(Usually, in classical algebraic and geometric topology, the here is taken to be a simplicial complex, but the difference does not really matter if one is considering uses in homotopy theoretic contexts. For the reason, see the discussion at subdivision. When considering polyhedral structure as such, for instance for PL manifolds, the simplical complex version would be needed. In such a case we may refer to a classical triangulation.)
Explicitly, is given by a coend formula
where is the standard affine simplex functor. Provided that is interpreted as a nice category of spaces (such as – see the discussion at geometric realization), the functor is left exact, and in particular preserves products.
There are various ways of understanding the affine simplex functor from a categorical perspective. (Note: in this article we will be working with the algebraist’s version of the simplex category , namely the category of finite ordinals and order-preserving maps, including the initial or empty object which represents a (-1)-dimensional simplex. The -element ordinal is conventionally, but perhaps unfortunately, denoted , to indicate the dimension.)
One way begins by regarding as isomorphic to the category of nonempty ordinals, for which maps are functions that preserve order and the top and bottom elements. In other words, as the category of finite intervals, where an interval is a totally ordered set with a top and bottom element. Indeed, for each ordinal , the hom-set inherits from an interval structure under the pointwise definitions, and
is an equivalence (this can also be seen as a restriction of the Stone duality between finite posets and finite distributive lattices).
Under this contravariant equivalence, the -element object of corresponds to the -element finite interval (again denoted ). Consider the functor that arises by homming into the standard unit interval :
This functor lifts to a functor that takes to the space of interval maps ,
topologized as a subspace of . This gives the affine simplex functor .
The category of intervals is an -accessible category where the finitely presentable objects are the finite intervals. It follows that each representable on , in particular , is a filtered colimit of representable presheaves on .
A second way of understanding is by taking advantage of the fact that the algebraist’s is the walking monoid. This means that given a monoidal structure on and a monoid therein, there is a unique monoidal functor which sends the generic monoid to the monoid . To this end, take the monoidal product on to be “topological simplicial join”: the join of two spaces , may be defined to be the pushout of the diagram
and now take the monoid in to be the 1-point space with its unique monoid structure.
The induced monoidal functor is the affine simplex functor . In effect, it identifies the -dimensional simplex with an iterated simplicial join of copies of :
because is itself the monoidal power of the 1-element ordinal . Equivalently, it can be regarded as the result of applying the cone functor times to .
where is the standard geometric cube functor.
The category may be regarded as a “walking interval” in a sense slightly different to the sense of interval above: it is initial among monoidal categories that are equipped with an object , two maps (where is the monoidal unit) and a map such that . The monoidal unit in is terminal, hence there is a unique map for any object . The interval of monoidally generates in the sense of PROS.
It follows that if is considered as a cartesian monoidal category equipped with in this sense of interval, we get an induced monoidal functor
The monoidal product on induces a monoidal product on by Day convolution. The cubical realization functor is, up to isomorphism, the unique cocontinuous monoidal functor which extends the monoidal functor along the Yoneda embedding; therefore takes -products of cubical sets to the corresponding cartesian products of spaces.
As explained below, there is a “cubulation” functor for standard simplices, , such that the affine simplex functor is naturally isomorphic to the composite
Given a triangulation of a space , we have isomorphisms
where in the last line we used the coend Fubini theorem? for interchange of coends. Thus, defining the cubical set by
we have a homeomorphism , i.e., we obtain a cubulation of .
There is also a triangulation functor for standard cubes, , which can be used to triangulate the realizations of cubical sets.
The functor effectively regards an -simplex as an iterated join of simplicial sets and then produces the analogous join in the category of cubical sets. This for instance regards the 2-simplex as a square with one degenerate edge.
In other words, to define , we mimic the second construction of the affine simplex functor given above, replacing by cubical sets and the topological simplicial join by a suitable “cubical simplicial join”. Formally, we define a monoidal structure on cubical sets by taking to be the pushout of the diagram
where the projection maps , are defined by taking advantage of the fact that the monoidal unit of is terminal:
The terminal cubical set is of course a monoid with respect to this monoidal product, so by the walking monoid property we obtain a monoidal functor
which plays a role analogous to the affine simplex functor into .
Observe that geometric realization takes cubical simplicial joins to topological simplicial joins, because sends -products to cartesian products, and preserves pushouts because it is cocontinuous. We conclude that both and take monoidal products in to topological simplicial joins, and both take the walking monoid of to the one-point space. By the universal property of , it follows that there is a natural isomorphism
(as monoidal functors), which is what we want.
Similarly, we can easily define a monoidal functor such that
In detail, regard the category of simplicial sets as a cartesian monoidal category equipped with the representable as an interval (with two face maps from and a projection to the terminal object ). By the walking interval property of , there is an induced functor
Finally, because is product-preserving and preserves the interval objects, the isomorphism (2) obtains by the universal property of .