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

Authors: Carlos Ruiz Salguero and Roberto Ruiz Salguero
Journal: Proc. Amer. Math. Soc. 81 (1981), 321-324
MSC: Primary 55U10
MathSciNet review: 593481
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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$.)

References [Enhancements On Off] (What's this?)

  • [1] John Milnor, The geometric realization of a semi-simplicial complex, Ann. of Math. (2) 65 (1957), 357–362. MR 0084138
  • [2] C. Ruiz Salguero and Roberto Ruiz, Remarks about the Eilenberg-Zilber type decomposition in cosimplicial sets, Rev. Colombiana Mat. 12 (1978), no. 3-4, 61–82. MR 533712
  • [3] R. Ruiz, Change of models in Top and $ {\Delta ^ * }S$, Doctoral Thesis, Temple University, Philadelphia, Pennsylvania, 1975.
  • [4] C. Ruiz and R. Ruiz, Conditions over a "Realization" functor in order for it to commute with finite products (to appear).

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Keywords: Simplicial set, cosimplicial topological space, geometric realization, Milnor's equivalence relation, optimal pairs, Eilenberg-Zilber $ (E-Z)$ decomposition, $ E-Z$ type cosimplicial set, Milnor's maps $ {M_1}$, $ {M_2}$, $ M$, Mac Lane decomposition, realization via $ Y({R_Y})$
Article copyright: © Copyright 1981 American Mathematical Society