category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
String diagrams are a graphical calculus for expressing operations in a monoidal category. The idea is roughly to think of objects in a monoidal category as “strings” and a morphism from one tensor product to another as a node which the source strings enter and the target strings exit. Further structure on the monoidal category is encoded in geometrical properties on these strings. For instance
putting strings next to each other denotes the monoidal product, and having no string at all denotes the unit;
braiding strings over each other corresponds to – yes, the braiding (if any);
bending strings around corresponds to dualities on dualizable objects (if any).
Many operations in monoidal categories that look rather unenlightening in symbols become very obvious in string diagram calculus, such as the trace: an output wire gets bent around and connects to an input.
There are many additional structures on monoidal categories, or similar structures, which can usually be represented by encode further geometric properties. For instance:
in monoidal categories which are ribbon categories the strings from above behave as if they have a small transversal extension which makes them behave as ribbons. Accordingly, there is a twist operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 180 degrees.
in a traced monoidal category, the trace can be represented by bending an output string around to connect to an input, even though if the objects are not dualizable the individual “bends” do not represent anything.
in monoidal categories which are spherical all strings behave as if drawn on a sphere.
string diagrams can be extended to represent monoidal functors in several ways. One nice way is described in these slides, and can also be done with “3D regions” as drawn here.
there is also a string diagram calculus for bicategories, which extends that for monoidal categories regarded as one-object bicategories. Thus, the strings now represent 1-cells and the nodes 2-cells, leaving the two-dimensional planar regions cut out by the strings to represent the 0-cells. This makes it manifest that in general, string diagram notation is Poincaré dual to the globular notation: where one uses $d$-dimensional symbolds the other uses $(2-d)$-dimensional symbols.
Similarly, one can extend this to “surface diagrams” for 3-categories (including monoidal bicategories) and so on; see for instance here.
As explained here, in the presence of certain levels of duality it may be better to work with diagrams on cylinders or spheres rather than in boxes. This relates to planar algebras and canopolises?.
A string diagram calculus for monoidal fibrations can be obtained as a generalization of C.S. Peirce’s “existential graphs.” The ideas are essentially contained in (Brady-Trimble 98) and developed in (Ponto-Shulman 12) , and was discussed here.
String diagrams for closed monoidal categories (see also at Kelly-Mac Lane graph) are similar to those for autonomous categories, but a bit subtler, involving “boxes” to separate parts of the diagram. They were used informally here, but can also be done in essentially the same way as the proof nets used in intuitionistic linear logic; see this paper. Proof nets for classical linear logic similarly give string diagrams for *-autonomous categories.
See the article by Selinger below for more examples.
John Baez, QG Seminar Fall 2000 (web), Winter 2001 (web), Fall 2006 (web).
John Baez and Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, arXiv
The Catsters (Simon Willerton), String diagrams (YouTube)
String diagrams appeared first in Max Kelly and Laplaza’s paper on coherence for compact closed categories
and again in Ross Street’s work with André Joyal in the mid-80’s:
Andre Joyal and Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112; MR92d:18011. (pdf)
Andre Joyal and Ross Street, The geometry of tensor calculus II. (pdf.
An early amplification of the use of string-diagram notation as an alternative for the traditional index-calculus for tensors is due to Roger Penrose.
Probably David Yetter was the first (at least in public) to write them with “coupons” (a term used by Nicolai Reshitikhin and Turaev a few months later) to represent maps which are not inherent in the (braided or symmetric compact closed) monoidal structure. See also these:
Peter Freyd, David Yetter, Braided compact closed categories with applications to low dimensional topology Advances in Mathematics, 77:156–182, 1989.
Peter Freyd and David Yetter, Coherence theorems via knot theory. Journal of Pure and Applied Algebra, 78:49–76, 1992.
David Yetter, Framed tangles and a theorem of Deligne on braided deformations of tannakian categories In M. Gerstenhaber and Jim Stasheff (eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics 134, pages 325–349. Americal Mathematical Society, 1992.
For more on the history of the notion see the bibliography in (Selinger 09).
Ross Street, Low dimensional topology and higher-order categories – talks about surface diagrams and includes some pictures (PS version only).
Ross Street, “Categorical structures” – discusses string diagrams for bicategories.
String diagrams for monoidal categories are discussed in
Andre Joyal and Ross Street, The geometry of tensor calculus I, Advances in Math. 88 (1991) 55-112; MR92d:18011. (pdf)
Andre Joyal and Ross Street, The geometry of tensor calculus II. (pdf.
Andre Joyal and Ross Street, Planar diagrams and tensor algebra, available here.
For traced monoidal categories in
For indexed monoidal categories in
Geraldine Brady, Todd Trimble, A string diagram calculus for predicate logic (1998)
Kate Ponto, Michael Shulman, Duality and traces for indexed monoidal categories, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 (arXiv:1211.1555)
The generalization of string diagrams to one dimension higher is discussed in
John Barrett, Catherine Meusburger, Gregor Schaumann, Gray categories with duals and their diagrams, available here.