Timeline for The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me)
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12 events
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| Jan 23, 2018 at 8:58 | comment | added | Yemon Choi | @DisappointedCategoricien Since you ask, no I don't think that was Cantor's original motivation. Look up his work on Fourier series and sets of uniqueness. Of course this is not an argument for or against the 'australian way' as you put it | |
| Jan 14, 2018 at 16:05 | comment | added | Disappointed Categoricien | I'm still thinking, from time to time, to this line: "History shows that the 'australian way' (whatever this might mean) is not the way to go". Can you explain why? The argument that category theory was born to explain [insert part of mathematics] seems quite weak in order to delegitimate the study of category theory per se. Wasn't set theory born because Cantor wanted to count algebraic VS transcendental numbers? | |
| Dec 26, 2017 at 13:37 | comment | added | user13113 | @DisappointedCategoricien: ... and that's why exploiting the homotopy theory of spaces (in whatever incarnation you want to view it) as a basic building block has been so effective for developing higher category theory, since the topologists already figured out how that works half a century ago. (and the process of disentangling the formal theory of equivalence from the topological application is slow-going) | |
| Dec 26, 2017 at 13:34 | comment | added | user13113 | @DisappointedCategoricien: Actually I was going to comment about natural isomorphisms and/or equivalences of categories. But that's the lens through which I see the word "homotopy" -- IMO, "abstract homotopy theory" is roughly a synonym for "formal theory of equivalences", by which I mean the more refined notion where we are interested in the equivalences (up to equivalence), not merely the proposition "is equivalent to". | |
| Dec 26, 2017 at 13:22 | comment | added | Disappointed Categoricien | And yet mathematics exhibits, at least to my eye, this curious behaviour for which form is substance. You cannot separate the thing you define from the way in which you define it; every encoding loses/gains something over the others, and thus is a different thing. (I'm waiting for someone to comment "but why care about homotopic encodings?") | |
| Dec 26, 2017 at 13:17 | comment | added | user13113 | @DisappointedCategoricien: How could they not know that "function" means "an arrow from $X$ to $Y$ in a well-pointed topos with NNO and choice?" Of course, I know you probably meant "a particular kind of subset of $X \times Y$ a Cartesian product", or maybe you meant "a term of type $X \to Y$" instead, but I hope my point is made that the definition is merely an encoding of the notion being defined. | |
| Dec 26, 2017 at 12:50 | comment | added | Disappointed Categoricien | That's precisely why I wanted to learn mathematics; I suspect we're losing the point, and it's also my fault, but I want to keep clear that I don't want to abandon mathematics; I want to avoid my mathematics to turn into an arid gloss | |
| Dec 26, 2017 at 12:40 | comment | added | Bernie | @DisappointedCategoricien the very notions of categories and functors, natural transformations were developped to formalise constructions in linear algebra, algebraic topology,... abelian categories were developped as a suitable framework for homological algebra. Kan extensions as generalisations of total derived functors (I know you don't like the word 'derived', but the notion of a derived functor is very useful in real maths). Do I need to say more? In short: the people who formed category theory were real mathematicians, not philosophers. | |
| Dec 26, 2017 at 12:35 | comment | added | Disappointed Categoricien | "History shows that the 'australian way' (whatever this might mean) is not the way to go" please, expand this interpretation of history. "I don't know whether you are joking" of course I am, in the specific; but overall, I'm damn serious. Several people engaged in philosophy of science do not know the definition of a function, and they claim to dig the very foundation of things. It's a nonsense, and not ofthe abstract kind. | |
| Dec 26, 2017 at 12:31 | review | First posts | |||
| Dec 26, 2017 at 12:50 | |||||
| S Dec 26, 2017 at 12:26 | history | answered | Bernie | CC BY-SA 3.0 | |
| S Dec 26, 2017 at 12:26 | history | made wiki | Post Made Community Wiki by Bernie |