Characterization of the set-theoretical geometric realization in the non-Euclidean case

Abstract:

A helpful feature in Milnor’s geometric realization [1] of a simplicial set $X$ is that each equivalence class admits one and only one element (optimal pair) of the form $(\bar x,\bar t)$ where $\bar x$ is nondegenerate and $\bar t$ is an interior point. This realization and several other aspects of Algebraic Topology admit generalizations (R. Ruiz [3]) changing the cosimplicial topological space of the ${\Delta ^n}$ by a general one, say $Y$. This paper is devoted to establishing conditions on $Y$ which guarantee the existence of such pairs on ${R_Y}(X)$ for every simplicial set $X$. ${R_Y}$ denotes the new realization via $Y$.)