| For a cofibrantly generated model category C , let A( C ) denote one of the following: (1) the category Mon( C ) of monoid objects in C , with respect to a general symmetric monoidal structure, (2) the category AbMon( C ) of abelian monoid objects in C , with respect to a general symmetric monoidal structure, (3) the category AbMon( C ) of abelian monoid objects in C , with respect to the cartesian structure, (4) the category Gr( C ) of group objects in C , with respect to the cartesian structure, or (5) the category Ab( C ) of abelian group objects in C (with respect to the cartesian structure).;For each case we provide a general criterion for the existence of A( C )ind (case (1) has already been dealt with by Shipley and Schwede in [18] and case (2) by Lurie in [15]). As an application, we prove the existence of AbMon( J )ind and Ab( J )ind, where J denotes the category of compactly generated topological spaces, endowed with the standard model category structure, and where abelian monoid objects and abelian group objects are defined with respect to the cartesian structure.;Another result is Theorem 1.2. Theorem 1.3 is an application of Theorem 1.2 to motivic homotopy theory: for the category C of symmetric T-spectrum objects on the Nisnevich site, endowed with the stable motivic model category structure, A( C )ind exists (here in all cases A( C ) is defined with respect to the cartesian structure) and the free functors C→FAC ind preserve weak equivalences. For A( C ) = AbMon( C ) or A( C ) = Ab( C ), A( C )ind is a monoidal model category. One may express motivic cohomology in terms of the homotopy category of Ab( C )ind.;Keywords: Algebra, Homotopy Theory;In this thesis we consider, for each of the cases (1)-(5), the question: under which conditions may one define an induced model category structure on A( C ), denoted A( C )ind if it exists, where a morphism is a weak equivalence/fibration in the induced structure iff the underlying morphism is a weak equivalence/fibration of C ?... |
