Fibres. Temesghen Kahsai. Fibres in Concrete category. Generalized Fibres. Fibres. Temesghen Kahsai 14/02/ 2007

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1 14/02/ 2007

2 Table of Contents

3 ... and back to theory Example Let Σ = (S, TF) be a signature and Φ be a set of FOL formulae: 1. SPres is the of strict presentation with: objects: < Σ, Φ >, morphism σ :< Σ, Φ > < Σ, Φ > is a sig. morph. σ : Σ Σ such that σ(φ) Φ. 2. Pres is the of presentation with: objects: < Σ, Φ >, morphism σ :< Σ, Φ > < Σ, Φ > is a sig. morph. σ : Σ Σ such that σ(c Σ (Φ)) c Σ (Φ ). 3. Theo is the of theories with: objects: < Σ, Φ >, where Φ = cσ (Φ). morphism σ :< Σ, Φ > < Σ, Φ > is a sig. morph. σ : Σ Σ such that σ(φ) Φ.

4 Closure of a set of formulas Definition Let L be an algebra logic [Loeckx et al], Σ a signature, φ L(Σ) a set of formulas. The closure of φ is the set of formulae: φ = {ϕ L(Σ) φ = ϕ} Definition (Logical consequence) Let L be an algebra logic, Σ a signature, φ L(Σ) a formula, Φ L(Σ) a set of formulas and U be a Σ-domain. φ is called logical consequence of Φ in U, if A = Σ φ, for each A Mod U,Σ (Φ); one writes Φ = U,Σ φ.

5 Monoid specification in CASL spec CommMonoid1 = sort Elem ops end spec CommMonoid2 = sort Elem ops n: Elem; _*_ : Elem x Elem -> Elem vars x,y,z: elem. n * x = x. (x * y) * z = x * (y * z). x * y = y * x n: Elem; _*_ : Elem x Elem -> Elem vars x,y,z: elem. x * n = x. (x * y) * z = x * (y * z). x * y = y * x

6 Concrete Definition Let X be a. A concrete over X is a pair < D, υ >, where υ : D X is a faithful functor. Concrete categories over SET are called constructs. X is sometimes called base of < D, υ >.

7 Example: concrete Examples Theo, Pres and Spres are concrete over the Sign. sign : SPres Sign sign :< Σ, Φ > < Σ > sign : Pres Sign sign : Theo Sign where Sign is the of signatures with: objects= < Σ >, and morphism= σ : Σ Σ are signature morphism.

8 Definition Given a concrete < D, υ > over C and a C-object c. The fibre of c is the preordered class consisting of objects d of D with υ(d) = c, ordered by d 1 d 2 iff id c : υ(d 1 ) υ(d 2 ) is a D-morphism. Definition A concrete < D, υ > over C is called: Amnestic provided its fibres are partially ordered: d 1 c d 2 and d 2 c d 1 implies d 1 = d 2 for all C-objects c and objects d 1, d 2 in the fibre of c. Fibre-complete if its fibres are complete lattices. Fibre-discrete if its fibres are ordered by equality.

9 Examples SPres and Theo are amnestic. Pres is not amnestic. Fibre-discrete categories they are such that the extension that D makes over the objects of C is inessential, i.e. it has no intrinsic structure or meaning.

10 Concrete functors Definition A concrete functor ϕ between two concrete categories < D 1, υ 1 > and < D 2, υ 2 > over the same underlying C is a functor ϕ : D 1 D 2 such that υ 1 = ϕ; υ 2. Examples ϕ :< Set, id Set > < Set, id Set > is a concrete functor. ϕ : Rng Ab that forgets multiplication is a concrete functor.

11 Proposition 1. Every concrete functor is faithful. 2. Given ϕ and ψ between two concrete categories < D 1, v 1 > and < D 2, v 2 >, ϕ = ψ if, for every D 1 -object d, ϕ(d) = ψ(d).

12 Concrete Subcategories Definition Let < D, v > be a concrete over X and A is a sub of D with inclusion i : A D, then < A, v; i > is a concrete sub of < D, v >.

13 Generalised definition of fibres Definition Consider a functor ϕ : D C Given a C-object c, the fibre of c is the sub of D that consists of all the objects d that are mapped to c, such that ϕ(d) = c, together with D-morphisms f : d 1 d 2 such that ϕ(f ) = id c The functor ϕ is said to be amnestic if, in its fibres, no two distinct objects are isomorphic. That is : given an isomorphism f : d 1 d 2 such that ϕ(f ) = id c for some object c of C, then f is itself an identity. D(c) : fibre of c

14 (Co)Cartesian morphisms Definition Let ϕ : D C be a functor and f : c c a C-morphism. 1. Let d : D(c), a D-morphism g : d d is co-cartesian of f and d iff: ϕ(g) = f g : d d and f : c ϕ(d ) such that ϕ(g ) = f ; f, there is a unique morphism h : d d such that ϕ(h) = f and g = g; h 2. Let d : D(c ), a D-morphism g : d d is cartesian of f and d iff: ϕ(g) = f g : d d and f : ϕ(d ) c such that ϕ(g ) = f ; f, there is a unique morphism h : d d such that ϕ(h) = f and g = g; h

15 (Co)Fibrations Definition Let ϕ : D C be a functor ϕ is a fibration if, for every C-morphism f : c c and D-object d in the fibre of c, there is a cartesian morphism for f and d. ϕ is a cofibration if, for every C-morphism f : c c and D-object d in the fibre of c, there is a co-cartesian morphism for f and d.

16 Specification as (Co)Fibrations Example Given a signature morphism f : Σ Σ. 1. In SPres: f :< Σ, f 1 (Φ ) > < Σ, Φ > is a cartesian morphism for < Σ, Φ >. f :< Σ, Φ > < Σ, f (Φ ) > is a co-cartesian morphism for < Σ, Φ >. 2. In Pres: f :< Σ, f 1 (c(φ )) > < Σ, Φ > is a cartesian morphism for < Σ, Φ >. f :< Σ, Φ > < Σ, f (Φ ) > is a co-cartesian morphism for < Σ, Φ >. 3. In Theo: f :< Σ, f 1 (Φ ) > < Σ, Φ > is a cartesian morphism for < Σ, Φ >. f :< Σ, Φ > < Σ, c(f (Φ )) > is a co-cartesian morphism for < Σ, Φ >.

17 Cleavages, cloven fibrations Definition Let ϕ : D C be a functor. A choice of a cartesian morphism for every C-morphism f : c ϕ(d ) and D-object d is called a cleavage. A fibration equipped with a cleavage is called cloven.

18 Proposition Let φ : D C be a cloven fibration and f : c c a C-morphism. 1. The morphism f defines a functor f 1 : D(c ) D(c) as follows: Given d : D(c ), f 1 (d ) is the source of the Cartesian morphism φ f,d : d d that the cleavage associates with the fibration. Given g : d1 d 2 in D(c ), f 1 (g) is the morphism f 1 (d 1 ) f 1 (d 2 ) that results from the universal property of the Cartesian morphism φ f,d2 : f 1 (d 2 ) d 2 when applied to φ f,d1 ; g and id c. 2. The morphism f defines a functor f : D(c) D(c ) in the dual way, i.e. by working on the target side of the co-cartesian morphism.

19 What if f = id c or f = f 1 ; f 2? Proposition Let φ : D C be a functor. 1. Given a C-object c and an object d in the fibre of c, the identity id d is both Cartesian and co-cartesian morphism for id c and d. 2. Given C-morphisms f 1 : c 1 c 2 and f 2 : c 2 c 3, an object d in the fibre of c1, and co-cartesian morphisms g1 : d f 1 (d) and g 2 : f 1 (d) f 2 (f 1 (d)), the composition g 1 ; g 2 provides a co-cartesian morphism for f 1 ; f 2 and d.

20 Next week... Fibre completness Grothendieck Construction

21 References José Luiz Fiadeiro. Categories for Software Engineering. Springer-Verlag, Germany, George E. Strecker Horst Herrlich. Category Theory. Allyn and Bacon Inc, Boston, George E. Strecker Ji rí Adámek, Horst Herrlich. Abstract and concrete categories (the joy of cats). Published under the GNU Free Documentation License, January Saunders Mac Lane. Categories for the Working Mathematician. Springer-Verlag, New York, second edition, 1998.