ON c. s. s. COMPLEXES.

doi:10.2307/2372558

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ON c. s. s. COMPLEXES.

Journal Article•DOI•

ON c. s. s. COMPLEXES.

01 Jul 1957-American Journal of Mathematics-Vol. 79, Iss: 3, pp 449

About: This article is published in American Journal of Mathematics.The article was published on 1957-07-01. It has received 124 citations till now.

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Book Chapter•DOI•

Algebraic K-theory of spaces

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Friedhelm Waldhausen¹•Institutions (1)

Bielefeld University¹

01 Jan 1985

TL;DR: The algebraic K-theory of spaces as mentioned in this paper is a variant of the standard algebraic ktheory for topology, introduced by F. Waldhausen in the late 1970s.

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Abstract: The algebraic K–theory of spaces is a variant, invented by F. Waldhausen in the late 1970’s, of the standard algebraic K–theory of rings. Until that time, applications of algebraic K–theory to topology mostly relied on the passage from a space X with base point to Z[π1X], the group ring of the fundamental group of X. Waldhausen discovered that for more sweeping applications of algebraic K–theory to topology (described below) it is necessary to replace the fundamental group π1X by the space of based loops ΩX, and the group ring Z[π1X] by the space Q(ΩX+) = lim i→∞ ΩΣ(ΩX+)

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615 citations

Book•

Simplicial homotopy theory

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Edward B Curtis¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Jan 1968

303 citations

Journal Article•DOI•

Riemann-Roch theorems for higher algebraic K-theory

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Henri Gillet¹•Institutions (1)

Princeton University¹

01 Jun 1981-Advances in Mathematics

286 citations

Journal Article•DOI•

Sheafifiable homotopy model categories

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Tibor Beke

11 Feb 2001-arXiv: Category Theory

TL;DR: In this paper, the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets, i.e., their satisfaction is a consequence of the satisfaction over the set of sets.

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Abstract: If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.

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186 citations

Journal Article•DOI•

Functors involving c.s.s. complexes

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Daniel M. Kan

01 Feb 1958-Transactions of the American Mathematical Society

183 citations

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References

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Journal Article•DOI•

Combinatorial homotopy. II

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J. H. C. Whitehead

01 Mar 1949-Bulletin of the American Mathematical Society

824 citations

Journal Article•DOI•

The geometric realization of a semi-simplicial complex

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John Milnor

01 Mar 1957-Annals of Mathematics

TL;DR: In this article, the face and degeneracy maps of K were denoted by d: Kn Kn-1 and si:K respectively, and the terminology for semi-simplicial complexes was followed.

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Abstract: homology and homotopy groups. The terminology for semi-simplicial complexes will follow John Moore [7]. In particular the face and degeneracy maps of K will be denoted by d: Kn Kn-1 and si:K. -> Kn+1 respectively. 1. The definition

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266 citations

"ON c. s. s. COMPLEXES." refers background in this paper

...Mlilnor (see [8]) ; I K I is a CW-complex of which the n-cells are in onle-to-onie correspondenice with the noin-degenerate n-simplices of K....
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Journal Article•DOI•

Functors involving c.s.s. complexes

[...]

Daniel M. Kan

01 Feb 1958-Transactions of the American Mathematical Society

183 citations

"ON c. s. s. COMPLEXES." refers background or methods in this paper

...It then may be verified that the functors Sd and Ex may be obtained by the general method of [7], Section 3 by puttinig 3 = j and X -/'....
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...7r'' A [7] A[ml A [n] It is readily verified that...
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...1) is an immediate consequence of the results of [7]....
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Journal Article•DOI•

Semi-simplicial complexes and singular homology

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Samuel Eilenberg, J. A. Zilber

01 May 1950-Annals of Mathematics

175 citations

"ON c. s. s. COMPLEXES." refers background or methods in this paper

...q), where ari is the unique non-degenerate simplex of A [n] for which (see [2]) there exist an epimorphism yt: [dim ri -e [dim ea] such that commutativity holds in the diagram...
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...The notation used will be that of [2] except that the face and degeneracy operators will be denoted by SE and vji (instead of e and ,7ni)....
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...map p:A' [2] -+K by p((0), (0, 1))-+((0), (02 1)), p((j), (O....
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...Let S: a -) SE be the simplicial sintgular funictor which assigns to a topological space X its simplicial singular complex SX (see [2]); an ?nsimplex of SX is any continuous map cr: A[n] ' X and for every map a: [m] -> [n] the n-simplex oao is the composite map...
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Journal Article•DOI•

Abstract homotopy

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Daniel M. Kan

15 Dec 1955-Proceedings of the National Academy of Sciences of the United States of America

99 citations

"ON c. s. s. COMPLEXES." refers background in this paper

...It was indicated in [3] how the usual notions of homotopy theory may be defined for cubical complexes which satisfy a certain extension condition....
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