A groupoid, also known as a virtual group, is a small category where every morphism is invertible. We can give a more explicit, algebraic definition: start with a set , and a partial binary operation on . Call a pair of elements of a composable pair if . A groupoid is the pair , together with two unary operations and on it, satisfying the following conditions:

1. is a composable pair iff .
2. and are composable pairs iff and are, and if one of these is true, then .
3. and are composable pairs and .
4. for each , there exists such that and are composable pairs, and and .

Below are some properties:

1. In condition 4 above, and . This is true by condition 1, since both and are composable pairs.
2. Again, in condition 4, is unique. To see this, suppose satisfies condition 4 (in place of ). Then . Notice property 1 is used in the proof. We call the inverse of , and write .
3. In view of condition 4, both and are unique. In other words, if are unary operators on satisfying conditions 3 and 4 above (in place of and ), then and . In fact, and .
4. Since , we see that is composable with itself, and that by the previous property. Similarly, . This shows that and are idempotent with respect to for every .
5. Since is a composable pair, for any . Similarly, . Hence . Similarly, . This shows that and are idempotent with respect to functional compositions.
6. (Cancellation property): if , then ; if , then .
   Proof. Since is a composable pair, . But , we have so that is a composable pair, hence is a composable pair and . Since is a composable pair, . As a result, . Similarly . By assumption, we deduce that . The other statement is proved similarly.
7. The algebraic definition given can be easily turned into a categorical definition (using objects and morphisms). The details are left for the reader.

If and are constant functions, then is a group.

Remark. There is also a group-theoretic concept with the same name.