The category of cubes
The category may be defined universally to be a walking interval: 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 may be shown that if , there are injections , the same as the number of -dimensional faces of the geometric -cube. There are no diagonal maps in the category of cubes as defined here.
- A different possibility is to consider the Lawvere theory of two constants, which gives a different category of cubes with diagonal maps.
Standard geometric cube functor
From the universal property of , 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.
In terms of an explicit formula, the cubical realization of a cubical set is given by the coend formula
A cubulation of a topological space is a cubical set together with a homeomorphism .
Relation between triangulation and cubulation
In rough terms, a space can be triangulated if and only if it can be cubulated. This can be shown by simple conceptual arguments, as follows.
Cubulating a triangulated space
In this section, may be taken to be the category of topological spaces, or otherwise any sufficiently convenient category of spaces (completeness and cocompleteness are baseline assumptions).
Simplices as cubical sets
We define a functor
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.
To define , we mimic the second definition of the affine simplex functor given at triangulation, 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 .
Cubulating geometric simplices
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), giving the canonical cubulation of affine simplices. In terms of an explicit formula, we have
Standard cubulation of a triangulated space
Given a triangulation of a space , we have isomorphisms
where in the last line we used the coend Fubini theorem?. Thus, defining the cubical set by
we have a homeomorphism , i.e., we obtain a cubulation of .
Triangulating a cubulated space
In this section we assume is a convenient category of spaces, so that geometric realization of simplicial sets is product-preserving.
Cubes as simplicial sets
Define a monoidal functor as follows: 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
Triangulating geometric cubes
Next, because is preserves cartesian products and preserves the interval objects, we have an isomorphism
by the universal property of . In terms of an explicit formula, we have
Standard triangulation of a cubulated space
Given a cubulation of a space , we have isomorphisms
where in the last line we used the coend Fubini theorem?. Thus, defining the simplicial set by
we have a homeomorphism , i.e., we obtain a triangulation of .