# Stacks and Homotopy Theory Seminar

[References]

• Note: of the talks so far, only 3 have taken place outside of the ULL topology seminar.

• Note: (a) We have now seen enough of the basics of the theory of presheaves, sheaves on spaces and sites, and Grothendieck topoi, so that we are ready to move on to the next topic. (b) One important tool that we haven't yet covered is the sheafification functor from Sets^{ (C, J)^{op} } to Shv(C, J); it would be good to cover this topic sometime.

• 4/8/09: "Sheaves on a site, the concepts of a subcanonical and canonical topology, how to take limits of sheaves," Daniel (9:52-10:45 pm; MDD 214). Note: This seminar was a make-up for last week's missed seminar (which was not held because the speaker was at a homotopy theory conference at the University of Nebraska). Synopsis: the title pretty much explains the content of the talk. (#,1)
• 3/28/09: "Sheaves of sets on a Grothendieck site," Daniel (3:30-4:45; MDD 214). Synopsis: the definition of sheaves of sets in terms of covering sieves; the category of sheaves of sets on a site; the definition of Grothendieck topos in terms of a category of sheaves; a few remarks about the motivation in algebraic geometry for studying sheaves on a site; the topology on G-Sets_df; a rough definition of stack (as a type of presheaf of groupoids); discrete G-sets form a Grothendieck topos; defining sheaves of sets on a space in terms of covering sieves; every small set is a Grothendieck site with the trivial topology; the example of sheaves on a discrete category with the trivial topology.
• 3/20/09: "Obtaining Grothendieck sites via bases," Daniel (1:04-1:54 pm; MDD 214). Note: This talk took place as a UL Topology seminar. Synopsis: recollection of the fact that Obj( (Top(X), J)|_{knock off second coordinate} ) = T_X, the topology on X, an arbitrary space; the trivial and atomic topologies; the axioms for a pretopology; the definition of site in terms of covering families; the Grothendieck topology associated to a pretopology; two examples of Grothendieck sites; motivating Grothendieck toposes via the example of discrete G-sets, each of which gives a representable functor that is a sheaf.
• 3/14/09: "Sieves, Grothendieck topologies, and Grothendieck sites," Daniel (3:10-4:10 pm). # of participants: 3. Synopsis: the definition of sieve and a construction and a lemma on them; the axioms for a Grothendieck topology (attempt to motivate them via "fluidity"); the definition of a Grothendieck site; an exposition of how every space X yields a Grothendieck site (Top(X), J), where Obj(Top(X)) is exactly the topology on X; introduction to the notion of a Grothendieck topos, and the example of the classifying topos for G as sheaves on a site.
• 3/6/09: "Fibered products, sieves, and pretopologies on categories," Daniel (1:05-1:55 pm). Note: This talk took place in the UL Topology seminar. Synopsis: review of definition of sheaf of sets on X in terms of an equalizer diagram; another example of a sheaf; sheaf(\varnothing) = {e_G}; the definition of fibered products in general and in Sets; some historical comments about Leray and Grothendieck.
• 2/27/09: "Defining sheaves with equalizer diagrams and Grothendieck sites," Daniel (1:10-2 pm). Note: This talk took place in the UL Topology seminar. Synopsis: review of definition of sheaf of abelian groups on a space; two examples; equalizers; sheaf of sets on a space defined with equalizer diagram.
• 2/6/09: "Sheaves of abelian groups on topological spaces," Daniel Davis (1-1:50 pm). Note: This talk took place in the UL Topology seminar. Synopsis: a few comments about the importance of stacks; presheaf of abelian groups on a space; a C-valued presheaf on D; the definition of sheaf of abelian groups on a space.

## References

• Edidin: the tantalizing section "Current Trends" in his "What Is ... A Stack?" Available in published form at www.math.missouri.edu/~edidin/Papers/whatisastack.pdf.
• Hartshorne: Algebraic Geometry.
• Section 6.2 of Jardine's "Generalized Etale Cohomology Theories."
• Mac Lane and Moerdijk: Sheaves in Geometry and Logic, A First Introduction to Topos Theory.
• Vistoli: Notes on Grothendieck topologies, fibered categories and descent theory. Available at http://homepage.sns.it/vistoli/descent.pdf. Note: the online version is more authoritative than the published version (see the footnote on pg. 5).

organized by Daniel (Davis).