TITLE: On Street's "australian conspectus"
QUESTION [8 upvotes]: 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?

REPLY [5 votes]: For 2., there is a sense in which you are right but that is `in hindsight'.  The nature of homotopy limits from a categorical point of view was not clear (at least to me). There were several approaches knocking around and sorting them out categorically was not clear. I remember reading B & K and trying to link it with Vogt's ideas. I learnt a lot from John Gray's article and from there Bourn and Cordier started clarifying things, but the ideas and terminology of weighted limits were slow to be fully understood outside a small group of enriched category theorists. Kelly's book appeared in 1982 but the ideas, although available earlier than that took time to be absorbed, and then Gray's insights on B & K's holims became clearer (to me). I am unable to comment on B & K's insights, but I always felt they came from work on the derived functors of the limit functor and various other situations outside a purely categorical context. They were doing homotopy theory and using categorical constructions, not doing category theory.  Were they aware of indexed / limits etc.?  I do not know. 
I chatted to John Gray about this but he had realised the observation that Ross mentioned whilst trying to understand what B & K were doing from a categorical viewpoint and had compared their approach with some points I had made in an article in the Cahiers. (I also noted that Illusie in his thesis had developed the total derived functor of the limit functor independently, but that is not an answer to your question, just a comment in passing!)