symmetric monoidal (∞,1)-category of spectra
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 notions such as affine spaces, principal homogeneous spaces are very similar, the concrete definition of heaps is more direct and simpler in the sense that it postulates just a single ternary operation satisfying just two axioms.
Concretely, for heaps underlying a group structure, this ternary operation is defined from the binary group operation $(\text{-})\cdot(\text{-})$ by
and every heap arises this way, up to isomorphism, from a group called its automorphism group (Prop. below).
The discussion there (3) shows that one may usefully think of this heap operation (1) as bracketed to the right
and thought of as the action of group elements $a$ on elements of its underlying torsor, which reduces to the group action on itself (2) once the otherwise arbitrary origin “$b$” of this torsor is (re-)identified with the neutral element.
There is also a dual version of this concept, see at quantum heap.
(disambiguation)
Heaps in the sense of algebra should not be confused with wither
nor with
(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.
A heap $(H,t)$ is a nonempty set $H$ equipped with a ternary operation $t \colon H \times H \times H\to H$ satisfying the following two relations
$t(b,b,c) = c = t(c,b,b)$,
$t\big(a,b,t(c,d,e)\big) = t\big(t(a,b,c),d,e\big)$.
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
A heap homomorphism, of course, is a function that preserves the ternary operations. This defines a category $Heap$ of heaps.
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.
As indicated in (1) a group $G$ becomes a heap by setting
This construction (1) defines a functor
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 a converse functor
such that
there exist natural group isomorphisms of the form
$Aut\big(Prin(G)\big) \,\cong\, G$
there exist heap isomorphisms of the form
$Prin\big(Aut(H)\big) \,\cong\, H$
which however are not natural (whence we do not have an equivalence of categories).
Given a heap $H$, the claimed automorphism group $Aut(H)$ is described, up to isomorphism, by any of the following constructions:
Choosing any element $\mathrm{e} \in H$, then the binary operation
constitutes a group structure on $H$, with neutral element $\mathrm{e}$.
This serves as the required automorphism group: $Aut(H) \,\coloneqq\, (H, \cdot)$.
Take the underlying set of $Aut(H)$ to be that of equivalence classes of pairs $(a,b) \in H\times H$, subject to the equivalence relation
(the idea is to think of the pair $(a,b)$ as the representative of $a^{-1} \cdot b$)
and take the binary operation on the group to be given on representatives by
This again defines a group $Aut(H)$.
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 $a$).
Finally, $Aut(H)$ 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 $Aut(H)$ to be set bijections of the form
where $a,b \in H$, with composition as the group’s binary operation.
Notice here that
so that $Aut(H)$ is closed under this operation.
The first axiom of a heap shows that $Aut(H)$ 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 $Aut(H)$ is a subgroup of the symmetric group of $H$.
It remains to see that these constructions all agree and are functorial.
First to see that the equivalence relations used in the second and third construction agree, notice that the following are equivalent:
(i) the bijections $t(\cdot,a,b)$ and $t(\cdot,a',b')$ coincide
(ii) $t(a,a',b') = b$,
(iii) $t(b,b',a') = a$.
Namely:
(ii) follows from (i) and $t(a,a,b) = b$.
(iii) follows from (ii) by applying $t(\cdot,b',a')$ on the right. Similarly (ii) follows from (iii).
(i) follows from (ii) by the calculation:
Since the composition laws are also easily seen to agree, we have that the second two constructions of $Aut(H)$ 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 constructions is manifestly functorial.
The automorphism group $Aut(H)$ in (?) comes equipped with a canonical group action on $H$, as is most clear from the third definition (4).
This action is
by $t(a,a,b) = b$
and
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 $Aut(H)$-torsor (over a point).
Conversely, any torsor $H$ over a group $G$ becomes a heap, by defining
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 $f:H\to H'$.
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.)
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 notions: torsor, affine space, Mal'cev variety, truss
Christopher D. Hollings, Mark V. Lawson, Wagner’s theory of generalised heaps, 2017, Springer. doi
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).
G.M. Bergman, A.O. Hausknecht, Cogroups and co-rings in categories of associative rings, Ch.IV, paragraph 22, p.95ff, AMS 1996.
Z. Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12, No. 1, pp. 1–9 (2007); arXiv:math.QA/0701749
Thomas Booker, Ross Street, Torsors, herds and flocks, J. Algebra 330 (2011) 346–374 pdf arXiv:0912.4551
Peter T. Johnstone, The ‘closed subgroup theorem’ for localic herds and pregroupoids, Journal of Pure and Applied Algebra 70 (1991) 97-106. doi90010-y).
wikipedia:heap (mathematics)
Lukáš Vokřínek, Heaps and unpointed stable homotopy theory, arXiv.
There is an oidification (horizontal categorification) of a heap, sometimes called a heapoid.
Last revised on March 20, 2024 at 04:17:24. See the history of this page for a list of all contributions to it.