What does "wedge" mean?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

A wedge refers to the shape of this object:

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.