This course M721 was given in Fall 2019 at Indiana University by Jim Davis. It investigates the algebraic topology of diagrams of spaces: functors from a fixed category to the category of topological spaces, generalizing the notion of a G-space. It is a panoramic view of categories, algebraic topology, simplicial methods, spectra, and algebraic K-theory. Exercises for the course are found here. The basic reference for the course is the paper Davis-Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory. Question and comments about the course are welcome.
It will be accessible to anyone who his familiar with algebraic topology at the level of Davis-Kirk, Lecture Notes in Algebraic Topology and, in particular, to those familiar with fibrations and cofibrations. Lück has numerous surveys on the subject of isomorphism conjectures on his webpage. Other useful references are May, A Concise Course in Algebraic Topology and Riehl, Category Theory in Context.
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The nonconnective K-spectrum of a small additive category constructed by Pedersen-Weibel, A Nonconnective Delooping of Algebraic K-theory.
Algebraic K-theory; Additive Categories
From James Davis
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Balanced products as left adjoints for mapping…
From James Davis
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C-sets and introduction to the orbit category
From James Davis
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The first three weeks of the course were not filmed (handwritten notes available upon request). Topics included the definition of a G-CW-complex (built from cells of…
Cohomology of Groups
From James Davis
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One can find a lot of info on Compactly Generated Weak Hausdorff spaces on the web (especially the notes of Strickland), but one can start with the books of May and…
Compactly Generated Weak Hausdorff
From James Davis
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Equivariant homology theories were defined in Lück, Chern characters for proper equivariant homology theories and applications to K- and L-theory, but…
Equivariant Homology Theories
From James Davis
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Lück has several surveys on the Farrell-Jones Conjecture on his website.
Farrell-Jones Isomorphism Conjectures
From James Davis
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Functorial models for BC, EC, ho(co)limits, and…
From James Davis
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Fundamental Groupoids and Local Coefficient…
From James Davis
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Groupoids and Equivalences of Categories
From James Davis
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For homotopy colimits and limits, I like the approach of Davis-Lück the best. The classical reference is Bousfield-Kan, Homotopy Limits, Completions and…
Homotopy Colimits
From James Davis
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The proof of the ladder and gluing lemmas can be found in May-Ponto More Concise Algebraic Topology.
Homotopy Colimits and Mapping Telescopes
From James Davis
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References for this material are given in the book by Strom mentioned above as well as Munson-Volic Cubical Homotopy Theory. I don't know the geodesic route to…
Homotopy Pushouts and Pullbacks
From James Davis
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A good reference for exactness properties of classical colimits and limits is Weibel's book An Introduction to Homological Algebra.
Limits and Colimits II
From James Davis
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ZC-Homological Algebra (Limits and Colimits VI)
From James Davis
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