Heap

Idea

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 $(a,b,c)\mapsto a{b}^{-1}c$. We can interpret that operation as shifting $a$ by the (right) translation in the group which translates $b$ into $c$. 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’.

Definition

A heap $(H,t)$ is a nonempty set $H$ equipped with a ternary operation $t:H\times H\times H\to H$ 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 $\mathrm{Heap}$ of heaps.

Automorphism group

As suggested above, if $G$ is a group and we define $t(a,b,c)=a{b}^{-1}c$, then $G$ becomes a heap. This construction defines a functor $\mathrm{Prin}:\mathrm{Grp}\to \mathrm{Heap}$. In fact, up to isomorphism, all heaps arise in this way; to every heap is associated a group $\mathrm{Aut}(H)$ called its automorphism group, unique up to isomorphism. There are a number of ways to define $\mathrm{Aut}(H)$ from $H$.

1. If we choose an arbitrary element $e\in H$, then we can define a multiplication on $H$ by $ab=t(a,e,b)$. It is straightforward to verify that this defines a group structure on $H$, whose underlying heap structure is the original one.

2. We can define $\mathrm{Aut}(H)$ to be the set of pairs $(a,b)\in H\times H$, modulo the equivalence relation $(a,b)\sim (a\prime ,b\prime )$ iff $t(a,a\prime ,b\prime )=b$. (We think of $(a,b)$ as representing $a^{-1}b$.) We then define multiplication by $(c,d)(a,b)=(c,t(d,a,b))$; the inverse of (the equivalence class of) $(a,b)$ is (the equivalence class of) $(b,a)$ and the identity element is (the equivalence class of) $(a,a)$ (for any $a$).

3. We can also define $\mathrm{Aut}(H)$ as an actual subgroup of the symmetric group of $H$, analogously to Cayley’s theorem for groups. We take the elements of $\mathrm{Aut}(H)$ to be set bijections of the form $t(-,a,b):H\to H$ where $a,b\in H$, with composition as the group operation. Note that

   (1)$$t(-,c,d){\cdot }_{\mathrm{Aut}(H)}t(-,a,b)=t(t(-,c,d),a,b)=t(-,c,t(d,a,b)),$$t(-,c,d) \cdot_{Aut(H)} t(-, a,b) = t(t(-,c,d),a,b) = t(-,c,t(d,a,b)),

   so $\mathrm{Aut}(H)$ is closed under this operation. The first axiom of a heap shows that $\mathrm{Aut}(H)$ contains the identity $t(-,x,x)$ for any $x$), and the inverse of $t(\cdot ,a,b)$ is $t(\cdot ,b,a)$; thus $\mathrm{Aut}(H)$ is a subgroup of the symmetric group of $H$.

Note that in both the second and third constructions, the elements of $\mathrm{Aut}(H)$ are determined by pairs of elements of $H$, modulo some equivalence relation. The following theorem shows that the two equivalence relations are the same.

The composition laws are also easily seen to agree, so the second two constructions of $\mathrm{Aut}(H)$ are canonically isomorphic. To compare them to the first construction, observe that for a fixed $e\in H$, any equivalence class contains a unique pair of the form $(e,a)$. (If $(b,c)$ is in the equivalence class, then $a$ is determined by $a=t(e,b,c)$.) 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 $\mathrm{Aut}:\mathrm{Heap}\to \mathrm{Grp}$. Note that we have $\mathrm{Aut}(\mathrm{Prin}(G))\cong G$ for any group $G$, and $\mathrm{Prin}(\mathrm{Aut}(H))\cong H$ for any heap $H$, but while the first isomorphism is natural, the second is not. In particular, the categories $\mathrm{Heap}$ and $\mathrm{Grp}$ are not equivalent.

Heaps and Torsors

Note that $\mathrm{Aut}(H)$ comes equipped with a canonical action on $H$ (this is most clear from the third definition). This action is transitive (by $t(a,a,b)=b$) and free (if $t(a,b,c)=a$ then by the previous statement $t(x,b,c)=x$ for each $x$, and in particular $t(b,b,c)=b$ and also $t(b,b,c)=c$). Therefore, $H$ is an $\mathrm{Aut}(H)$-torsor (over a point). Conversely, a torsor $H$ over any group $G$ can be made into a heap, by defining $t(a,b,c)=g\cdot c$, where $g\in G$ is the unique group element such that $g\cdot b=a$.

In fact, the category $\mathrm{Heap}$ is equivalent to the following category $\mathrm{Tors}$: its objects are pairs $(G,H)$ consisting of a group $G$ and a $G$-torsor $H$, and its morphisms are pairs $(\varphi ,f):(G,H)\to (G\prime ,H\prime )$ consisting of a group homomorphism $\varphi :G\to G\prime$ and a $\varphi$-equivariant map $f:H\to H\prime$.

References and remarks

• 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)

• wikipedia:heap

• Heaps and torsors

There is an oidification (horizontal categorification) of a heap, sometimes called a heapoid.