This is nothing but the category of presheaves on Δ. As such, it is a topos. Face and degeneracy maps and simplicial identities The simplex category Δ is generated by two particularly important families of morphisms (maps), whose images under a given simplicial set functor are called face maps and degeneracy maps of that simplicial set. The face maps of a simplicial set X are the images in that simplicial set of the morphisms , where is the only (order-preserving) injection that "misses" . Let us denote these face maps by respectively, so that is a map . If the first index is clear, we write instead of . The degeneracy maps of the simplicial set X are the images in that simplicial set of the morphisms , where is the only (order-preserving) surjection that "hits" twice. Let us denote these degeneracy maps by respectively, so that is a map . If the first index is clear, we write instead of . The defined maps satisfy the following simplicial identities: if i < j. (This is short for if 0 ≤ i < j ≤ n.) if i < j. if i = j or i = j + 1. if i > j + 1. if i ≤ j. Conversely, given a sequence of sets Xn together with maps and that satisfy the simplicial identities, there is a unique simplicial set X that has these face and degeneracy maps. So the identities provide an alternative way to define simplicial sets. Examples Given a partially ordered set (S,≤), we can define a simplicial set NS, the nerve of S, as follows: for every object [n] of Δ we set NS([n]) = hompo-set( [n] , S), the order-preserving maps from [n] to S. Every morphism φ:[n]→[m] in Δ is an order preserving map, and via composition induces a map NS(φ) : NS([m]) → NS([n]). It is straightforward to check that NS is a contravariant functor from Δ to Set: a simplicial set. Concretely, the n-simplices of the nerve NS, i.e. the elements of NSn=NS([n]), can be thought of as ordered length-(n+1) sequences of elements from S: (a0 ≤ a1 ≤ ... ≤ an). The face map di drops the i-th element from such a list, and the degeneracy maps si duplicates the i-th element. A similar construction can be performed for every category C, to obtain the nerve NC of C.