What does the first arrow in an exact sequence represent?

Question

An exact sequence is written $0 \rightarrow X \rightarrow Y \rightarrow 0$, but how is the $\rightarrow$ from $0$ to $X$ defined? $0$ has exactly one element, how can there exist a map to $X$, seeing as $\rightarrow$ is a mapping. Does this imply that $X$ contains a single element as well?

Answer 1

Assuming $X$ is abelian there is only one homomorphism from 0 to X because homomorphisms in groups preserves the identity. It doesn't mean that $X$ is a group with one element at all it just means that the map to $X$ is in most cases not on to. It is only on to if X is the trivial group.

Answer 2

The arrow $0 \rightarrow X$ is a homomorphism from the group with one element $0$ (the set $\{0\}$ with the obvios product) to the group $X$ (not necessarily an abelian group, although "Zero group" or $0$ is additive notation, mostly used for abelian groups and $1,e,\{e\}$ for multiplicative notation and groups in general).

Since the arrow is a homomorphism, it takes the neutral element from $0$ to the neutral element in $X$, but beacuase $0$ has only one element, such homomorphism is unique, but it doesn't mean $|X|=1$.