Tese de Doutorado

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 $\\mathcal Ho\\texttt^ earrow_0$.