We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, Jun. 2003. / Includes bibliographical references (p. 25-26). Also available on the Internet.

In this thesis we propose a description of the operad defining post-Lie algebras in terms of rooted trees and we discuss some applications of such a construction. In particular, we re-derive both the free post-Lie algebra defined in [22] and the main result of the paper [8]. Furthermore, a possible extension of the concept of symmetric brace algebra to the category of the post-Lie algebras is proposed. / Nessa tese propomos a descrição da operad que define as álgebras pós-Lie em termos de árvores enraizadas e discutimos algumas aplicações dessa construção. Em particular, nós obtemos novamente a álgebra pós-Lie livre definida em [22] e o resultado principal do artigo [8]. Além disso, uma possível extensão do conceito de álgebra brace simétrica à categoria de álgebras pós-Lie é apresentada.

In this thesis we propose a description of the operad defining post-Lie algebras in terms of rooted trees and we discuss some applications of such a construction. In particular, we re-derive both the free post-Lie algebra defined in [22] and the main result of the paper [8]. Furthermore, a possible extension of the concept of symmetric brace algebra to the category of the post-Lie algebras is proposed. / Nessa tese propomos a descrição da operad que define as álgebras pós-Lie em termos de árvores enraizadas e discutimos algumas aplicações dessa construção. Em particular, nós obtemos novamente a álgebra pós-Lie livre definida em [22] e o resultado principal do artigo [8]. Além disso, uma possível extensão do conceito de álgebra brace simétrica à categoria de álgebras pós-Lie é apresentada.

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O princípio de reconhecimento de espaços de $\\infty$-laços é que o funtor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ dado por $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induz uma equivalência entre a categoria homotópica de espectros conectivos e a categoria homotópica de $\\mathcal E^\\infty$-álgebras grouplike para qualquer resolução cofibrante $\\mathcal E^\\infty$ do operad $\\mathcal Com$ de monóides comutativos. Nesta tese é provado um princípio de reconhecimento de 2-espaços de $N$-laços para $2<N\\leq\\infty$. Quando $N=\\infty$ esse princípio afirma o seguinte: Um espectro relativo é um par de espectros $B_\\bullet$ e $Y_\\bullet$ equipados com uma sequências de aplicações pontuadas $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatíveis com as estruturas de espectros. Um espectro relativo é conectivo se o par de espectros subjacentes forem conectivos. Denotamos a categoria de espectros relativos por $\\texttt^ earrow$ e de espectros relativos conectivos por $\\texttt^ earrow_0$. Um $2E_\\infty$-operad é uma resolução cofibrante $\\mathcal E_2^\\infty$ do 2-operad $\\mathcal Com^\\shortrightarrow$ de homomorfismos de monóides comutativos. Uma $\\mathcal E^\\infty_2$-álgebra $(X_c,X_o)$ é grouplike se $X_c$ e $X_o$ forem grouplike. Denotamos a categoria de $\\mathcal E^\\infty_2$-álgebras por $\\mathcal E^\\infty_2[\\texttt]$ e a categoria de $\\mathcal E^\\infty_2$-álgebras grouplike por $\\mathcal E^\\infty_2[\\texttt]_$. O 2-espaço de $\\infty$-laços de um espectro relativo é o par de espaços $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega^{\\bullet}_{\\text} \\iota_\\bullet)$. Temos que as imagens do funtor $\\Omega^\\infty_2$ admitem uma estrutura natural de $\\mathcal E^\\infty_2$-álgebra, logo $\\Omega^\\infty_2$ define um funtor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. Existe um funtor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ e uma adjunção $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ e $\\mathcal Ho\\texttt^ earrow$ que induzem uma equivalência entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ e $\\mathcal Ho\\texttt^ earrow_0$. / The recognition principle of $\\infty$-loop spaces is that the functor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ defined by $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induces an equivalence between the homotopy category of connective spectra and the homotopy category of grouplike $\\mathcal E^\\infty$-algebras for any cofibrant resolution $\\mathcal E^\\infty$ of the commutative monoid operad $\\mathcal Com$. In this thesis a relative recognition principle of $N$-loop 2-spaces is proved for $2<N\\leq\\infty$. For $N=\\infty$ this principle states the following: A relative spectrum is a pair of spectra $B_\\bullet$ and $Y_\\bullet$ equipped with a sequence of pointed maps $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatible with the spectrum structures. A relative spectrum is connective if the underlying pair of spectra are connective. The category of relative spectra is denoted by $\\texttt^ earrow$ and the category of connective relative spectra by $\\texttt^ earrow_0$. A $2E_\\infty$-operad is a cofibrant resolution $\\mathcal E_2^\\infty$ of the commutative monoid homomorphism 2-operad $\\mathcal Com^\\shortrightarrow$. An $\\mathcal E^\\infty_2$-algebra $(X_c,X_o)$ is grouplike if $X_c$ and $X_o$ are grouplike. The category of $\\mathcal E^\\infty_2$-algebras is denoted by $\\mathcal E^\\infty_2[\\texttt]$ and the category of grouplike $\\mathcal E^\\infty_2$-algebras by $\\mathcal E^\\infty_2[\\texttt]_$. The $\\infty$-loop 2-space of a relative spectrum is the pair of pointed spaces $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega_{\\text}^{\\bullet} \\iota_\\bullet)$. The images of the functor $\\Omega^\\infty_2$ admit an $\\mathcal E^\\infty_2$-algebra structure, therefore $\\Omega^\\infty_2$ defines a functor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. The infinite relative recognition principle is that there is a functor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ and a derived adjunction $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ between the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ and $\\mathcal Ho\\texttt^ earrow$ that induce an equivalence beteween the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ and $\\mathcal Ho\\texttt^ earrow_0$.

O princípio de reconhecimento de espaços de $\\infty$-laços é que o funtor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ dado por $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induz uma equivalência entre a categoria homotópica de espectros conectivos e a categoria homotópica de $\\mathcal E^\\infty$-álgebras grouplike para qualquer resolução cofibrante $\\mathcal E^\\infty$ do operad $\\mathcal Com$ de monóides comutativos. Nesta tese é provado um princípio de reconhecimento de 2-espaços de $N$-laços para $2<N\\leq\\infty$. Quando $N=\\infty$ esse princípio afirma o seguinte: Um espectro relativo é um par de espectros $B_\\bullet$ e $Y_\\bullet$ equipados com uma sequências de aplicações pontuadas $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatíveis com as estruturas de espectros. Um espectro relativo é conectivo se o par de espectros subjacentes forem conectivos. Denotamos a categoria de espectros relativos por $\\texttt^ earrow$ e de espectros relativos conectivos por $\\texttt^ earrow_0$. Um $2E_\\infty$-operad é uma resolução cofibrante $\\mathcal E_2^\\infty$ do 2-operad $\\mathcal Com^\\shortrightarrow$ de homomorfismos de monóides comutativos. Uma $\\mathcal E^\\infty_2$-álgebra $(X_c,X_o)$ é grouplike se $X_c$ e $X_o$ forem grouplike. Denotamos a categoria de $\\mathcal E^\\infty_2$-álgebras por $\\mathcal E^\\infty_2[\\texttt]$ e a categoria de $\\mathcal E^\\infty_2$-álgebras grouplike por $\\mathcal E^\\infty_2[\\texttt]_$. O 2-espaço de $\\infty$-laços de um espectro relativo é o par de espaços $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega^{\\bullet}_{\\text} \\iota_\\bullet)$. Temos que as imagens do funtor $\\Omega^\\infty_2$ admitem uma estrutura natural de $\\mathcal E^\\infty_2$-álgebra, logo $\\Omega^\\infty_2$ define um funtor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. Existe um funtor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ e uma adjunção $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ e $\\mathcal Ho\\texttt^ earrow$ que induzem uma equivalência entre as categorias homotópicas $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ e $\\mathcal Ho\\texttt^ earrow_0$. / The recognition principle of $\\infty$-loop spaces is that the functor $\\Omega^\\infty:\\textttightarrow \\mathcal E^\\infty[\\texttt]$ defined by $\\Omega^\\infty Y_\\bullet=\\text_{\\bullet\\shortrightarrow\\infty}\\Omega^\\bullet Y_\\bullet$ induces an equivalence between the homotopy category of connective spectra and the homotopy category of grouplike $\\mathcal E^\\infty$-algebras for any cofibrant resolution $\\mathcal E^\\infty$ of the commutative monoid operad $\\mathcal Com$. In this thesis a relative recognition principle of $N$-loop 2-spaces is proved for $2<N\\leq\\infty$. For $N=\\infty$ this principle states the following: A relative spectrum is a pair of spectra $B_\\bullet$ and $Y_\\bullet$ equipped with a sequence of pointed maps $\\iota_\\bullet:B_\\bulletightarrow Y_{\\bullet+1}$ compatible with the spectrum structures. A relative spectrum is connective if the underlying pair of spectra are connective. The category of relative spectra is denoted by $\\texttt^ earrow$ and the category of connective relative spectra by $\\texttt^ earrow_0$. A $2E_\\infty$-operad is a cofibrant resolution $\\mathcal E_2^\\infty$ of the commutative monoid homomorphism 2-operad $\\mathcal Com^\\shortrightarrow$. An $\\mathcal E^\\infty_2$-algebra $(X_c,X_o)$ is grouplike if $X_c$ and $X_o$ are grouplike. The category of $\\mathcal E^\\infty_2$-algebras is denoted by $\\mathcal E^\\infty_2[\\texttt]$ and the category of grouplike $\\mathcal E^\\infty_2$-algebras by $\\mathcal E^\\infty_2[\\texttt]_$. The $\\infty$-loop 2-space of a relative spectrum is the pair of pointed spaces $\\Omega^\\infty_2\\iota_\\bullet:=\\text_{\\bullet\\shortrightarrow\\infty}(\\Omega^\\bullet Y_\\bullet,\\Omega_{\\text}^{\\bullet} \\iota_\\bullet)$. The images of the functor $\\Omega^\\infty_2$ admit an $\\mathcal E^\\infty_2$-algebra structure, therefore $\\Omega^\\infty_2$ defines a functor $\\texttt^ earrowightarrow \\mathcal E^\\infty_2[\\texttt]$. The infinite relative recognition principle is that there is a functor $B^\\infty_2:\\mathcal E^\\infty_2[\\texttt]ightarrow \\texttt^ earrow$ and a derived adjunction $(\\mathbb L B^\\infty_2\\dashv\\mathbb R\\Omega^\\infty_2)$ between the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]$ and $\\mathcal Ho\\texttt^ earrow$ that induce an equivalence beteween the homotopy categories $\\mathcal Ho\\mathcal E^\\infty_2[\\texttt]_$ and $\\mathcal Ho\\texttt^ earrow_0$.

A prop profile of a differential geometric structure is a minimal resolution of an algebraic prop such that representations of this resolution are in one-to-one correspondence with structures of the given type. We begin this thesis with a detailed account of the algebraic tools necessary to construct prop profiles; we treat operads and props, and resolutions of these through Koszul duality. Our main results can be summarized as follows. Firstly, we contribute to the work of S.A. Merkulov on the prop profiles of Poisson and Nijenhuis structures. We prove that the operad of the latter prop profile is Koszul by showing that it has a PBW-basis, and we provide a geometrical interpretation of the former in terms of an L-infinity structure on the structure sheaf of a manifold. Secondly, we construct prop profiles of compatible Poisson and Nijenhuis structures. Representations of minimal resolutions of props correspond to Maurer-Cartan elements of certain Lie algebras associated to the resolved props. Also the differential geometric structures are defined as solutions of Maurer-Cartan equations. We show the correspondence between props and differential geometry by providing explicit isomorphisms between these Lie algebras. Thirdly, in order to construct the prop profiles of compatible Poisson and Nijenhuis structures we study operads of compatible algebraic structures. By studying Cohen-Macaulay properties of posets associated to such operads we prove the Koszulness of a large class of operads of compatible structures.

Thesis (Ph.D.) - University of Glasgow, 2008. / Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, Department of Mathematics, University of Glasgow, 2008. Includes bibliographical references. Print version also available.