Enriched Categories. Stephen Fitz

Transcription

1 Enriched Categories Stephen Fitz Abstract I will introduce some of the basic concepts of Enriched Category Theory with examples of enriched categories. Contents 1 Enriched Categories Introduction Examples: M=Set M=Ab M=R-Mod M=Vect M=Mat M= M=pSet M=R Enriched Functors Introduction Examples: V=R Enriched Natural Transformations Generalized Elements Examples: Enriched Natural Transformations

2 1 Enriched Categories 1.1 Introduction DEFINITION 1 (Enriched Category). Let (M,, 1) be a monoidal category (e.g. Grp, Vect). We can define a category C enriched in M in the following way: C consists of Collection ObC A, B ObC, Hom(A, B) ObM A, B, C ObC, : Hom(B, C) Hom(A, B) Hom(A, C) that is a morphism in M A ObC, id A : 1 Hom(A, A) such that the associativity and unit laws hold, i.e. the following diagrams commute: α (Hom(C, D) Hom(B, C)) Hom(A, B) Hom(C, D) (Hom(B, C) Hom(A, B)) 1 Hom(B, D) Hom(A, B) 1 Hom(C, D) Hom(A, C) I Hom(A, B) id A 1 Hom(A, D) Hom(A, A) Hom(A, B) λ Hom(A, B) Hom(A, B) I ρ Hom(A, B) 1 id B Hom(A, B) Hom(B, B) 1.2 Examples: M=Set If we take M to be the category of sets with cartesian product as the monoidal operation (i.e. M = (Set,, 1)) then the M-category (i.e. the category enriched over M) is just the ordinary category (i.e. an object of Cat). 2

3 1.2.2 M=Ab The category of abelian groups is enriched over itself. The set of homomorphisms betwen two abelian groups is itself an abelian group. Let G where φ and ψ are homomorphisms between abelian groups G and H. We can impose an abelian group structure on Hom(G, H) by defining multiplication in Hom(G, H) in the following way: g G, (φ ψ)(g) := φ(g)ψ(g). It is straightforward to check that this is a group with identity element being the trivial e homomorphism G,H G H, and inverse for each φ Hom(G, H) defined by φ 1 (g) = φ. Remark: this is an additive category M=R-Mod The category R mod of left R-modules is enriched over Z mod, i.e. enriched over Ab (since Z-modules are abelian groups) M=Vect The category of vector spaces over a field k is enriched over itself. This is because if U, V are vector spaces over a field k, then the set of linear transformations Hom(U, V) is itself a vector space over k. In fact it inherits the structure of V in the following way: u U, [S + T](u) = S(u) + T(u) and [ct](u) = ct(u) for linear transformations S, T : U V. Remark: this is a linear category. φ ψ H, M=Mat The category of matrices is enriched over Vect as a special case of the preceding example M=2 The category 2 consisting of 2 objects and one nontrivial morphism can be represented as ObM = {0, 1}, MorM = {id 0, id 1, 0 1}. We can then make it into a monoidal category by defining tensor product to be the multiplication of numbers (in Z since ObM Z). Then a category C enriched over M can be interpreted as a preordered set (preorder is a binary relation that is reflexive and transitive, e.g. any equivalence relation is a preorder). Under this interpretation we have A, B ObC, A B Hom(A, B) = {1} M=pSet If M is the category of pointed sets with smash product for the monoidal operation, then any category C enriched over M has zero morphisms. The zero morphism A B is the special point in the pointed set Hom(A, B) 3

4 1.2.8 M=R+ Consider the following category (which we will denote R + ): Objects: positive numbers a [0, ] together with the point at infinity Morphisms: Hom(a, b) = { { } if a b, if a < b. (1) We can make this into a monoidal category (R +, +, 0). Now let C be a category enriched in R +. Then we have the following: a, b ObC, Hom(a, b) [0, ] a, b, c ObC, Hom(a, b) + Hom(b, c) Hom(a, c) a ObC, 0 Hom(a, a) [0, ], i.e. Hom(a, a) = 0 Here associativity and unit laws follow vacuously. This category is called the generalized metric space (note that the metric is not necessarily symmetric here, i.e. in general Hom(a, b) Hom(b, a)) 2 Enriched Functors 2.1 Introduction DEFINITION 2 (Enriched Functor). Let (M,, 1) be a monoidal category. If C, D are M-categories, then an M-enriched functor T :C D is a map which assigns to each object of C and object of D and for each pair of objects A, B ObC provides a morphism in M, T AB : C(A, B) D(T (A), T (B)) between Hom-objects of C and D, such that identities and compositions are preserved. Since Hom-objects are no longer necessarily sets we can t speak of particular morphisms (i.e. identity) or particular compositions. Instead we think of selecting identity on a ObC as a map in M, id a : 1 C(a, a), and of composition as the monoidal product. Therefore the usual functorial axioms are replaced by the following commutative diagrams: 1 C(a, a) id a T(a,a) id T(a) D(T(a), T(a)) i.e. T a a id a = id T (a), where 1 is the unit object of M (Remark: this is analogous to F(id a ) = id F(a) for an ordinary functor F), and 4

5 C(b, c) C(a, b) C(a, c) T bc T ab D(T(b), T(c) D(T(a), T(b)) D(T(a), T(c)) Remark: this is analogous to F(f g) = F(f) F(g) for an ordinary functor F. 2.2 Examples: V=R+ Let V = (R +, +, 0), the monoidal category defined in section 1. Then if Θ : C D is a V-enriched functor between two V-enriched categories C, D we must have a, b ObC, Hom(a, b) Hom(Θ(a), Θ(b)), so for categories enriched over V all functors have to be non-expanding maps between the respective generalized metric spaces. 3 Enriched Natural Transformations 3.1 Generalized Elements DEFINITION 3 (Generalized Element). Let (M,, 1) be a monoidal category. We define a functor Γ : M Set by m M, m Hom(1, m). We think of generalized elements of a given object m M to be elements of the set Γ(m), i.e. elements of the image of m under Γ Examples: M 1 Γ Set { } identity, so the generalized elements are just elements of sets Vect C C Γ(M) = M, the underlying set of M R + 0 Γ(m) ={{ } if m = 0; otherwise } 3.2 Enriched Natural Transformations DEFINITION 4 (Enriched Natural Transformation). Let F, G :C D be two functors enriched over a monoidal category (M,, 1). Then θ : F G is an enriched natural transformation if x C, θ x Γ(Hom(Fx, Gx)), such that x, y ObC, the following diagram commutes: T ac 5

6 Hom(x, y) 1 Hom(x, y) Hom(x, y) 1 θ y F Hom(Fy, Gy) Hom(Fx, Fy) G θ x Hom(Gx, Gy) Hom(Fx, Gy) Hom(Fx, Gy) (Remark: this is equivalent to the naturality square Fx Ff α x Gx Gf Fy α y Gy for an ordinary natural transformation α and a morphism x f y ) 6

7 References [1] F. Borceux: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications Volume 2, Cambridge University Press (1994) [2] G.M. Kelly: Basic Concepts of Enriched Category Theory, Reprints in Theory and Applications of Categories No. 10 (2005) [3] Catsters on YouTube 7