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Since the birth of rational homotopy theory, the possibility of
extending the Quillen approach - in terms of Lie algebras - to a
more general category of spaces, including the non-simply connected
case, has been a challenge for the algebraic topologist community.
Despite the clear Eckmann-Hilton duality between Quillen and
Sullivan treatments, the simplicity in the realization of algebraic
structures in the latter contrasts with the complexity required by
the Lie algebra version. In this book, the authors develop new
tools to address these problems. Working with complete Lie
algebras, they construct, in a combinatorial way, a cosimplicial
Lie model for the standard simplices. This is a key object, which
allows the definition of a new model and realization functors that
turn out to be homotopically equivalent to the classical Quillen
functors in the simply connected case. With this, the authors open
new avenues for solving old problems and posing new questions. This
monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize,
a prestigious award for books of expository nature presenting the
latest developments in an active area of research in mathematics.
Since the birth of rational homotopy theory, the possibility of
extending the Quillen approach - in terms of Lie algebras - to a
more general category of spaces, including the non-simply connected
case, has been a challenge for the algebraic topologist community.
Despite the clear Eckmann-Hilton duality between Quillen and
Sullivan treatments, the simplicity in the realization of algebraic
structures in the latter contrasts with the complexity required by
the Lie algebra version. In this book, the authors develop new
tools to address these problems. Working with complete Lie
algebras, they construct, in a combinatorial way, a cosimplicial
Lie model for the standard simplices. This is a key object, which
allows the definition of a new model and realization functors that
turn out to be homotopically equivalent to the classical Quillen
functors in the simply connected case. With this, the authors open
new avenues for solving old problems and posing new questions. This
monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize,
a prestigious award for books of expository nature presenting the
latest developments in an active area of research in mathematics.
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