A History of Duality in Algebraic Topology
نویسندگان
چکیده
Duality in the general course of human affairs seems to be a juxtaposition of complementary or opposite concepts. This frequently leads to poetical sounding uses of language, both in the common language and in the precision of mathematical theorems. Thus the duality of Projective Geometry: Two points determine a line; two lines determine a point. Gergonne first introduced the word duality in mathematics in 1826. He defined it for Projective Geometry. By the time of Poincaré’s note in the Comptes Rendus of 1893, duality was very much in vogue.
منابع مشابه
STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES
$Rsb{0}$-algebras, which were proved to be equivalent to Esteva and Godo's NM-algebras modelled by Fodor's nilpotent minimum t-norm, are the equivalent algebraic semantics of the left-continuous t-norm based fuzzy logic firstly introduced by Guo-jun Wang in the mid 1990s.In this paper, we first establish a Stone duality for the category of MV-skeletons of $Rsb{0}$-algebras and the category of t...
متن کاملCategorically-algebraic topology and its applications
This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...
متن کاملA duality between fuzzy domains and strongly completely distributive $L$-ordered sets
The aim of this paper is to establish a fuzzy version of the dualitybetween domains and completely distributive lattices. All values aretaken in a fixed frame $L$. A definition of (strongly) completelydistributive $L$-ordered sets is introduced. The main result inthis paper is that the category of fuzzy domains is dually equivalentto the category of strongly completely distributive $L$-ordereds...
متن کاملFunctorial semantics of topological theories
Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology. The cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. We introduce t...
متن کاملAlgebraic and topological aspects of quasi-prime ideals
In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...
متن کامل