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In Allen Hatcher's Algebraic Topology, $X\vee Y$ means the "wedge sum" of two (topological) spaces $X$ and $Y$. However, in $\LaTeX$, \wedge is the notation for $\wedge$, while $\vee$ is wirtten \vee. Where does this inconsistency come from?

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The inconsistency comes from different mathematicians doing different branches of mathematics.

In exterior algebra, a wedge product is written with an upward-pointing wedge, like this: $\alpha \wedge \beta.$

The wedge sum in topology is written with a downward-pointing wedge.

Only one of those symbols can be called \wedge in LaTeX. Evidently the terminology from exterior algebra won in this case.

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  • $\begingroup$ I truly think this is not correct, as in, $\wedge$ in latex is not called \wedge because of wedge product or anything mathematical at all. $\endgroup$ – ThoAppelsin Nov 17 '18 at 12:13
  • $\begingroup$ @ThoAppelsin I suppose if you look for it, you might find evidence for that claim and it would be the basis for a good answer. $\endgroup$ – David K Nov 17 '18 at 15:27
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A wedge refers to the shape of this object:

enter image description here

It doesn't matter if it's facing up or down, it's still a wedge. But since there was a need for two commands to produce $\wedge$ and $\vee$, and since $\vee$ conveniently looks like the letter V, it was probably decided that one should get the command "vee" and the other "wedge" (instead of something like "upwedge" and "downwedge", which would be harder to remember). But both are called "wedge" and it's up to the context to disambiguate.

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  • $\begingroup$ I don't think this answer addresses the real issue here since he's referring to a particular operation called 'wedge product'. Not just the symbol. $\endgroup$ – hjhjhj57 Oct 15 '14 at 7:29
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    $\begingroup$ @Javier Uh, I don't see what more you could want. Both the operations and the symbols are called wedge, and it's only a quirk of LaTeX that they had to have different commands. (And considering what you answered, saying that my answer doesn't address the "real issue" is a bit rich...) $\endgroup$ – Najib Idrissi Oct 15 '14 at 7:32
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    $\begingroup$ It may be a matter of opinion, but I think the fact that there are wedge products in algebra and topology denoted with different symbols ($\wedge$ and $\vee$, resp.) must have something to do with it. $\endgroup$ – hjhjhj57 Oct 15 '14 at 7:35
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I believe the reason for the command for $\wedge$ being \wedge in $\LaTeX$ is not related to any mathematical definition, but rather related to the English dictionary meanings of those words themselves or their homonyms.

Wedge, according to Oxford Dictionaries is the word for the tapered objects that are driven between two other objects to secure or separate them. The object described essentially and usually is a triangular prism. In our world where gravity is ubiquitous, wedges lay on one of their flat edges, which looks like the $\wedge$ symbol, hence, I believe, \wedge is the command in $\LaTeX$ for the symbol. It would be rather weird to call $\vee$ the wedge, since you almost never see a wedge staying like that.

There are many other mathematical symbols in $\LaTeX$ named in a similar fashion, such as $\cup$, $\cap$, $\smile$, $\frown$, and $\diamond$, with their corresponding commands \cup, \cap, \smile, \frown, and \diamond.

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The thing is that 'wedge' may also refer to the wedge product in the exterior algebra (or differential forms), which is also writen as $\wedge$.

I'm unaware of the real reason, you may try on TeX.SE, but coming from a geometrical background I believe this is how it should be, even though some topologists may disagree.

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