Kristen Schemmerhorn | Concordia University Chicago (original) (raw)
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Papers by Kristen Schemmerhorn
There are many ways to present model categories, each with a different point of view. Here we’d l... more There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We’re going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subje...
Contemporary Mathematics, 2007
There are many ways to present model categories, each with a different point of view. Here we'd l... more There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and Spalinski [16] is a superb and short introduction, and the books of Hovey [22] and Hirschhorn [21] provide much more in-depth analysis. For a focus on simplicial model categories-model categories enriched over simplicial sets in an appropriate way-one can read [20]. Reaching back a bit further, there's no harm in reading the classics, and Quillen's original monograph [30] certainly falls into that category. Contents 1. Model Categories and Resolutions 2 2. Quillen Functors and Derived Functors 12 3. Generating New Model Categories 17 4. Simplicial Algebras and Resolutions in Non-abelian Settings 21 5. Resolutions in Model Categories 36 References 46
There are many ways to present model categories, each with a different point of view. Here we’d l... more There are many ways to present model categories, each with a different point of view. Here we’d like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We’re going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subje...
Contemporary Mathematics, 2007
There are many ways to present model categories, each with a different point of view. Here we'd l... more There are many ways to present model categories, each with a different point of view. Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions in non-abelian settings; for example, in his work on the homology of commutative algebras [29], it was important to be very flexible with the notion of a free resolution of a commutative algebra. Similar issues arose in the paper on rational homotopy theory [31]. (This paper is the first place where the now-traditional axioms of a model category are enunciated.) We're going to emphasize the analog of projective resolutions, simply because these are the sort of resolutions most people see first. Of course, the theory is completely flexible and can work with injective resolutions as well. There are now any number of excellent sources for getting into the subject and since this monograph is not intended to be complete, perhaps the reader should have some of these nearby. For example, the paper of Dwyer and Spalinski [16] is a superb and short introduction, and the books of Hovey [22] and Hirschhorn [21] provide much more in-depth analysis. For a focus on simplicial model categories-model categories enriched over simplicial sets in an appropriate way-one can read [20]. Reaching back a bit further, there's no harm in reading the classics, and Quillen's original monograph [30] certainly falls into that category. Contents 1. Model Categories and Resolutions 2 2. Quillen Functors and Derived Functors 12 3. Generating New Model Categories 17 4. Simplicial Algebras and Resolutions in Non-abelian Settings 21 5. Resolutions in Model Categories 36 References 46