Re: Simplicial versus (cubical with connections)

[Todd Trimble <trimble1@optonline.net>]

My impression is that there are at least two distinct notions of
cubical set which have entered this discussion. One version
describes cubical sets as presheaves on the Lawvere theory
generated by two constants or 0-ary operations; this is close
to what Vaughan described. More precisely, instead of taking
the category whose objects are finite sets equipped with two
distinct points (which is opposite to the Lawvere theory), he
adds in a terminal object (where the two constants are forced
to coincide), giving a category C.  Anyway, whether one takes
the Lawvere theory or C^{op}, the result is a category with
finite cartesian products and an interval object, and one notion
of cubical set is that of presheaf on this category.

Whereas cubical sets in the sense described by Ross are
different: they are presheaves on the free *monoidal* category
with an interval object. This category does not include diagonal
maps. I expect this is the notion of cubical set that Dmitry and
Urs were actually concerned with, but in any event, both the
cartesian version and the monoidal version of the cubical site
appear in the literature, and it is important to clarify which
notion is meant.

Todd Trimble


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