On Street's "australian conspectus"

Question

Skimming the australian conspectus of higher category theory I noticed I have a few questions, both mathematical and historical.

1. 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?
2. 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?
3. 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?

Answer 1

For your first question, that is somehow the point with KZ-monads: they don't look idempotent (in the definition) but they do behave a little bit like idempotent monad in practice: For a KZ-monad,

• an object can only have one structure of algebra.
• It is not true that $TT X$ is the same as $ TX$
• Not every morphism between the underlying objects of two algebras is a morphisms of algebra.
• Every morphism between the underlying objects of two algebras is in a unique way a lax morphisms of algebra.

While for a really idempotent monad, every morphism of the underlying object is a morphism of algebra.

The typical example of KZ-monad is the free co-completion monad (let say under finite colimits to avoid size problems, but they are not very important):

An algebra is a finitely co-complete category, a morphism of algebra a finite co-limit preserving functor. For every functor between co-complete category you have a natural comparison maps between $Colim F(X_i)$ and $F(colim X_i)$ (which makes $F$ into a lax morphism of algebras) but it is not always true that this map is an isomorphism (this is the case when $F$ is a morphism of algebra). Also the co-completion of a co-complete category (seen as a category) is not the category itself (hence there is not real "idempotency" of the monad").

Note, that this definition in terms of finite ordinal is exactly A.Kock's definition in 'Monads for which structures are adjoint to units' (other link to the paper), which is given in terms of the existence of a certain $2$-cell satisfying some identity, coupled to A.Kock's description of the monoidal 2-category $\Delta$ en term of generator and relations given in Generators and relations for $\Delta$ as a monoidal $2$-category.

A way to explain the relation to idempotency directly on the definition, is that you can say that an idempotent monad is exactly a mondad $T$ where the two natural maps $T (\epsilon_X), \epsilon_{TX} : T(X) \rightrightarrows TT(X)$ are equal. A 'pseudo-idempotent' monad (on a 2-category) would be a monad where there is an isomorphism between these two $1$-cell $T(X) \rightrightarrows TT(X)$ satisfying some coherence condition. A lax-idempotent monad (or KZ-monad) is when you just have a non-invertible $2$-cell between these two $1$-cell also satisfying some coherence conditions. (this is the definition in Kock's paper)

Only a pseudo-idempotent monad would really be "2-dimensional generalization of idempotent monad" in the sens that they would satisfies that $TT(X)$ is isomorphic to $T(X)$ for all $X$.

For more detail, have a look to A.Kock's paper linked above where he develop the theory of such monad (he calls them KZ-doctrines)

Answer 2

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!)