Profunctorial Semantics I TAL Fosco Loregian TECH March 19, 2020 Algebraic structures A group is a set equipped with operations • m : G × G ! G • i : G ! G • e : 1 ! G … you know the drill Algebraic structures Theorem (Higman-Neumann 1953) A group is a set equipped with a single binary operation = : G × G ! G subject to the single equation x=((((x=x)=y)=z)=(((x=x)=x)=z)) = y for every x; y; z 2 X. Well. This is awkward. The theory of equationally definable classes of alge- bras, initiated by Birkhoff in the early thirties, is[…] hampered in its usefulness by two defects. […T]he sec- ond is the awkwardness inherent in the presentation of an equationally definable class in terms of operations and equations. Quite recently, Lawvere, by introducing the notion - closely akin to the clones P. Hall - of an algebraic the- ory, rectified the second defect. Definition An operator domain is a sequence Ω = (Ωn j n 2 N); the elements of Ωn are called operations of arity n. Definition An interpretation E of an operator domain Ω consists of a n pair (E; (f! j ! 2 Ωn; n 2 N)) where f! : E ! E is an n-ary operation on the set E called the carrier of E. An operator domain can be represented as a (rooted) graph: for example, for groups i 0 e 1 m 2 Way better to use functors. A Lawvere theory is an identity-on-objects functor p : Finop !L that commutes with finite products. Unwinding the definition: • L is a category with the same objects as Fin, the category of finite sets and functions; • p is a functor that acts trivially on objects • The only thing that can change between Fin and L is the number of morphisms [n] ! [m]. Equivalently: p is a promonad on the opposite of Fin, regarded as an object of the bicategory of profunctors, that preserves the monoidal structure. L is the Kleisli object of p. ( ) identity on obj n o monads in Prof left adjoints ⇆ p:Finop Finop p:[L;Set]![Finop;Set] • The trivial theory is the identity funtor op ! op 1Fin : Fin Fin • Since p preserves products, it is uniquely determined by its value on [1]. This means that if p : Finop !L is a Lawvere theory, then every object of L is Xn if p[1] = X. • The only difference between Fin and L is thus the set of morphisms [n] ! [m], added on top of those in Fin. i L Grp = [0] [1] [2] e m A model for a Lawvere theory p is a product-preserving functor ` : D! Set. The category Mod(p) for a Lawvere theory is a full, reflective subcategory of the category [D; Set] of all functors D! Set. Theorem The following conditions are equivalent: • ` is a model for a Lawvere theory D; • The composition ` ◦ X preserves finite products; • The composition ` ◦ X is representable (with respect to the inclusion J : Fin ! Set), i.e. ∼ `(X[n]) = Set(J[n];X1) for some X1 2 Set. As a consequence, the square (p) −−−−!r [D; ] Mod? ?Set ? ? uy y_◦X Set −−−−! [Finop; Set] [J;1] is a pullback. • Mod(p) is a reflective subcategory of [D; Set]. We write r! a r for the resulting adjunction. • The functor u is monadic, with left adjoint f. • This sets up a functor M : ThL(Fin) ! Mnd<!(Set) because the monad uf above is finitary. There is an equivalence of categories between ThL(Fin) and Mnd<!(Set). We have to construct a functor in the opposite direction, Z : Mnd<!(Set) ! ThL(Fin); given T , we consider the F T composition Fin ,! Set −−! SetT and its bo-ff factorization, in a square ff / T LopO SetO bo F T Fin / Set J • the left vertical arrow is a Lawvere theory almost by definition. • SetT has the universal property of the category of L-models. There is a 2-monad S~ : Prof ! Prof whose algebras are exactly promonoidal categories. Given a profunctor p : A B between promonoidal categories (A; P;JA); (B; Q;JB): • p is a pseudo-S~-algebra morphism; • The cocontinuous left adjoint p^ associated to p is strong monoidal with respect to the convolution monoidal product on presheaf categories; Assume the promonoidal structures P; Q on A; B are representable; then, the conditions above are in turn equivalent to C B ∗ • Both mates p : A ! PB che p : B ! P A are strong monoidal wrt convolution on their codomains. Theorem There is a strong monoidal equivalence of categories ∼ [Fin; Set] = End<!(Set) If the LHS is endowed with the monoidal structure induced by composition of endofunctors; this is called the substitution monoidal product of functors F; G : Fin ! Set: Z n F ∗ G : m 7! F n × (Gm)n The substitution monoidal product is a highly non-symmetric, right closed monoidal structure (not left closed). The category [Fin; Set] works as base of enrichment. From [Garner] From now on we blur the distinction between the ∼ categories [Fin; Set] = End<!(Set): • A finitary monad is a monoid in End<!(Set), i.e. a End<!(Set)-category with a single object, i.e. a [Fin; Set]-category with a single object. • A Lawvere theory is a [Fin; Set]-category that is absolute (Cauchy-, Karoubi-)complete as an enriched category and generated by a single object. • Lawvere theories form a reflective subcategory in finitary monads; reflection is the enriched Cauchy completion functor. Equivalently, • A Lawvere [Fin; Set]-category is an enriched category where every object A is the tensor y[n] ⊙ X for a ∼ distinguished object X = y[1] ⊙ X. All such categories are enriched-Cauchy complete. • A [Fin; Set]-category is a special kind of cartesian multicategory: one where a multimorphism f : X1 :::Xn ! Y is such that X1 = X2 = ··· = Xn. Generalisations/extensions: • let N be the discrete category over natural numbers; • let P be the groupoid of natural numbers; The categories [N; Set] and [P; Set] become monoidal with respect to substitution products ∗N ; ∗P : a a F ∗N G = Gk × Xn × · · · × Xn P 1 k k2N ~nj n =n Z i k;~n ( P ) ∗ × × · · · × × F P G = Yk Xn1 Xnk P ni; n PRO(P)S ∗N and ∗P -monoids are respectively non-symmetric and symmetric operads. • A PRO is an identity-on-objects strong monoidal functor p : N !P. P is possibly non-cartesian. • A PROP is an identity-on-objects strong monoidal functor p : N !P. P is symmetric monoidal. These are, of course, other examples of promonoidal promonads. PRO(P)s and operads Every PRO p : Finop !T gives rise to the operad O(T ) = (T (n; 1) j n 2 N). 2. Conversely, any operad (O(n) j n 2 N) gives rise to a pro T (O), where a T (O)(n; m) = O(k1) × · · · × O(km): k1+:::+km=n (It would be helpful to imagine a picture of m trees stacked vertically.) If we begin with an operad O, we have O = O(T (O)). (This is because T (O)(n; 1) = O(n), according to the above formula.) On the other hand, if we start with a PRO T , then there exists a canonical map of PROs T (O(T )) !T , given by, for each n and m, a canonical function a T (k1; 1) × · · · × T (km; 1) ! T (n; m)(?) k1+···+km=n induced from the monoidal product on T. This sets up an adjunction T : Opd[S] ⇆ PRO[P]: O with fully faithful left adjoint, so that [symmetric] operads can be regarded as a PRO[P]s T such that each function (*) is bijective. Re-enact [Garner] away from Set. Let V be a locally presentable base of enrichment; let F(V) be the subcategory of finitely presentable objects: • F(V) is the free finite weighted cocompletion of the point; • There is a strong monoidal equivalence of categories ∼ [F(V); V] = [V; V]<! between functors F(V) !V and finitary endo-V-functors; • The V-substitution product on LHS is Z B B F ∗ G = A 7! FB ⊗V (GA) • There is an equivalence of categories between finitary V-monads and enriched-Cauchy-complete categories generated by a single object under iterated finite powers. • Models for a Lawvere theory correspond to algebras for the associated finitary monad; free models are free agebras are representables in Alg(T; C) = [F(V); V]-Cat(T; C) ∼ = [F(V); V]-Cat(T;^ C) = Mod(T;^ C) class of lims finite × D-limts finite powers weighted D-limits bicat × theory Finop completion of {∗} completion of {∗} completion of {∗} completion of {∗} semantics Set Set V V Prof eq. with finitary D-accessible [F(V );V ]-monoids [?;V ]-monoids ??? Profunctorial semantics • Characterise the free carbicat CB(∗) on a singleton: see link here); • Check if the univ property of Fin remains true for CB(∗); • Take CB(∗) = F , and consider its free cocompletion in the bicolimit sense • Prove that ∼ [P F; P F ] = [CB(∗);PF ] ∼ = PF monoidally; ⊙-monoids := monoids in PF wrt composition in [P F; P F ]. Profunctorial semantics • Prove that there is a syntax-VS-semantics adjunction here: theories are promonoidal promonads T on (a 1-skeleton of) CB(∗), and models are carbicat homomorphisms Kl(T ) ! Prof. There is an equivalence ∼ ftheoriesg = f??? monadsg • Let PROs come into play: analogue of the adjunction between PROs and operads. Bibliography • Lawvere, F. William. ”Functorial semantics of algebraic theories.” Proceedings of the National Academy of Sciences of the United States of America 50.5 (1963): 869. • Linton, Fred EJ. ”Some aspects of equational categories.” Proceedings of the Conference on Categorical Algebra.