Calculating with string diagrams Ross Street Macquarie University Workshop on Diagrammatic Reasoning in Higher Education University of Newcastle Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 1 / 32
Reasons for choice of this topic § A conviction that string diagrams can be understood better than algebraic equations by most students § My experience with postgraduate students and undergraduate vacation scholars using strings § As seen by the general public, knot theory for mathematics seems a bit like astronomy for physics § A belief that string diagrams are widely applicable and powerful in communicating and in discovery § That this is “advanced mathematics from an elementary viewpoint” (to quote Ronnie Brown’s twist on Felix Klein) Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 2 / 32
Intentions § Moving from linear algebra, we will look at braided monoidal categories (bmc) and explain the string diagrams for which bmc provide the environment. § Familiar operations from vector calculus will be transported to bmc where the properties can be expressed in terms of equalities between string diagrams. § Geometrically appealing arguments will be used to prove the scarcity of multiplications on Euclidean space, a theorem of a type originally proved using higher powered methods. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 3 / 32
Arrows and categories § Already introduced in undergraduate teaching is the notation f : X Ñ A for a function taking each element x in the set X to an element f p x q of the set A . g f § In the situation X Ý Ñ A Ý Ñ K we can follow f by g and obtain a new function, called the composite of f and g , denoted by g ˝ f : X Ñ K . § There is an identity function 1 X : X Ñ X for every set X : 1 X p x q “ x . § If we now ignore the fact that X , A , K are sets (just call them vertices or objects) and that f , g are functions (just call them edges or morphisms) we are looking at a big directed graph. § If we admit the existence of a composition operation ˝ which is associative and has identities 1 X , we are looking at a category. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 4 / 32
Euclidean space § The set of real numbers is denoted by R . § A vector of length n is a list x “ p x 1 , . . . , x n q of real numbers. The set of these vectors is n -dimensional Euclidean space, denoted R n . § Algebra is about operations on sets. We can add vectors x and y entry by entry to give a new vector x ` y . We can scalar multiply a real number r by a vector x to obtain a vector rx . § For example, R 3 is ordinary 3-dimensional space. We have three particular unit vectors: e 1 “ p 1 , 0 , 0 q , e 2 “ p 0 , 1 , 0 q , e 3 “ p 0 , 0 , 1 q . Every vector x in R 3 is a unique linear combination x “ x 1 e 1 ` x 2 e 2 ` x 3 e 3 . Similarly in R n Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 5 / 32
Linear algebra § A function f : R m Ñ R n is linear when it preserves linear combinations: f p x ` y q “ f p x q ` f p y q , f p rx q “ rf p x q . § Thus we have a category E : objects are Euclidean spaces and morphisms are linear functions. We write E p V , W q for the set of morphisms from object V to object W . § For this category E , we can add the morphisms in E p V , W q : define f ` g by p f ` g qp x q “ f p x q ` g p x q . Composition distributes over this addition. Such a category is called additive. § Notice that the only linear functions f : R Ñ R are those given by multiplying by a fixed real number. So E p R , R q can be identified with R . Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 6 / 32
Multilinear algebra § Categorical algebra is about operations on categories. The category E has such an operation called tensor product: R m b R n – R mn . However, when thinking of the mn unit vectors of R mn as being in the tensor product they are denoted by e i b e j for 1 ď i ď m , 1 ď j ď n . Every element of R m b R n is a unique linear combination of these. § Bilinear functions U ˆ V Ñ W are in bijection with linear functions U b V Ñ W . § Note that R acts as unit for the tensor. § For linear functions f : R m Ñ R m 1 and g : R n Ñ R n 1 , we have a linear function f b g : R m b R n Ñ R m 1 b R n 1 defined by p f b g qp e i b e j q “ f p e i qb g p e j q . Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 7 / 32
Monoidal categories and their string diagrams § A category V is monoidal when it is equipped with an operation called tensor product taking pairs of objects V , W to an object V b W and pairs of morphisms f : V Ñ V 1 , g : W Ñ W 1 to a morphism f b g : V b W Ñ V 1 b W 1 . There is also an object I acting as a unit for tensor. Composition and identity morphisms are respected in the expected way. An example is V “ E with I “ R . § A morphism such as f : U b V b W Ñ X b V is depicted as U V W f X V § Composition is performed vertically with splicing involved; tensor product is horizontal placement. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 8 / 32
D C d c ② ❏ ② ❏ ② ✞ ❏ C B ② ❏ ✞ ② ❏ ② ✞ ❏ ② ❏ D ✞ ② ❏ ❏ ② ✞ ② ✞ b ✞ ✞ B ✞ ✞ C ✞ ✞ ✞ a B A a b c d B b B Ý Ñ A , C b D Ý Ñ B , C Ý Ñ B b C , D Ý Ñ D b C . The value of the above diagram Γ is the composite 1 B b c b d � B b B b C b D b C v p Γ q “p B b C b D 1 b 1 b b b 1 � B b B b B b C a b 1 b 1 � A b B b C q . Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 9 / 32
Here is a deformation of the previous Γ ; the value is the same using monoidal category axioms. C B c ✷ t D t ✷ t ✷ t t ✷ t t ✷ t B ✷ t t ✷ t ✷ a ✷ C ✷ ✷ ✷ ✷ d ✷ t t ✷ t ✷ t t ✷ t ✷ t t D ✷ t A t t b C B 1 B b c b 1 � B b B b C b D a b 1 b 1 � A b C b D v p Γ q “p B b C b D 1 b 1 b d � A b C b D b C 1 b b b 1 � A b B b C q . Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 10 / 32
a b The geometry handles units well: if I Ý Ñ A b B and C Ý Ñ I , then the following three string diagrams all have the same value. C C C a a b ✴ ✴ ✎✎✎✎✎✎✎ ✴ ✎✎✎✎✎✎✎ ✴ b a b ✴ ✴ A ✴ B ✾ A ✴ B ✾ ✴ ✆✆✆✆✆ ✴ A ✾ B ✴ ✴ ✾ ✴ ✴ ✾ , , The straight lines can be curved while the nodes are really labelled points. There is no bending back of the curves allowed: the diagrams are progressive. These planar deformations are part of the geometry of monoidal categories. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 11 / 32
Progressive graph on Mollymook Beach Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 12 / 32
Duals A morphism ε : A b B Ñ I is a counit for an adjunction A % B when there exists a morphism η : I Ñ B b A satisfying the two equations: A A B B η η B “ “ A ε ε A B We call B a right dual for A . Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 13 / 32
Backtracking When there is no ambiguity, we denote counits by cups Y and units by caps X . So the duality condition becomes the more geometrically “obvious” operation of pulling the ends of the strings as below. A A B B B “ “ A A B The above are sometimes called the snake equations . The geometry of duality in monoidal categories allows backtracking in the plane. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 14 / 32
Dot product, vector product and the quaternions § For any x and y in R n , the dot product x ‚ y “ x 1 y 1 ` ¨ ¨ ¨ ` x n y n defines a bilinear function R n ˆ R n Ñ R and so a linear function ‚ : R n b R n Ñ R . § For any x and y in R 3 , the vector product x ^ y “ p x 2 y 3 ´ x 3 y 2 , x 3 y 1 ´ x 1 y 3 , x 1 y 2 ´ x 2 y 1 q defines a bilinear function R 3 ˆ R 3 Ñ R 3 and so a linear function ^ : R 3 b R 3 Ñ R 3 . § The quaternions is the non-commutative ring H “ R ˆ R 3 p– R 4 q with componentwise addition and associative multiplication defined by p r , x qp s , y q “ p rs ´ x ‚ y , ry ` sx ` x ^ y q Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 15 / 32
Braiding Now suppose the monoidal category is braided . Then we have isomorphisms c X , Y : X b Y Ý Ñ Y b X which we depict by a left-over-right crossing of strings in three dimensions; the inverse is a right-over-left crossing. X Y Y X Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 16 / 32
The braiding axioms reinforce the view that it behaves like a crossing. g f “ “ “ g f Z X Y Z X X Y Z X b Y Y b Z “ “ Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 17 / 32
The following Reidemeister move or Yang-Baxter equation is a consequence. “ We will refer to these properties as the geometry of braiding. Ross Street Macquarie University Calculating with string diagrams 9 Nov 2018 18 / 32
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