A Simple Approach to Geometric Realization of Simplicial and Cyclic Sets

The theory of simplicial sets and their realization is perhaps the basis to the combinatorial approach to homotopy theory. The construction of the geometric realization is very natural except for one thing: It begins with the postulation of the geometric realization of each of the “standard simplices”, to which no justification is given. It occurred to us that by “explaining” the nature of this realization, the theory could become even more natural. In particular, we wanted to find a completely obvious proof to the fact that geometric realization commutes with products in the right topology. This explanation is achieved here in the first two sections. We interpret the geometric realization of the standard simplex ∆n as an appropriately topologization of the space of order preserving maps from the unit interval to the ordered set [n] := {1, . . . , n}. The definition makes perfect sense for any finite partially ordered set (and in fact more generally for categories). On the other hand, a partially ordered set P gives rise to a natural simplicial set and we show that its realization is the same as that of P . On the point set level this boils down to the fact that an order preserving map from the unit interval to P has to factor through a map [n] → P for some [n]. On the point set level it is again clear that for partially ordered set the constructions of geometric realization and of associated simplicial sets commute with products. Once we check that this remains true in topology, the proof that geometric realization of simplicial sets commutes with products is attained. It turns out that a very similar idea works also for cyclic sets. One needs to introduce a notion that plays the same role in the theory of cyclic sets as that of partially ordered sets in the theory of simplicial sets. This notion, introduced in section 4, is that of a periodic partially ordered set, which is just a partially ordered set with an action of a free abelian group of finite rank, but the definition of morphisms is a