In ordinal arithmetic, the ordinal sum is a natural addition on ordered sets. This is particularly well behaved when considering finite ordinals and so provides a useful tool when manipulating simplicial sets. With infinite ordinals as well the properties are subtler and so care has to be taken.
The basic idea is that in ordinal sum you ‘first put one of the two ordinals and then the other’, so that the elements of the second ordinal are all bigger than those in the first one.
Restricted to and thus to finite ordinals, the ordinal sum induces the operation of join of simplicial sets endowing the category, , of augmented simplicial sets with the structure of a monoidal category.
The objects of the augmented simplex category can be identified with the finite totally ordered sets, including the empty set, which we write in this context as
so that then
is the singleton set, as usual and
and so on, so that
This counting is off by one compared to the cardinality of these sets.
(We will restrict for the moment to finite ordinals and thus to the category .)
The monoidal structure on is, at the level of the sets, just the disjoint union, but we consider the order on that union.
If we have and , we form the union of the two sets, where we know the order on two elements we keep it, but if we have two elements one, , say, from the and the other, , from we put .
As an example, consider and , where the overlines are just so that we can keep track of where the different elements come from. We form the union of the two sets and the rule says that anything without an overline is less than anything with one. This gives a linear order
which is isomorphic as a poset to . Similarly , which helps explain the picture of the related join of simplicial sets given there.
We can thus think of the operation as the addition of cardinalities, but must remember that has elements. In terms of the counting ‘off-by-one’, this reads
but must remember there is also the order to keep track of.
This operation extends to give the ordinal sum structure on (for details see the discussion in the entry simplex category) making it a monoidal category, whose monoidal product is the operation described above. Note however that and , although isomorphic are not naturally so.
If we consider ordinals that may not be finite then clearly the idea of ordinal sum generalises, so, for ordinals, and , is the set with total order in which elements in either part are ordered as originally, but where any is less than and .
If either ordinal is infinite, this may lead to and not being isomorphic. For instance, is isomorphic to itself, but has a greatest element, i.e. the in -summand, so clearly there can be no isomorphism between the two ordinals.
We thus give a warning: the monoidal structure given by ordinal sum is not a symmetric monoidal category structure even when restricted to the subcategory of finite ordinals. If one extends to consider infinite ordinals, there may even be no order preserving maps at all from to for , let alone isomorphisms.
In “Ordinal Sums and Equational Doctrines”, Lawvere defines the ordinal sum of two categories and as the pushout
where is the coproduct of and , 2 is the interval category, is its underlying discrete category of objects, and the left vertical arrow is defined by
Concretely, may be described as the category together with exactly one arrow adjoined for every object (in other words, as the collage of and along the terminal profunctor).
Equivalently, may be described as the colimit of the diagram
Last revised on August 10, 2023 at 13:30:04. See the history of this page for a list of all contributions to it.