Diagram rewriting and operads

We introduce an explicit diagrammatic syntax for PROs and PROPs, which are used in the theory of operads. By means of diagram rewriting, we obtain presentations of PROs by generators and relations, and in some cases, we even get convergent rewrite systems. This diagrammatic syntax is useful for practical computations, but also for theoretical results. Moreover, rewriting is strongly related to homotopy theory. For instance, it can be used to compute homological invariants of algebraic structures, or to prove coherence results. Lecture given at the Thematic school : Operads, CIRM, Luminy (Marseille), 20-25 April 2009. 1 1 PROs and PROPs Definition 1 A PRO (abbreviation for product category) is a strict monoidal category, that is a small category C equipped with some associative functor ∗ : C ×C → C and a unit object, such that the set of objects of C is N, and p ∗ q = p+ q for all p, q ∈ N. In particular, the unit object of C is 0. Since objects are already known, it suffices to give cells f : p→ q for all p, q ∈ N, together with: • a vertical composition g ◦ f : p→ r defined for any f : p→ q and g : q → r; • a horizontal composition f ∗ f ′ : p+ p → q + q defined for any f : p→ q and f ′ : p → q. This terminology will be explained in the next section. Of course, those two compositions must be associative, with units, and satisfy the interchange laws: • (h ◦ g) ◦ f = h ◦ (g ◦ f) for any f : p→ q, g : q → r, and h : r → s; • f ◦ idp = f = idq ◦ f for any f : p→ q; • (f ∗ f ) ∗ f ′′ = f ∗ (f ′ ∗ f ) for any f : p→ q, f ′ : p → q, and f ′′ : p → q; • f ∗ id0 = f = id0 ∗ f for any f : p→ q; • (g ◦ f) ∗ (g ◦ f ) = (g ∗ g) ◦ (f ∗ f ) for any f : p→ q, g : q → r, f ′ : p → q, and g : q → r; • idp ∗ idq = idp+q for all p, q ∈ N. Here are typical examples: • the PRO F, where a cell f : p→ q is a map from {1, . . . , p} to {1, . . . , q}; • the PRO ∆ ⊂ F, where a cell f : p→ q is a monotone map from {1, . . . , p} to {1, . . . , q}; • the PRO Lin(K), where a cell f : p→ q is a K-linear map from K to K (or equivalently, a q × p matrix). Here, K stands for any commutative field. In all examples, ◦ is composition of maps (or product of matrices), whereas ∗ is disjoint union (for F), ordered sum (for ∆), or direct sum (for Lin(K)). Definition 2 A PRO C is reversible if all C(p, p) are groups, and C(p, q) = ∅ for p 6= q. Hence, C is a groupoid. In that case, it suffices to give a group Cp = C(p, p) for all p, together with a horizontal composition f ∗g ∈ Cp+q defined for any f ∈ Cp and g ∈ Cq . Here are typical examples: • the reversible PRO S ⊂ F, where Sp is the p-th symmetric group; • the reversible PRO B, where Bp is the p-th braid group; • the reversible PRO GL(K) ⊂ Lin(K), where GLp(K) is the p-dimensional linear group over K; • the reversible PRO O ⊂ GL(R), where Op ⊂ GLp(R) is the p-dimensional orthogonal group. Definition 3 A PROP (abbreviation for product category with permutations) is a PRO which contains S. For instance, both F and Lin(K) are PROPs, but not ∆. 2 2 Diagrams We introduce the diagrammatic syntax [La03]: • a cell f : p→ q is pictured as a box with p inputs and q outputs: