Timeline for Why does mathematics seem to have a polarity bias?
Current License: CC BY-SA 4.0
32 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 30 at 11:52 | comment | added | user3840170 | The sine and cosine ought to have their names swapped. Cosine is the more basic operation. | |
| Mar 18 at 12:40 | comment | added | Stabilo | Take $x$. Subtract and add $1$ and then multiply. I am still very confused by the fact that it gives $x^2-1$ and not $x^2$. | |
| Mar 17 at 23:55 | answer | added | dan l | timeline score: 3 | |
| Jan 9 at 23:10 | comment | added | Jeff Strom | I have pondered the fact that a group object is one for which the morphisms into it have a group structure, while a cogroup object is one for which the morphisms out of it have a group structure. Both are defined in terms of groups. | |
| Jan 9 at 22:21 | comment | added | Cameron Zwarich | Yes, after thinking about the different examples I am familiar with, I concur that to the extent that this is a real technical phenomenon, it is generally inherited from sets having elements rather than co-elements. | |
| Jan 9 at 22:15 | comment | added | Dan Piponi | One thing I picked up from Vicious Circles by Barwise and Moss is that sometimes standard Set Theory nudges you towards "things" rather than "cothings". For example coalgebras often lead to operations that unpack things out of something (eg. the opposite of a monoid that takes two things and packs them down to one). An obvious way to represent this unpacking is to identify an object with the set of things it unpacks to (or similar schemes). But that leads to non-wellfounded sets. | |
| Jan 9 at 19:08 | comment | added | sds | It seems that cohomology is more common/important than homology. | |
| Jan 8 at 15:12 | answer | added | Will Sawin | timeline score: 17 | |
| Jan 8 at 15:05 | answer | added | Max New | timeline score: 8 | |
| Jan 8 at 14:47 | comment | added | LSpice | @LeeMosher, re, to the contrary, according to the OED, co-ed has been around since 1886. 😄 | |
| Jan 8 at 13:44 | answer | added | Timothy Chow | timeline score: 6 | |
| Jan 8 at 8:14 | comment | added | N. Virgo | I learned a lot about coalgebras (of functors) before I ever thought much about algebras of functors, so I tend to think of alegbras as co-coalgebras. (And presheaves as co-copresheaves.) | |
| Jan 8 at 2:02 | comment | added | Lee Mosher | Back in the olden days, before the co-ification of mathematics, there was just the odd co-thing here and there, even as category theory was slowly growing. There were certainly cocycles and coboundaries and a few which we think of has having the modern meaning of co, but then there were others like cosets which were just tolerated, maybe the oddballs at the party, but then it wasn't a very big party... yet. And then, all of a sudden, it caught on! Everything that hadn't been co-ed already suddenly got co-ed! And here we are. | |
| Jan 7 at 21:49 | history | became hot network question | |||
| Jan 7 at 19:28 | comment | added | Deane Yang | As others have said, there is usually a clear asymmetry. The simplest examples are a vector space versus its dual and a linear map versus its transpose . In fact, many of the examples cited are special cases or variants of this. | |
| Jan 7 at 19:14 | comment | added | LSpice | Another pair of related questions, one already surfaced in the sidebar links: Why is Set, and not Rel, so ubiquitous in mathematics?; Why are inverse images more important than images in mathematics?. | |
| Jan 7 at 19:13 | comment | added | Pietro Majer | It may also be, in some particularly symmetric cases, that in lack of other reasons one is led to prefer the shorter name. E.g. I would rather use "sec" and "tan", than "cosec" and "cotan". | |
| Jan 7 at 19:02 | comment | added | LSpice | I could have sworn we had almost exactly this question, with a different title, about 10 years ago. Do any other old-timers remember it? One related question, that I think isn't the one I mean: Is the dual notion of a presheaf useful?. \\ @PietroMajer, re, surely colimits are more mmon? | |
| Jan 7 at 18:31 | comment | added | Yemon Choi | Without thinking about all the particular examples already mentioned: one source of asymmetry might be that we privilege certain "relations" and call them "functions" (so we have a bias towards working in $\textsf{Set}$ and $\textsf{Grp}$ rather than $\textsf{Set}^{\rm op}$ and $\textsf{Grp}^{\rm op}$) | |
| Jan 7 at 18:23 | comment | added | Daniel Asimov | Because the things without the "co-" were the things discovered first, and therefore were named first. When the dual-type objects arose, they got the co- prefix. | |
| Jan 7 at 18:18 | history | edited | David White | CC BY-SA 4.0 |
I fixed a typo and added a link because I don't think "cotopos" is all that well understood.
|
| Jan 7 at 18:17 | answer | added | David White | timeline score: 25 | |
| Jan 7 at 18:12 | comment | added | Donu Arapura | I suppose that most of us learn how to take the product of two groups, for example, before we learn how to take their coproduct, or free product. Some constructions are just simpler to describe. | |
| Jan 7 at 17:05 | comment | added | David White | Should this be community wiki? It feels like answers will be primarily opinion based, and it's hard to imagine any one answer being objectively "more correct" than any other. | |
| Jan 7 at 16:22 | comment | added | Noah Schweber | @PietroMajer I thought mmons were more colimital? | |
| Jan 7 at 16:10 | comment | added | Pietro Majer | but colimits are more cocommon | |
| Jan 7 at 15:15 | comment | added | Jochen Wengenroth | Mor$(X,\lim Y_i) \cong \lim$ Mor$(X,Y_i)$ and Mor$($co$\lim X_i,Y)\cong \lim$Mor $(X_i,Y)$. Limits win 2-1 against the colimit. | |
| Jan 7 at 14:44 | comment | added | Sean Sanford | I like to think of products as constructions, and coproducts as deconstructions. I think that as humans we have a bias towards wanting to build new things, and only bother to deconstruct things after they have already been built. | |
| Jan 7 at 14:10 | history | edited | Cameron Zwarich | CC BY-SA 4.0 |
added 76 characters in body
|
| Jan 7 at 14:04 | comment | added | JoshuaZ | This may be an unsatisfying answer, but to some extent this is likely due to what order humans discover things. If one side of any pair of a formal duality is going to show up more often in easy contexts, then that side will get named first. So that one gets named as the main object class and the other gets the co associated with it. | |
| Jan 7 at 14:03 | comment | added | Maxime Ramzi | Presentability is clearly also a big asymmetry - from this perspective, one might argue that colimits are "more common" than limits :) But my guess is that this is precisely where the polarity comes from: the category of sets is presentable, and simply not co-presentable. This ultimately comes down to elements (as opposed to co-elements) | |
| Jan 7 at 13:49 | history | asked | Cameron Zwarich | CC BY-SA 4.0 |