What are Picard categories, where can I learn more about them, and why should I care to?

Question

I have the category-theoretic background of the occasional stroll through MacLane's text, so excuse my ignorance in this regard. I was trying to learn all that I could on the subject of tensor algebras, and higher exterior forms, and I ran into the notion of cohomological determinants. Along this line of inquiry, I ran into the general use of the notion of a Picard category, and kept running into frustration in trying to find some sort of exposition of what these structures are. So where can I find out more about these structures, and about (cohomological) determinants in K-theory, which seem to be a hot topic among AG, AT, and RT researchers alike at the moment.

Answer 1

I ran across Picard categories in a totally different area of mathematics, but maybe it helps.

In short, a Picard category is a group object in the category of groupoids.

Picard categories come up when you study Picard stacks. Roughly, a Picard stack is a sheaf of Picard categories. The classical example is taking a two-term (perfect??) complex of sheaves, and associating to such a complex the groupoid quotient of one term by the other. This is important when you want to produce a geometric object from such a complex. This is an important tool in defining virtual fundamental classes as in http://arxiv.org/abs/alg-geom/9601010.

Before I tell you too many things that are not true, here are the references I know of:

Lecture notes of Martin Olsson, Lecture 5. You can even watch it on video.

The definitive reference is Exposé XVIII of SGA 4.

And finally there are very friendly and down-to-earth lectures of Barbara Fantechi at http://www.openeya.org/sissa/ [link dead]. I think lecture 3 or lecture 4 is about Picard categories.

Answer 2

Determinants are discussed (in a language relevant to this current question) in this MO question.

One place Picard categories naturally appear is as fundamental (aka Poincare) groupoids -- specifically those of infinite loop spaces (which are a refined homotopical version of abelian group objects in spaces, so form a natural source of abelian groups in categories. In fact Picard categories are equivalent to spectra which have only two consecutive homotopy groups, which up to shift we may as well take to be $\pi_0$ and $\pi_1$ -- one direction is given by the fundamental groupoid.

The important example of the Picard category of graded lines over a field arises this way from the algebraic K-theory spectrum of the field, via the determinant line construction (see eg Beilinson's paper referred to in the answers to the above MO link).

Another example important in rep theory is the Picard category of sheaves of twisted differential operators. This is discussed in detail in the famous "Proof of Jantzen Conjectures" paper of Beilinson-Bernstein.

Answer 3

Regarding the transcription of H X Sinh’s thesis [pdf] (still in progress), professor J Baez pointed out [link]:

It may be worth mentioning that Hoàng Xuân Sính , often called "Madame Sinh" in the literature, was a Vietnamese student of Grothendieck who in her thesis studied what we now call 2-groups (monoidal categories where all objects and morphisms are invertible). She called them Gr-categories, and she classified them in terms of group cohomology.

She showed that any Gr-category gives an element of $H^3(G,A)$ where G is the group of isomorphism classes of objects and A is the group of endomorphisms of the unit object. This is sometimes called the Sinh invariant. It comes from the associator, which gives a 3-cocycle thanks to the pentagon identity. Replacing the Gr-category by a monoidally equivalent one changes this 3-cocycle by a cohomologous one, so the Sinh invariant is a cohomology class.

[Gr-catégories], H X Sinh

The theory of Picard categories appears prominently in the thesis.

Answer 4

Pursuing stacks should fit here.

There are plenty of uses of Picard categories (see the appendix), and in fact, the appendix list some examples of Picard 2, 3-categories.