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Multisorted Lawvere Theories Contents 1 Todd Trimble Home Page All Pages Feeds multisorted Lawvere theories Contents 1. Background 2. Crude monadicity Existence of the left adjoint 3. Further consequences 4. References This material is in reply to a query of John Baez at the n-Category Café. 1. Background This section is just to provide some background for notions of multisorted Lawvere theories, to state a form of John’s desired theorem, and to fix some general notation. Let i: Fin → Set be the inclusion of the category of finite sets {1, …, n} into the category of all sets. Let Λ be a set, and let i ↓ Λ, or just Fin/Λ, be the comma category whose objects are pairs (S, x: S → Λ) where S is a finite set. We can think of such a pair (S, x) as a finite set fibered over Λ, so that all but finitely many fibers are empty. The category Fin/Λ has finite coproducts (they are just coproducts in each separate fiber). A key fact we need is that Fin/Λ is the free category with finite coproducts generated by Λ. More formally: treat the set Λ as a discrete category; there is an evident functor ι: Λ → Fin/Λ which takes λ ∈ Λ to the evident fibered set λ: 1 → Λ. Lemma 1.1. Let C be a category with finite coproducts, and let f: Λ → C be a functor. There is (up to unique isomorphism) a unique coproduct-preserving functor ˜f : Fin/Λ → C that extends f along ι: Λ → Fin/Λ. If C has chosen coproducts, we can demand preservation on the nose. ˜ The idea of proof is that f takes a fibered set x: S → Λ to the coproduct ∑i ∈ Sf(x(i)) in C. Dually, the free category with finite products generated by Λ is (Fin/Λ)op. For C a category with finite products, the product-preserving extension of f: Λ → C to (Fin/Λ)op → C will be denoted f − . Definition 1.2. A Λ-sorted Lawvere theory is a category with finite products Θ equipped with a finite- product preserving functor k: (Fin/Λ)op → Θ that is the identity on objects. (Thus we can think of Θ as having chosen products.) Yes, this is “evil”, but it can be a convenient (and harmless) evil. Even in the usual one-sorted case, if we purge this evil, we can have a Lawvere theory in which the generic object x is isomorphic to its square x2. Whereas it is convenient to think of them as distinct so that n-ary operations in the theory are identified with morphisms xn → x, and the arity is well-defined. But anyone who is offended by this evil can rewrite matters, at the cost of some extra words (replacing “identity/bijective on objects” with “essentially surjective”). As usual, a model of a Lawvere theory Θ in a category with finite products C is a product-preserving functor F: Θ → C. A homomorphism of models is a natural transformation between such functors. Thus we define the category of models, ≔ ModC(Θ) Prod(Θ, C), Λ and there is a forgetful functor ModC(Θ) → C obtained by an obvious composition: Prod ( k , C ) Prod(Θ, C) → Prod((Fin/Λ)op, C) ≃ CΛ. We will prove Theorem 1.3. Let C be a category that admits general colimits and finite products that distribute over Λ colimits. Then the forgetful functor ModC(Θ) → C is monadic. There are a number of ways of proving this theorem. One idea is to invoke a monadicity theorem, say a crude monadicity theorem, and in fact we will pursue this first. If all we are after is monadicity, then this approach is arguably overkill: the monadicity can be proved in a much “softer” and more direct fashion (by adapting some of Kelly’s ideas on operad theory). This approach is touched upon here; perhaps at some point we’ll go through it in more detail. However, an extra bonus of the approach via crude monadicity is that it makes cocompleteness of ModC(Θ) immediate, by an old result of Linton. 2. Crude monadicity Recall the crude monadicity theorem: a functor U: A → B is monadic if 1. U has a left adjoint F: B → A, 2. The category A has reflexive coequalizers, 3. The functor U: A → B preserves reflexive coequalizers and reflects isomorphisms. In the case of the forgetful functor U: Prod(Θ, C) → CΛ, these conditions are not difficult to check, although for our purposes the existence of the left adjoint will require some preface, so we save this for later. Lemma 2.1. U: Prod(Θ, C) → CΛ reflects isomorphisms. Proof. On account of Prod((Fin/Λ)op, C) ≃ CΛ, this is equivalent to saying the functor G: Prod(Θ, C) → Prod((Fin/Λ)op, C), induced from the product-preserving functor k: (Fin/Λ)op → Θ, reflects isomorphisms. Suppose M, N: Θ → C are product-preserving functors and ψ: M → N is a natural transformation, such that the whiskering ψk: Mk → Nk is invertible. But such a natural transformation is invertible iff all its components ψk(x) are invertible. Since k is the identity on objects, this means each component ψ(x) is invertible, so ψ itself is invertible, as was to be shown. ▮ Lemma 2.2. Suppose C has finite products and reflexive coequalizers, and that products distribute over reflexive coequalizers. Then the product functor C × C → C preserves reflexive coequalizers. The following proof is based on a neat argument given by Steve Lack. Another proof may be based on lemma 0.17 from Johnstone’s Topos Theory, page 4 (you can see it by looking inside the book at Amazon). → Proof. The walking reflexive fork a → b → a is a sifted category D, meaning precisely that the diagonal functor Δ: D → D × D is a final functor. (See Adámek, Rosický, Vitale for basic information on sifted categories and sifted colimits; see particularly their example 1.2.) Now suppose we have a reflexive fork diagram D → C × C given by two reflexive fork diagrams F, G: D → C in our category C. We have ≅ (colimd ∈ DF(d)) × (colimd ′ ∈ DG(d′ )) colimd ∈ D[F(d) × colimd ′ ∈ DG(d′ )] ≅ colimd ∈ Dcolimd ′ ∈ D[F(d) × G(d′ )] ≅ colim ( d , d ′ ) ∈ D × DF(d) × G(d′ ) ≅ colimd ∈ DF(d) × G(d) where the first two isomorphisms are come from products distributing over reflexive coequalizers, the third comes from a “Fubini theorem”, and the last from the finality of Δ: D → D × D. This shows the product applied to a reflexive coequalizer of ⟨F, G⟩: D → C × C is canonically isomorphic to the reflexive coequalizer of the product F × G: D → C, as was to be shown. ▮ Proposition 2.3. If products distribute over reflexive coequalizers in C, then Prod(Θ, C) admits reflexive coequalizers and U: Prod(Θ, C) → CΛ preserves them. Proof. Let [Θ, C] denote the category of all functors Θ → C; this certainly has reflexive coequalizers if C has them. For any category A, let AD be the category of reflexive fork diagrams in A. Let j: Prod(Θ, C) ↪ [Θ, C] be the full inclusion. We show Prod(Θ, C) is closed under reflexive coequalizers as computed in [Θ, C], i.e., that the composite (where “colim” means coequalizer) D D Prod(Θ, C) ↪ jD[Θ, C] → colim[Θ, C] factors through the full inclusion j: Prod(Θ, C) ↪ [Θ, C]. This will mean both that Prod(Θ, C) has reflexive coequalizers and that j preserves them, whence U which is the composite ↪ Λ Prod(Θ, C) j[Θ, C] → [ kι , 1C ] [Λ, C] = C also preserves them. But if θ1, θ2 are two objects of Θ, and if F: D → Prod(Θ, C) is a reflexive fork, then colim jF preserves the product θ1 × θ2 since ≅ [colimd ∈ DF(d)](θ1) × [colimd ′ ∈ DF(d′ )](θ2) colimd[F(d)(θ1)] × colimd ′ [F(d′ )(θ2)] ≅ colimd[F(d)(θ1) × F(d)(θ2)] ≅ colimdF(d)(θ1 × θ2) ≅ [colimdF(d)](θ1 × θ2) where the first and last isomorphisms hold since colimits in [Θ, C] are computed pointwise, the second isomorphism holds by the lemma, and the third holds since each F(d): Θ → C is product-preserving. ▮ Existence of the left adjoint Now we show that U: Prod(Θ, C) → CΛ has a left adjoint Free: CΛ → Prod(Θ, C), under the following assumptions C has all colimits (we don’t actually need all, but that’s certainly the most natural assumption), C has finite products, and Products distribute over colimits. We will call such a category cartesian monoidally cocomplete, or CMC. (Another name for it could be ‘cartesian 2-rig’, and indeed some of the material below is a cartesian analogue of the formalism of typed operads in the context of 2-rigs; see Baez-Dolan, section 2.3.) It’s possible to dive right in and write down a coend formula for the asserted left adjoint Free, but it is clarifying to introduce it by telling a little abstract story. The basic theme is the idea of the free cartesian monoidally cocomplete category on a set Λ and suitable monads thereon as being “Λ-typed cartesian operads”; this kind of idea has been used by various authors (Kelly, Baez-Dolan) in the symmetric monoidal case instead of the cartesian monoidal case undertaken here, but the development is conceptually almost identical. We will see that typed or multisorted Lawvere theories canonically give rise to such typed cartesian operads. Proposition 2.4. Let D be a category with finite products. For CMC categories A, B, let CMC(A, B) denote the category of finite-product preserving, cocontinuous functors between them.
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