Why do Wikipedia and nLab seem to give completely incompatible definitions of the term "simplex"?

Question

nLab seems to define that a simplex is an inhabited finite totally-ordered set.

Wikipedia seems to define that simplex is a subset of real affine space satisfying certain conditions.

Q. What's going on here?

Answer 1

That nLab article is not really defining a simplex; what it is defining is the simplex category $\Delta$. This is the diagram category used to define simplicial objects and cosimplicial objects. When you talk about "concrete simplices" as particular topological spaces, what you are really doing is describing a particular cosimplicial object, namely the functor $\Delta \to \text{Top}$ sending the "abstract $n$-simplex" as defined by the nLab to the "concrete $n$-simplex" as defined by Wikipedia.

(In general, the question to ask is not "what is an X?" but "what is the category of Xes?" It's possible to present the same category in many different ways, and in particular it's possible to define the simplex category by taking its objects to be "concrete $n$-simplices" provided that you define the morphisms appropriately.)

By abstract nonsense, this cosimplicial object determines an adjunction between the category $[\Delta^{op}, \text{Set}]$ of simplicial sets and $\text{Top}$, given in one direction by geometric realization and in the other direction by taking the singular simplicial set. This is the starting point of the simplicial (as in simplicial sets) approach to homotopy theory, and in fact was a major motivating example for the invention / discovery of adjoint functors.