Timeline for Why is Set, and not Rel, so ubiquitous in mathematics?
Current License: CC BY-SA 3.0
35 events
| when toggle format | what | by | license | comment | |
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| Mar 29 at 16:11 | comment | added | Terry Tao | In probability theory we have the notion of a coupling (or joining), which is to probability-preserving maps as relations are to sets. See for instance the Monge and Kantorovich formulations of the optimal transport problem en.wikipedia.org/wiki/Transportation_theory_(mathematics) : the Monge version requires a measure-preserving transport map, whereas the Kantorovich problem merely asks for a transportation plan that need not arise from a map. (But, if there is enough convexity and smoothness present, one can show that the plan must be a map.) | |
| Mar 29 at 15:54 | answer | added | David Ellerman | timeline score: 1 | |
| May 2, 2018 at 21:36 | answer | added | Tim Campion | timeline score: 5 | |
| May 2, 2018 at 21:05 | comment | added | Tim Campion | Basically the same question on math.se -- though this MO question came first. | |
| Mar 25, 2016 at 18:14 | answer | added | Todd Trimble | timeline score: 19 | |
| Aug 28, 2014 at 20:54 | comment | added | Michał Masny | A question from math.se about homomorphic relations: math.stackexchange.com/questions/148715/… | |
| Aug 28, 2014 at 20:00 | comment | added | Jonathan Beardsley | I think the main reason we stick so close to functions is purely historical. Set theory and category theory are natural generalizations of things that already existed - that is, geometry, space and time, as was said above. However, I'd also say that there seems to be a general theme of looking at relation-eqsue objects when one starts working with motives. That is, subsets of $X\times Y$ satisfying some property, rather than functions $X\to Y$. | |
| Aug 19, 2014 at 8:29 | answer | added | Lehs | timeline score: 1 | |
| May 20, 2013 at 6:23 | answer | added | Włodzimierz Holsztyński | timeline score: 1 | |
| May 18, 2013 at 23:34 | answer | added | Włodzimierz Holsztyński | timeline score: 4 | |
| May 18, 2013 at 23:05 | answer | added | Włodzimierz Holsztyński | timeline score: 2 | |
| May 18, 2013 at 21:17 | answer | added | Ronnie Brown | timeline score: 6 | |
| Feb 21, 2013 at 17:05 | vote | accept | Qfwfq | ||
| Feb 8, 2013 at 19:07 | comment | added | Zhen Lin | @Qfwfq: $\textbf{Set}$ is to toposes as $\textbf{Rel}$ is to power allegories. | |
| Feb 8, 2013 at 18:03 | history | edited | Qfwfq | CC BY-SA 3.0 |
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| Feb 8, 2013 at 11:15 | comment | added | Jérôme JEAN-CHARLES | Too short and too partial for an answer: I think that functions are on the technical side whereas relation are on the conceptual side. A function (partial) "is" a partition whereas a relation thought as a bipartite graph is much more complicated. A related (no pun) question is that partial functions should be used instead of function. | |
| Feb 7, 2013 at 21:44 | comment | added | Qfwfq | @Ronnie Brown: maybe a modern way to conceptualize partial-function solutions of ODEs and PDEs is just sheaf theory. If the "sheaf of local solutions" of a differential operator is taken as a subsheaf of the sheaf of germs of continuous/smooth/analytic functions, it allows you to express that the solutions vary continuously/smoothy/analytically etc. | |
| Feb 7, 2013 at 17:24 | comment | added | Ronnie Brown | @Zhen Lin: Partial functions are useful as a web search shows. Also real analysis is nicely expressed in terms of partial functions. Solutions of first order differential equations with a parameter $y$ are often partial functions whose domain varies with $y$. How to express that this partial function varies continuously, smoothly, etc? Where is the functional analysis of partial functions? The change from monoid to category, group to groupoid, is algebraically the change from a total to a partial algebraic structure, and makes a big difference. The step is not easy, and often not welcomed. | |
| Feb 7, 2013 at 16:38 | comment | added | Qfwfq | Another question could be: $\mathsf{Set}$ is to a general topos as $\mathsf{Rel}$ is to what structure? | |
| Feb 7, 2013 at 16:36 | comment | added | Qfwfq | * Typo: "[...] a Yoneda's lemma for Rel Wich is about [...]" should read "[...] a Yoneda's lemma for Rel is about [...]" in the above comment. | |
| Feb 7, 2013 at 16:32 | comment | added | Qfwfq | @DanielMoskovich: Yoneda's lemma states that there's a fully faithful embedding of a category $\mathcal{C}$ into the category $\mathsf{Cat}(\mathcal{C}^{\mathrm{op}},\mathsf{Set})$. Perhaps a Yoneda's lemma for Rel Which is about the relationship between $\mathcal{C}$ and $\mathsf{Cat}(\mathcal{C}^{\mathrm{op}},\mathsf{Rel})$, or maybe something less naif. Also, simplicial sets are objects of $\mathsf{Set}^\Delta$; what about $\mathsf{Rel}^\Delta$? | |
| Feb 7, 2013 at 14:44 | answer | added | Joël | timeline score: 4 | |
| Feb 7, 2013 at 13:31 | comment | added | Pietro Majer | Just a remark: functions, not relations, usually inherit the algebraic structure of the co-domain. The sum and product of relations on $\mathbb{R}$ can be defined too, but looses many algebraic properties. | |
| Feb 7, 2013 at 13:24 | answer | added | Mike Shulman | timeline score: 45 | |
| Feb 7, 2013 at 11:16 | answer | added | Martin Brandenburg | timeline score: 11 | |
| Feb 7, 2013 at 10:45 | answer | added | Dirk | timeline score: 2 | |
| Feb 7, 2013 at 10:41 | comment | added | Eric Wofsey | I like to think of the role of functions as "the arrow of time" of mathematics. They make Set not equivalent to its opposite the way Rel is, and this is the root of an enormous amount of duality symmetry-breaking in category theory. | |
| Feb 7, 2013 at 9:49 | history | edited | Charles Matthews | CC BY-SA 3.0 |
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| Feb 7, 2013 at 8:19 | comment | added | Zhen Lin | Before jumping all the way to relations, one might take an intermediate step and ask why mathematics focuses on total functions instead of partial functions. I think the answer to both questions is the same, however. | |
| Feb 7, 2013 at 8:00 | answer | added | user80744 | timeline score: 10 | |
| Feb 7, 2013 at 4:32 | comment | added | Daniel Moskovich | I don't really understand (3)- what kind of analogue to Yoneda's Lemma would you envision? | |
| Feb 7, 2013 at 1:34 | comment | added | David Roberts♦ | Not really answering the question, but: In the foundational theory SEAR (ncatlab.org/nlab/show/SEAR), relations are a primitive object, and functions are defined as special relations. From the first four axioms (numbered 0,..,3) one arrives at the fact Rel is a power allegory (ncatlab.org/nlab/show/allegory) and the subcategory of sets and functions is a well-pointed topos. There are some remarks at the nLab page on allegories which may be useful. | |
| Feb 7, 2013 at 1:31 | history | made wiki | Post Made Community Wiki by Qfwfq | ||
| Feb 7, 2013 at 1:12 | answer | added | Benjamin Steinberg | timeline score: 8 | |
| Feb 7, 2013 at 0:51 | history | asked | Qfwfq | CC BY-SA 3.0 |