# Identity map

## Definition

The "identity map", written on this project as [ilmath]\text{Id} [/ilmath], is a map which maps every item (in the domain) to itself, that is if [ilmath]\text{Id}:X\rightarrow X[/ilmath] is a function / map on some set [ilmath]X[/ilmath], then:

- [ilmath]\forall x\in X[\text{Id}(x)\eq x][/ilmath]

## Conventions

If we are dealing with two sets [ilmath]X[/ilmath] and [ilmath]Y[/ilmath], then technically we must use differing notation for the identity map on each, for example [ilmath]\text{Id}_X[/ilmath] and [ilmath]\text{Id}_Y[/ilmath] however this is rarely needed and we (even I, Alec) usually just write [ilmath]\text{Id} [/ilmath] for both

An "identity map" between different sets, for example [ilmath]f:X\rightarrow Y[/ilmath] such that [ilmath]\forall x\in X[f(x)\eq x][/ilmath] and as a result we must have [ilmath]X\subseteq Y[/ilmath], then [ilmath]f[/ilmath] is called an inclusion map

## Other notations

Sometimes [ilmath]I[/ilmath] is used for the identity map.

## See also

- Inclusion map, which is a map [ilmath]i:A\rightarrow B[/ilmath] where [ilmath]A[/ilmath][ilmath]\subseteq[/ilmath][ilmath]B[/ilmath] such that [ilmath]i:a\mapsto a[/ilmath] for all [ilmath]a\in A[/ilmath] - a sort of identity map in some sense.