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Model categories provide a general abstract framework in which to do homotopy theory. This will be a purely introductory, expository talk in which we will introduce the definition of a model category and run through the axioms at a leisurely pace giving examples and motivation. After covering the minimal definition we will briefly discuss cofibrantly generated model structures, simplically enriched model structures, and left Bousfield localization. Time permitting, the goal will be to describe a useful simplicial model category structure on simplicial presheaves on Cartesian spaces, a large family of objects that encompasses smooth manifolds and stacks. The basic concepts from category theory that we need will be briefly reintroduced as necessary. |