Symmetric Group $S_n$ and Artin Braid Group $B_n$: Differences in definition of composition?

Question

I have been starting to look more closely at braid groups recently and I am slightly confused on the notion of "composition" in $S_n$ versus $B_n$. Function composition in $S_n$ is the usual, where $f\circ g$ means to feed $g$ into $f$. Like for cycles in $S_5$: $$(1 \; 2\; 3)\circ(3\; 5)$$ However, for $B_n$, it seems that "braid composition" means to feed the first braid into the second as the order would imply (say we're working in $B_4$): $$\sigma_1\circ \sigma_2$$ Is this what is actually going on? Or do I understand this exactly backwards? I have read that if we forget braid twists, there is a surjective group homomorphism $B_n\to S_n$. How do we conserve the different notions of composition (if such a difference actually exists) in this homomorphism? Are we forced to reverse the order of products?

Answer 1

In both $S_n$ and $B_n$, the group operation is essentially composition of functions, and there's no inherent reason $f \circ g$ has to mean "$g$, then $f$" as opposed to "$f$, then $g$". Either convention is fine if it is done consistently.

When reading, you have to pay attention to the author's conventions, but it is not part of the inherent definition of the groups.

When writing, if working with both $S_n$ and $B_n$ in the same text, it would be less confusing to the reader to choose conventions such that the order of composition does not need to be reversed to obtain the homomorphism you indicated in the original post.