A heap is an algebraic structure which is basically equivalent to a group when one forgets about which element is the unit. Similar notions are affine space, principal homogeneous space and so on. However, the notion of a heap has a directness and simplicity in the sense that it is formalized as an algebraic structure with only one ternary operation satisfying a short list of axioms. If we start with a group the ternary operation is defined via . We can interpret that operation as shifting by the (right) translation in the group which translates into . There is also a dual version, quantum heap.
Heaps in the sense of algebra should not be confused with heaps in the sense of theoretical computer science. There are also a number of synonyms for the term “heap;” below we consider “torsor” in this light. In Russian one term for a heap is “gruda” meaning a ‘heap of soil’; this is a pun as it is parallel to the russian word “gruppa” meaning a ‘group’: forgetting the unit element is sort of creating an amorphous version. This term also appears in English as ‘groud’.
A heap is a nonempty set equipped with a ternary operation satisfying the relations
More generally, a ternary operation in some variety of algebras satisfying the first pair of equations is called a Mal'cev operation. A Mal’cev operation is called associative if it also satisfies the latter equation (i.e. it makes its domain into a heap).
A heap homomorphism, of course, is a function that preserves the ternary operations. This defines a category of heaps.
As suggested above, if is a group and we define , then becomes a heap. This construction defines a functor . In fact, up to isomorphism, all heaps arise in this way; to every heap is associated a group called its automorphism group, unique up to isomorphism. There are a number of ways to define from .
If we choose an arbitrary element , then we can define a multiplication on by . It is straightforward to verify that this defines a group structure on , whose underlying heap structure is the original one.
We can define to be the set of pairs , modulo the equivalence relation iff . (We think of as representing .) We then define multiplication by ; the inverse of (the equivalence class of) is (the equivalence class of) and the identity element is (the equivalence class of) (for any ).
We can also define as an actual subgroup of the symmetric group of , analogously to Cayley's theorem? (see Wikipedia) for groups. We take the elements of to be set bijections of the form where , with composition as the group operation. Note that
so is closed under this operation. The first axiom of a heap shows that contains the identity for any ), and the inverse of is ; thus is a subgroup of the symmetric group of .
Note that in both the second and third constructions, the elements of are determined by pairs of elements of , modulo some equivalence relation. The following theorem shows that the two equivalence relations are the same.
The following are equivalent
bijections and are the same maps,
(ii) follows from (i) and .
(iii) follows from (ii) by applying on the right. Similarly (ii) follows from (iii).
(i) follows from (ii) by the calculation:
The composition laws are also easily seen to agree, so the second two constructions of are canonically isomorphic. To compare them to the first construction, observe that for a fixed , any equivalence class contains a unique pair of the form . (If is in the equivalence class, then is determined by .) This sets up a bijection between the first two constructions, which we can easily show is an isomorphism.
The second two constructions are clearly functorial, so we have a functor . Note that we have for any group , and for any heap , but while the first isomorphism is natural, the second is not. In particular, the categories and are not equivalent.
Note that comes equipped with a canonical action on (this is most clear from the third definition). This action is transitive (by ) and free (if then by the previous statement for each , and in particular and also ). Therefore, is an -torsor (over a point). Conversely, a torsor over any group can be made into a heap, by defining , where is the unique group element such that .
In fact, the category is equivalent to the following category : its objects are pairs consisting of a group and a -torsor , and its morphisms are pairs consisting of a group homomorphism and a -equivariant map .
If we wish to be an algebraic category, then we must remove the clause that the underlying set of a heap must be nonempty. Then the empty set becomes a heap in a unique way. However, in this case, the various theorems relating heaps to groups above all break down. For this reason, one usually requires a heap to be inhabited.
On the other hand, we could generalize the notion of group to allow for an empty group. This even remains a purely algebraic notion: we can define a group as a (traditionally nonempty) set equipped with a binary operation (to be thought of as ) satisfying these laws:
Then any possibly-empty-group is a possibly-empty-heap, and every possibly-empty-heap arises in this way from its automorphism possibly-empty-group (defined by either method (2) or (3)); the category of possibly-empty-heaps is equivalent to the category of possibly-empty-groups equipped with torsors over the point; etc.
This is even constructive; the theorems can be proved uniformly, rather than by the (rather trivial) method of treating the empty and nonempty cases separately.
G.M. Bergman, A.O. Hausknecht, Cogroups and co-rings in categories of associative rings, Ch.IV, paragraph 22, p.95ff – Providence, R.I. : AMS 1996.
Z. Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12, No. 1, pp. 1–9 (2007); (math.QA/0701749)
There is an oidification (horizontal categorification) of a heap, sometimes called a heapoid.