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Newsgroups: sci.physics.research
From: t...@ugcs.caltech.edu (Toby Bartels)
Date: 1998/11/29
Subject: Re: Just Categories now
John Baez <b...@galaxy.ucr.edu> wrote: Yeah, that's kind of the point of these definitions; >Toby Bartels <t...@ugcs.caltech.edu> wrote: >>Given a functor U: C -> D, interpret U as a forgetful functor. >Of course, part of the point of this puzzle is that the term >"forgetful" is usually given no precise definition, so here we >are seeking a precise definition. Usually people don't bother >to define "forgetful functor" very precisely - like pornography, >you're just supposed to know a forgetful functor when you see it. it's reasonable to regard U as a forgetful functor if that allows you to think of C as D with something extra. >>Then C is D with extra *structure* if U is surjective on the objects Point taken. >>and, given a pair of objects, injective on the morphisms between them. >However, the more serious reason I'm unhappy is that sometimes C-objects >are D-objects with extra structure but *not every D-object can be made >into a C-object*. In this case U: C -> D is not essentially surjective. >Note that a fully faithful functor always reflects isomorphisms - Yeah, that was fun. >this is a fun little exercise. >Also note that the way I'm setting things up, "extra properties" I know you've been saying before that the two concepts are related, >is a special case of "extra structure". and I thought I found the difference between them, that properties was all about adding extra requirements, while structure was about doing everything except extra requirements. But I see that was wrong, because we don't hesitate to add extra requirements, even to the old structure, when we add new structure. Thus, a ring is not just an Abelian group with a new monoid structure; we add new requirements to the Abelian group, to wit, that the new monoid should distribute over it. Similarly, we can start with a group, add a new group structure, and then require that the two group structures be compatible in that they commute with each other and have the same identity -- but now all we've really done is require that the original group be Abelian! So extra properties really are a case of extra structure. >Now I forget if there is *any* property we should demand of U: C -> D Well, I just chose the generalization of the other concepts I'd come up with. >when D-objects are supposed to be C-objects with "extra stuff"! >Maybe James Dolan will remind me what he told me about this case. If someone can think of something useful to require of U, then we can call that "extra stuff" if we like, but otherwise we can just let that be perfectly general. (There is one concept left: the case where U is faithful but not full. >Fun stuff, eh? But I'm afraid it's drifting rather far afield If that's good enough for you, it's good enough for me. >from physics, except insofar as every mathematical physicist >should spend a little time thinking about "properties" vs >"structure". s.p.r is more fun than s.math.r anyway -- and I say that as a mathematician. (Just like you, I guess -- and yet you're even a moderator here.) -- Toby You must Sign in before you can post messages.
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