nLab heap

Idea

In algebra, a heap is an algebraic structure which is almost that of a group structure but without the specification of a neutral element.

While closely related to notions such as affine space, principal homogeneous space and torsor, the concept of heap is simpler in the sense that it postulates just a single set with a single ternary operation obeying two axioms (see Def. below).

Concretely, any group gives a heap where this ternary operation is defined from the group’s binary operation $(\text{-})\cdot(\text{-})$ and inversion $(-)^{-1}$ by

(1)$$t \,\colon\, (a,b,c) \;\mapsto\; a \cdot b^{-1} \cdot c$$

and every heap arises this way, up to isomorphism (cf. Prop. below) from a group that may be called its structure group.

However, the category of heaps (Def. below) is not equivalent to the category of groups. Instead:

Proposition

The category of heaps (Def. ) is equivalent to the evident category of pairs consisting of a group $G$ and a $G$-torsor $H$.

Proof

On the one hand, for any heap $H$, choosing any element $h' \in H$, the endofunctions $t(h,h',-) \colon H \to H$ for $h \in H$ constitute a group $G$ under composition, and the underlying set $H$ of the heap manifestly carries the structure of a $G$-torsor. On the other hand, given a group $G$ and a $G$-torsor $H$ we can make $H$ into a heap as follows: given elements $a,b,c \in H$ let $t(a,b,c) = g c$ where $g$ is the unique $g \in G$ with $g b = a$.

There is also a dual version of the concept of heap, see at quantum heap.

Remark

(disambiguation)
Heaps in the sense of algebra should not be confused with either

• heaps in the sense of theoretical computer science,

nor with

• heaps of pieces in the combinatorial sense of Viennot 2006.

Remark

(synonyms)
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 ‘груда’ (‘gruda’) meaning a heap of soil; this is a pun as it is parallel to the Russian word ‘группа’ (‘gruppa’) meaning a group: forgetting the unit element is sort of creating an amorphous version. This term also appears in English as ‘groud’. In universal algebra the standard name is associative Malcev algebra (in various spellings, including Mal’cev, Mal’tsev and Maltsev), other names include herd.

There is an oidification (horizontal categorification) of heaps, sometimes called heapoids.

Definition

Definition

A heap $(H,t)$ is a inhabited set $H$ equipped with a ternary operation $t \colon H \times H \times H\to H$ satisfying the following two relations

1. $t(b,b,c) = c = t(c,b,b)$,

2. $t\big(a,b,t(c,d,e)\big) = t\big(t(a,b,c),d,e\big)$.

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 is abelian if it additionally satisfies the relation

$$t(a,b,c) = t(c,b,a)$$

Definition

A homomorphism of heaps $f \colon H \to H'$ is, of course, a function of the underlying set which respects the ternary operations.

This defines a category $Heap$ whose objects are heaps and whose morphisms are heap homomorphisms.

The hom-sets of the full subcategory $AbHeap$ of abelian heaps inherit an abelian heap structure from the pointwise operation in the codomain: given $f,g,h\colon H \to G$, the function $a\mapsto t_G(f(a),g(a),h(a))$ is again a heap homomorphism.

Properties

Structure groups of heaps

As indicated in (1) a group $G$ becomes a heap by setting

$$a,b,c \,\in\, G \;\;\;\;\;\; \vdash \;\;\;\;\;\; t(a,b,c) \,\coloneqq\, a \cdot b^{-1} \cdot c \,.$$

Proposition

This construction (1) defines a functor

$$Prin \,\colon\, Grp \longrightarrow Heap$$

from the category Grp of groups to that of heaps, which is essentially surjective, meaning that, up to isomorphism all heaps arise in this way.

In fact, there is also a functor

$$Str \,\colon\, Heap \longrightarrow Grp$$

such that

1. there exist natural group isomorphisms of the form

   $Str\big(Prin(G)\big) \,\cong\, G$

2. there exist heap isomorphisms of the form

   $Prin\big(Str(H)\big) \,\cong\, H$

   which however are not natural (whence we do not have an equivalence of categories).

This group $Str(H)$ is called the structure group of the heap $H$.

Proof

Given a heap $H$, the claimed structure group $Str(H)$ is described, up to isomorphism, by any of the following constructions:

1. Choosing any element $\mathrm{e} \in H$, then the binary operation

   (2)$$a \cdot b \,\coloneqq\, t(a, \mathrm{e} ,b) \,.$$

   constitutes a group structure on $H$, with neutral element $\mathrm{e}$.

   This serves as the required structure group: $Str(H) \,\coloneqq\, (H, \cdot)$.

2. Take the underlying set of $Str(H)$ to be that of equivalence classes of pairs $(a,b) \in H\times H$, subject to the equivalence relation

   $$(a,b) \,\sim\, (a',b') \;\;\;\;\text{iff}\;\;\;\; t(a,b,b') \,=\, a'$$

   (the idea is to think of the pair $(a,b)$ as the representative of $a b^{-1}$)

   and take the binary operation on the group to be given on representatives by

   (3)$$(c,d) \cdot (a,b) \,\coloneqq\, \big( c, t(d,a,b) \big) \,.$$

   This again defines a group $Str(H)$.

   Notice that:

   • the inverse of (the equivalence class of) $(a,b)$ is (the equivalence class of) $(b,a)$

   • the neutral element is (the equivalence class of) $(a,a)$ (for any $a$).

3. Finally, $Str(H)$ is realized also as an actual subgroup of the symmetric group on the underlying set of $H$, analogously to Cayley's theorem for groups. We take the elements of $Str(H)$ to be set bijections of the form

   (4)$$t(a,b,-) \,\colon\, H \rightarrow H \,,$$

   where $a,b \in H$, with composition as the group’s binary operation.

   Notice here that

   $$t(c,d,-) \cdot_{Str(H)} t(a,b,-) \;=\; t\big(c,d,t(a,b,-)\big) \;=\; t\big(t(c,d,a),b,-\big) \,,$$

   so that $Str(H)$ is closed under this operation.

   The first axiom of a heap shows that $Str(H)$ contains the neutral element $t(-,x,x)$, for any $x$), and the inverse element of $t(\cdot,a,b)$ is $t(\cdot,b,a)$; thus $Str(H)$ is a subgroup of the symmetric group of $H$.

It remains to see that these constructions all agree and are functorial.

First to show that the equivalence relation used in the second construction is symmetric, and that it coincides with that implicitly used in the first construction, notice that the following are equivalent:

• (i) the bijections $t(a,b,-)$ and $t(a',b',-)$ coincide

• (ii) $t(a,b,b') = a'$

• (iii) $t(a',b',b) = a$

Namely:

(ii) follows from (i) and $t(a,a,b) = b$.

(ii) implies (iii) as follows. Given $t(a,b,b') = a'$, we have $t(t(a,b,b'),b',b) = t(a',b',b)$, but

$$t(t(a,b,b'),b',b) = t(a,b,t(b',b',b)) = t(a,b,b) = a$$

so $t(a',b',b) = a$. (iii) implies (ii) by a similar argument.

(i) follows from (ii) by the calculation:

$$t(a',b',x) = t(a',b',t(b,b,x))= t(t(a',b',b),b,x) t(a,b,x)$$

Since the composition laws can also be seen to agree, the second two constructions of $Str(H)$ are canonically isomorphic.

To compare them to the first construction, observe that for a fixed $\mathrm{e} \in H$, any equivalence class contains a unique pair of the form $(\mathrm{e},a)$. (If $(b,c)$ is in the equivalence class, then $a$ is determined by $a = t(\mathrm{e},b,c)$.) This sets up a bijection between the first two constructions, which we can easily show is an isomorphism.

Finally, the second construction is manifestly functorial.

The following proposition expands on the fact that a group is the same as a pointed heap, i.e. a heap with a chosen element:

Proposition

The category of groups is equivalent to the slice category $1 \downarrow \mathrm{Heap}$ where $1$ is the terminal heap and $\mathrm{Heap}$ is the category of heaps (Def. ).

Proof

The one-element set $1$ is a heap in a unique way, and for any heap $H$, any function $f: 1 \to H$ or $f: H \to 1$ is a heap homomorphism, so $1$ is the terminal object in $\mathrm{Heap}$ and $1 \downarrow \mathrm{Heap}$ is the category of pointed heaps. As noted in the proof of the previous proposition, a heap $H$ with a chosen point $\mathrm{e}$ becomes a group with multiplication $a \cdot b \,\coloneqq\, t(a, \mathrm{e} ,b)$, and a morphism of pointed heaps then becomes a group homomorphism. This gives a functor $1 \downarrow \mathrm{Heap} \to \mathrm{Grp}$, which is an equivalence thanks to the functor $\mathrm{Grp} \to 1 \downarrow \mathrm{Heap}$ sending any group to the corresponding pointed heap.

Relation to torsors

Given a group $G$, a $G$-torsor is a nonempty set $X$ with a free and transitive action of $G$, which we may write as

$$\begin{array}{ccl} G \times X &\to& X \\ (g, x) &\mapsto& g x \end{array}$$

Equivalently, a $G$-torsor is a set $X$ with an action of the group $G$ but also a “division” operation

$$\begin{array}{ccl} X \times X &\to& G \\ (x, y) &\mapsto& x/y \end{array}$$

obeying

$$(x/y) y = x$$

Given a group $G$ together with a $G$-torsor $X$, we can make $X$ into a heap by giving it the ternary operation

$$\begin{array}{ccl} X \times X \times X &\to& X \\ (x, y, z) &\mapsto& (x/y) z \end{array}$$

Conversely, from any heap $H$ we can construct a group $G$ together with a $G$-torsor whose underlying set is $H$ itself. To see this, note that the group $G = Str(H)$ (?) comes equipped with a canonical group action on $H$, as is most clear from the third definition (4).

This action is

• transitive

  by $t(a,a,b) = b$

and

• free

  since 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 $Str(H)$-torsor (over a point).

Conversely, any torsor $H$ over a group $G$ becomes a heap, by defining

$$t(a,b,c) \,\coloneqq\, g \cdot c \,,$$

where $g\in G$ is the unique group element such that $g\cdot b = a$.

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

Groups as pointed heaps

As indicated in the discussion above, a heap $H$ acquires a group structure $(a, b) \mapsto t(a, e, b)$ as soon as an element $e \in H$ is selected (and this $e$ is then the identity element of the group). Notice that every function $1 \to H$ from a terminal set $1$ is in fact a heap homomorphism when we consider that $1$ carries a unique heap structure. In different words: the free heap on a terminal set $1$ is $1$ itself with its unique heap structure.

It follows that the category of groups Grp is equivalent to the comma category $1 \downarrow Heap$, with both categories considered as categories over $Heap$ (via $Prin: Grp \to Heap$ and the forgetful functor $U: 1 \downarrow Heap \to Heap$ that forgets the point). This says that groups are equivalent to pointed heaps.

Both of these forgetful functors are monadic over $Heap$. In the case of $U: 1 \downarrow Heap \to Heap$, the monad is $1 \sqcup -$, where the monad structure is induced from the monoid structure that $1$ carries with respect to the cocartesian monoidal product $\sqcup$. The free pointed heap on $H$ is therefore the heap $1 \sqcup H$, equipped with the pointing given by the coproduct coprojection $1 \to 1 \sqcup H$.

It follows that the free group on a heap $H$ can be constructed abstractly as $1 \sqcup H$. This is analogous to how the free vector space on an affine space $A$ can be described as $1 \sqcup A$, although it is true that coproducts of affine spaces are vastly easier to describe concretely than coproducts of heaps.

Here is a more group-theoretic description of the free group on $H$. Let $Str(H)$ be the structure group, given as the group of heap automorphisms of the form $t(a, b, -): H \to H$. Let

$$\alpha: Str(H) \times H \to H$$

denote the tautological action. Let $F(H)$ denote the free group on the underlying set of $H$. For groups $G, G'$, let $G \ast G'$ denote their free product (coproduct). Then the free group on the heap $H$ is the quotient of $Str(H) \ast F(H)$ given by the coequalizer of two maps

$$\beta, \gamma: F(Str(H) \times H) \rightrightarrows Str(H) \ast F(H)$$

where $\beta$ is the composition of $F(\alpha): F(Str(H) \times H) \to F(H)$ followed by the coproduct coprojection $F(H) \to Str(H) \ast F(H)$, and $\gamma$ is the unique group homomorphism that extends the function

$$Str(H) \times H \to Str(H) \ast H$$

mapping a pair $(\theta, h)$ to their product $\theta \cdot h$ in the group $Str(H) \ast H$.

The meaning of the construction is that group homomorphisms $Str(H) \ast F(H) \to G$ that factor through the coequalizer are in natural bijection with pairs

$$(\phi: Str(H) \to G, f: H \to G)$$

where $\phi$ is a homomorphism and $f$ is a function, satisfying the torsor homomorphism condition

$$f(\theta(h)) = \phi(\theta) \cdot f(h)$$

which is an equivalent way of describing a heap homomorphism $H \to Prin(G)$.

The empty heap

If we wish $Heap$ to be an algebraic category, then we must remove the clause that the underlying set of a heap must be nonempty (see also at pseudo-torsor). 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 an inhabited invertible semigroup $(S,\cdot,(-)^{-1})$ or an inhabited associative quasigroup. Then any invertible semigroup or associative quasigroup is a possibly-empty-heap, and every possibly-empty-heap arises in this way from its automorphism invertible semigroup or automorphism associative quasigroup (defined by either method (2) or (3)); the category of possibly-empty-heaps is equivalent to the category of invertible semigroups equipped with torsors over the point, and the category of associative quasigroups equipped with torsors over the point; etc.

This is even constructive; the theorems can be proved uniformly, rather than by treating the empty and inhabited cases separately. (This rather trivial method is obvious to a classical mathematician, but it's not constructively valid, since a invertible semigroup/associative quasigroup/heap as defined here can't be constructively proved empty or inhabited; it can only be proved empty iff not inhabited. Indeed, taking any group $G$ and any truth value $P$, the invertible subsemigroup or associative subquasigroup $\{x \in G \;|\; P\}$ is empty or inhabited iff $P$ is false or true.)

Examples in homotopy theory

The map $[0,1]\to[0,1]$ that sends $0\mapsto 0$, $1/3\mapsto 1$, $2/3\mapsto 0$, $1\mapsto 1$, and interpolates linearly between these points yields an associative Malcev operation on $[\Sigma X,Y]$, where $X$ and $Y$ are (unpointed) spaces, $\Sigma$ is the suspension functor, and $[-,-]$ denotes the set of morphisms in the homotopy category.

Thus, $[\Sigma X,Y]$ is a (nonabelian) heap. Likewise, the full mapping space $Map(\Sigma X,Y)$ can be turned into an (∞,1)-heap, defined as an (∞,1)-algebra (in spaces) over the algebraic theory of heaps.

See Vokřínek for more information.

Related concepts

• torsor

• affine space

• Mal'cev variety

• truss

References

• A. K. Sushkevich, Theory of generalised groups, DNTVU, Kharkov-Kiev (1937) (Russian original: А. К. Сушкевич, Теория обобщенных групп, Государственное научно-техническое издательство Украины, 1937).

• Peter T. Johnstone, The ‘closed subgroup theorem’ for localic herds and pregroupoids, Journal of Pure and Applied Algebra 70 (1991) 97-106 [doi:10.1016/0022-4049(91)90010-y]

• G. M. Bergman, A.O. Hausknecht, Ch.IV §22 pp 95 in: Cogroups and co-rings in categories of associative rings, AMS (1996) [ams:surv-45]

• Christopher D. Hollings, Mark V. Lawson, Wagner’s theory of generalised heaps, Springer (2017) [doi:10.1007/978-3-319-63621-4]

• Zoran Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12 1 (2007) 1-9 [math.QA/0701749]

• Thomas Booker, Ross Street, Torsors, herds and flocks, J. Algebra 330 (2011) 346-374 [pdf arXiv:0912.4551]

Review:

• Wikipedia: Heap (mathematics)

• John Baez, Torsors made easy (2009)

• John Baez, The group with no elements (2020)

In the context of stable homotopy theory:

• Lukáš Vokřínek, Heaps and unpointed stable homotopy theory, Archivum Mathematicum 50 5 (2014) 323-332 [arXiv:1312.1709, doi:10.5817/AM2014-5-323]

category: algebra