What is the definition of a heapoid?

Question

There is a notion of 'oidification' in category theory which characterises many object versions of mathematical objects. For example:

• magmas $\rightarrow$ magmoids

• loops $\rightarrow$ loopoids

• groups $\rightarrow$ groupoids

• rings $\rightarrow$ ringoids

And in reverse, one object magmoids are magmas and so on. This is also referred to as horizontal categorification in contrast to vertical categorification.

NLab mentions that heaps have a many object oidification: heapoids. But do not give an explicit characterisation. It's not immediately obvious to me what this is, unlike the mathematical objects mentioned above, heaps are characterised by a ternary operation.

Hence I'm just looking for pointers to a definition, preferably accessible online as I don't have access to a math library.

Answer 1

The claim that heapoids exist was added to the nLab page in revision 3 by Toby Bartels, so you could ask him what he had in mind.

I can speculate that a heapoid would have

• a set of objects.
• families of morphims $f:x\to y$.
• for any $f:x\to y$, $g:z\to y$, and $h:z\to w$, a ternary composite $t(h,g,f) : x\to w$ (note the reversal of direction in $g$).
• for any $f:x\to y$ and $g:x\to z$, we have $t(g,f,f) = g$.
• for any $f:x\to y$ and $g:z\to y$, we have $t(g,g,f) = f$.
• for any $f:x\to y$, $g:z\to y$, $h:z\to w$, $k:u\to w$, and $\ell : u\to v$, we have $t(\ell,k,t(h,g,f)) = t(t(\ell,k,h),g,f)$.

Then any groupoid would become a heapoid with $t(h,g,f) = h\circ g^{-1}\circ f$, just as any group becomes a heap in a similar way.