Subject: Re: Delta or facial sets or spaces Date: Mon, 07 Feb 2000 08:56:31 -0100 From: "Carlos.SIMPSON" > ... was hoping for a reason to > ignore them. ... Here is a possible reason not to ignore them. Simplicial objects occur in Segal's delooping machine. The most basic example is taking a monoid and forming the corresponding category with one object, then taking its nerve. In this setup, the degeneracy maps correspond to the identity element of the monoid. If one wants to look at non-unitary monoids (there must be a terminology for that but I don't know what it is, maybe semigroup???) then the corresponding Segal-type object is a Delta set (facial set in my proposed terminology). This of course is a silly way to look at semigroups but if we go to topological semigroups (i.e. topological monoids without identity) from a homotopy-theoretic approach, one way to look at them would be as facial (Delta) spaces. The thing which seems to complicate this picture is that the product of two facial sets doesn't usually have the right homotopy type. In fact there seems to be a closed model structure on facial sets with the property that the product of two fibrant facial sets does have the right homotopy type. Going back to the previous paragraph, this relates to the somewhat amusing following phenomenon: if you take two presentations of monoids considered as semigroups, by generators and relations (in the semigroup world) and take the ``product'' presentation, then the semigroup generated by it is *not* the product of the two semigroups. However, if both of the presentations (maybe only one suffices?) have the identity elements together with their obvious relations vis-a-vis the other elements, then the product presentation *does* give back the product of the monoids. ---Carlos