:58, Peter Selinger wrote:
Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
Indeed: Segal's (1974) Gamma-spaces are contravariant functors on the opposite of the category of finite pointed sets. Joachim. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 14/09/2017 17:58, Peter Selinger wrote:
Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
Indeed: Segal's (1974) Gamma-spaces are contravariant functors on the opposite of the category of finite pointed sets. Joachim.
On 15 Sep 2017, at 1:58 AM, Peter Selinger <selinger@mathstat.dal.ca> wrote:
Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek.
Yes, typical of Fred's word-game humour. ==Ross
Well, one could look to the various power set functors for guidance there. When you refer to the contravariant power set functor, it may be contrary but it's reliably so and people know what you mean. But "the" covariant power set functor??? What's that??? It's like Trump: one day he's agreeing with the Elephants, the next with the Scientific Consensus. If there are more than two covariant power set functors maybe we'll see yet another side of Trump, perhaps a side from another dimension. Give me good old reliable contravariant.???? Mind the pence and the pounding will take care of itself. Oh but wait, there's profunctors,?? ??: A' x B --> V.???? Which way do /they /go? The Elephant <https://www.amazon.com/Sketches-Elephant-Theory-Compendium-Oxford/dp/019852496X/ref=sr_1_1?ie=UTF8&qid=1505521975&sr=8-1&keywords=sketches+of+an+elephant> (kindle edition $4.99 <https://www.amazon.com/Draw-Animals-Step-Step-Elephants-ebook/dp/B007WKEMBE/ref=sr_1_4?ie=UTF8&qid=1505522093&sr=8-4>) says they go from A to B.?? The Consensus, being a bunch of Deniers, says ("bunch" is singular) they go from B to A. Who to believe??? It's enough to make anyone lose their composure. (Oh but wait, there's left Kan extensions.) Vaughan PS?? How many covariant power set functors according to the Elephant??? Does every element of ?? get one? On 09/14/17 8:58 AM, Peter Selinger wrote:
Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category. This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek.
-- Peter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Robert, Peter and all, We often turn covariant functors into contravariant ones: If C is a small category, then the category [C,Set] of covariant set valued functors on C is the topos of presheaves on C^{op}. Recall that the category \Gamma introduced by Graeme Segal is the opposite of the category Fin_\star of finite pointed sets. https://ncatlab.org/nlab/show/Segal%27s+category A Gamma-space was not defined by Segal to be a covariant functor Fin_\star --->Space but as a contravariant functor \Gamma---->Space https://ncatlab.org/nlab/show/Gamma-space -André ________________________________________ From: Peter Selinger [selinger@mathstat.dal.ca] Sent: Thursday, September 14, 2017 11:58 AM To: Categories List Subject: categories: Re: opposite category Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Robert, Peter and all, We often turn covariant functors into contravariant ones: If C is a small category, then the category [C,Set] of covariant set valued functors on C is the topos of presheaves on C^{op}. Recall that the category \Gamma introduced by Graeme Segal is the opposite of the category Fin_\star of finite pointed sets. https://ncatlab.org/nlab/show/Segal%27s+category A Gamma-space was not defined by Segal to be a covariant functor Fin_\star --->Space but as a contravariant functor \Gamma---->Space https://ncatlab.org/nlab/show/Gamma-space -André ________________________________________ From: Peter Selinger [selinger@mathstat.dal.ca] Sent: Thursday, September 14, 2017 11:58 AM To: Categories List Subject: categories: Re: opposite category Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek. -- Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Robert Pare wrote:
He said there may come a time when we have to consider covariant functors as contravariant ones on the opposite category.
This anecdote seems to have prompted a few posts about opposite categories, but I thought the point of the original anecdote was that Fred said that *covariant* functors should be considered as contravariant functors on the opposite category, i.e., that he considered contravariant functors to be the more fundamental concept. An interesting thought, and obviously tongue-in-cheek. -- Peter
participants (7)
-
Joachim Kock -
Joyal, André -
Peter Selinger -
Robert Pare -
Ross Street -
selinger -
Vaughan Pratt