On Operads

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Title: On Operads
Other Titles: Ãœber Operaden
Authors: Brinkmeier, Michael
Thesis advisor: Prof. Dr. Rainer Vogt
Thesis referee: Dr. habil. Martin Markl
Abstract: This Thesis consists of four independent parts. In the first part I prove that the delooping, i.e.the classifying space, of a grouplike monoid is an $H$-space if and only if its multiplication is a homotopy homomorphism. This is an extension and clarification of a result of Sugawara. Furthermore I prove that the Moore loop space functor and the construction of the classifying space induce an adjunction on the corresponding homotopy categories. In the second part I extend a result of G. Dunn, by proving that the tensorproduct $C_{n_1}\otimes\dots \otimes C_{n_j}$ of little cube operads is a topologically equivalent suboperad of $C_{n_1 \dots n_j}$. In the third part I describe operads as algebras over a certain colored operad. By application of results of Boardman and Vogt I describe a model of the homotopy category of topological operads and algebras over them, as well as a notion of lax operads, i.e. operads whose axioms are weakened up to coherent homotopies. Here the W-construction, a functorial cofibrant replacement for a topological operad, plays a central role. As one application I construct a model for the homotopy category of topological categories. C. Berger claimed to have constructed an operad structure on the permutohedras, whose associated monad is exactly the Milgram-construction of the free two-fold loop space. In the fourth part I prove that this statement is not correct.
URL: https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2001051822
Subject Keywords: Operads; Little Cubes; Tensorproduct of Operads; H-spaces; Strongly Homotopy Commutative; Homotopy Homomorphism; Permutohedron
Issue Date: 18-May-2001
Type of publication: Dissertation oder Habilitation [doctoralThesis]
Appears in Collections:FB06 - E-Dissertationen

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