In reply to P. May's message. I do not know of any work on "pseudo triple category". On the other hand, the pseudo double category Rng of rings, homomorphisms and bimodules is briefly considered in our first work on pseudo double categories, subsection 5.3 M. Grandis - R. Paré, Limits in double categories, Cah. Topol. Géom. Diff. Catég. 40 (1999), 162-220 as a substructure of the pseudo double category of Ab-categories, Ab- functors and Ab-profunctors. As an interesting construction, in this pseudo double category: - the "tabulator" of a bimodule u: R -+-> S (u is a left-R, right- S bimodule) (i.e., its double limit) can be constructed as a ring of triangular 2x2 matrices, with "matrix product" r x r' x' rr' rx'+xs' . = 0 s 0 s' 0 ss' (with r. r' in R, s, s' in S and x, x' in u). Tabulators are crucial for double limits, since all of them can be constructed from double products, double equalisers and tabulators. [ In a bicategory, the tabulator (of the vertical identity of A) is the cotensor 2*A, and the previous result amounts to the construction of weighted limits, in R.H. Street, Limits indexed by category valued 2-functors, J. Pure Appl. Algebra 8 (1976), 149-181. ] Best regards Marco Grandis On 17 Nov 2005, at 05:26, Peter May wrote:
In his posting today, John Baez advertised the slogan:
FIBRATIONS OVER THE BASE SPACE B WITH FIBER F ARE "THE SAME" AS HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.
He hedged it with a ``dose of vagueness'', but in fact I proved a completely precise and general version of exactly this result in ``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. Using Moore loops on B, LB, one has a topological monoid, and one also has the topological monoid Aut(F) of homotopy equivalences of $F$. A ``transport'' is a homomorphism of topological monoids from LB to Aut(F). Allowing F to vary by a homotopy equivalence, one can define an equivalence relation on transports such that the equivalence classes are in natural bijective correspondence with the equivalence classes of `fibrations over the base space B with fiber F'. One can generalize the context by allowing fibers in some nice category and prove the same result. See opus cit, Theorem 14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking about categorification, but I would imagine that the methods categorify.
Some questions from more recent work (in progress in fact):
1. In work on (equivariant, stable) parametrized homotopy theory, Johann Sigurdsson and I need and develop duality in ``symmetric bicategories B'', which are not to be confused with the reasonably standard symmetric monoidal bicategories. Rather there must be a prescribed involution on the bicategory B, a pseudo-equivalence t between B and its opposite bicategory (not completely general: we find it helpful to require tt = id on 0-cells). For example, the standard bicategory whose 0-cells are rings, whose 1-cells R >--> S are (S,R)-bimodules, and whose 2-cells are maps of bimodules is symmetric; t takes R to its opposite ring and takes an (S,R)-bimodule to the same Abelian group regarded as a (tR,tS)-bimodule. Is there a pre-existing theory of such bicategories and their duality theory, analogous to duality theory in symmetric monoidal categories?
2. The example in 1 is additionally a symmetric monoidal bicategory under the tensor product over Z, and there is an analogous bicategory starting with a commutative ground ring replacing Z. These assemble nicely into a tricategory of commutative rings, algebras, bimodules, and maps of bimodules. Moreover, the bicategory in 1 is actually part of a pseudo double category with maps of algebras as vertical 1-cells. Promoting this to the tricategory just mentioned, one has maps of commutative rings as vertical 1-cells and maps of algebras as vertical 2-cells. I don't know a name for the resulting notion, something like a pseudo triple category. Here again, what is most important is duality theory. Has anybody studied such structures? There are derived versions of the cited example, and such structures also appear naturally in our work on parametrized homotopy theory. Thankfully, we do not (yet) seem to need tetracategories!