Skimming the australian conspectus of higher category theory I noticed I have a few questions, both mathematical and historical.
- At about the middle of page 6, Kock-Zoberlein monads are defined as “strict monoidal 2-functors $ \text{Ord}_\text{fin}\to [{\cal K,K}]$ “, where $ \text{Ord}_\text{fin}$ is the 2-category of finite ordinals, monotone maps and pointwise order between maps. This definition surprises me, as I thought KZ-monads were the 2-dimensional analogue of idempotent monad. Where is the idempotency here?
- p. 7: “Gray then pointed out that, for $ \mathcal V = [\Delta°, Set]$ (the category of simplicial sets), homotopy limits of $ \cal V$ -functors could be obtained as limits weighted by the composite $ A\xrightarrow{L_A}{\bf Cat}\xrightarrow{N} [\Delta°, Set]$ .” Really? Wasn’t this first outlined in Bousfield and Kan’s book on homotopy limits and completions?
- p. 8: “In sheaf theory there are various ways of approaching the associated sheaf. Grothendieck used a so-called “L” construction. Applying L to a presheaf gave a separated presheaf (some “unit” map became a monomorphism) then applying it again gave the associated sheaf (the map became an isomorphism). I found that essentially the same L works for stacks. This time one application of L makes the unit map faithful , two applications make it fully faithful , and the associated stack is obtained after three applications when the map becomes an equivalence .” I’ve always found the associated sheaf construction quite striking: it’s a left exact localization which is “quadratic” in the sense that it is $ L^2$ for some endofunctor $ L$ . Here Street is telling that $ L^3$ works for stacks as it gradually builds faithfulness, fully faithfulness, and essential surjectivity of a map. What’s going on? Why is it so?
