By Mark Hovey
This booklet offers an axiomatic presentation of reliable homotopy concept. It begins with axioms defining a "stable homotopy category"; utilizing those axioms, you will make numerous constructions---cellular towers, Bousfield localization, and Brown representability, to call a couple of. a lot of the booklet is dedicated to those structures and to the research of the worldwide constitution of sturdy homotopy different types.
Next, a couple of examples of such different types are awarded. a few of those come up in topology (the traditional good homotopy class of spectra, different types of equivariant spectra, and Bousfield localizations of these), and others in algebra (coming from the illustration idea of teams or of Lie algebras, as good because the derived type of a commutative ring). consequently one can follow a few of the instruments of strong homotopy conception to those algebraic occasions.
Provides a reference for normal effects and structures in good homotopy thought.
Discusses purposes of these effects to algebraic settings, equivalent to staff concept and commutative algebra.
Provides a unified therapy of a number of various events in sturdy homotopy, together with equivariant solid homotopy and localizations of the reliable homotopy classification.
Provides a context for nilpotence and thick subcategory theorems, similar to the nilpotence theorem of Devinatz-Hopkins-Smith and the thick subcategory theorem of Hopkins-Smith in strong homotopy conception, and the thick subcategory theorem of Benson-Carlson-Rickard in illustration idea.
This publication provides solid homotopy concept as a department of arithmetic in its personal correct with purposes in different fields of arithmetic. it's a first step towards making strong homotopy thought a device necessary in lots of disciplines of arithmetic.
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Additional info for Axiomatic stable homotopy theory
2 B 0 W - ) 1 / 2 ± ^ ' ^ f™ = [2(a2+g2B2^^)}-'/2[^±(a2+g2B^+^)l/2Ta]1/2. -M. -L. Ge, K. -B. Zhang where |Xu(t)> = cos2 ^ | x i i ) + - ^ s i n f l e - ^ l x i o ) + s i n 2 ie-^lxi-i), |Xi-iW> = sin2 ^ - o ' l x i i ) - - ^ s i n ^ e - ^ l x i o ) +cos 2 °-\Xi-i), \x±(t)) = ^ / P H - s i n ^ e ^ ^ l x i i ) + V^cos^Xio) + s i n 0 e - i - o t | x 1 _ 1 ) } +#)|Xoo>. 28) We then obtain ( X n ( t % l x i i ( * ) > = -«"o(l - cos )>
10) and hence j 1 w +1) = ^(i) + E i = w>+w) fc=i V>(-j) = V(l) + Hj) - lim - . -M. -L. Ge, K. -B. 12) j Separating the finite part from the infinity the H is nothing but the 6D derived in super Yang-Mills (TV = 4) with the approximation. Of course, the derivation of SD based on super Yang-Mills (N = 4) explores much larger symmetry than Lipatov model. Therefore, DNW's result shows that the Lipatov's model possesses Y(SO(6)) symmetry. 13) where u is spectrum parameter and a a free parameter allowed by YBE.
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