On kan's condition

Abstract

In the category Δo Ens , of the simplicial sets, there exists a "Unit interval" : the simplicial set Δ[1], with extremities εi : Δ[0] → Δ[1] (i = 0, 1) ([1], [2] ). On the other hand, since in Δo Ens, the products are representable and Δ[0] is a final object, these exists in Δo Ens, a completely natural notion of homotopy : two arrows fo , f1 : X → Y are "simply homotopic" if there is an arrow F which makes the following diagram commutative.