Categories and their Algebra James A. Overton B.Math Honours Thesis Presentation September 12, 2005
Overview • Category Theory • Division of Categories • V arieties of Categories B. Tilson, Categories as algebra: An essential ingredient in the theory of monoids , Journal of Pure and Applied Algebra 48 ( 1987 ) , 83 – 198.
Category Theory • Categories as Graphs • Examples • Categories and Monoids • Functors • Natural T ransformations
Graphs • A Graph G is: • A set of vertices V(G) G c • Edge - sets connecting g v x each pair of vertices G(v,w) a f d • Graph concepts b • path e w y • loop • ( strongly ) connected • component
Category • Terminology: Object set Obj(C) , Arrow set Arr (C) , Hom - sets C(a,b) • A Category C is a graph satisfying: Composition Given a pair of arrows f : a → b, g : b → c ∈ Arr( C ) , a, b, c ∈ Obj( C ), there exists an arrow fg : a → c ∈ Arr( C ). Moreover, composition is associative. Identity For every object b ∈ Obj( C ), there ex- ists an identity arrow 1 b : b → b ∈ Arr( C ) satisfying the compositions f 1 b = f and 1 b g = g (with f and g as above).
Basic Categories 1 a 1 a 1 b 1 a 1 b 1 a 1 2 3 f a f 1 c a a b a b 1 b h f g b c g a, aba 1 c 0 1 c 1 c 0 c 1 ab ba b, bab ab=(ab) 2 , ba=(ba) 2
Examples Abbr. Name Objects Arrows Empty Category None None 0 One Object 1 a (identity arrows are assumed) 1 a Two Objects a, b f : a → b 2 Triangle f : a → b, g : b → c, fg : a → c 3 a, b, c Parallel f, g : a → b ↓↓ a, b ∆ Simplicial Category all finite ordinals all order preserving functions Sets all small sets all functions between them Set Pointed Sets small sets with a selected base point all base-point preserving functions Set ∗ Ensembles over a set V all sets within V all functions between them Ens V Category of Categories all small categories all functors Cat Monoids all small monoids all morphisms of monoids Mon Groups all small groups all morphisms of groups Grp Abelian Groups all small (additive) Abelian groups all morphisms between them Ab Rings all small rings all morphisms of rings Rng Commutative Rings all small commutative rings all morphisms of rings CRng
Monoids • A Monoid (M,•) is a set M combined with a binary operation • satisfying: • Associativity : The binary operation is associative • Identity : There is an identity element 1 M • A category with one object is a monoid. For this reason, categories can be seen as a way to extend monoids.
Functors A functor • A Functor is a T : C → D morphism of consists of an object function categories. and hom - set functions • Examples: T : Obj( C ) → Obj( D ) • Power Set P T : C ( a, b ) → D ( aT, bT ) • Group of Units * of a Ring where the latter satisfy • General Linear 1 c T = 1 cT Group GLn of a Ring ( fg ) T = fTgT
Types of Functor Type Object Function Hom-Set Functions Isomorphism Bijection Bijection Embedding Injection Injection Faithful Arbitrary Injection Quotient Bijection Surjection Full Arbitrary Surjection
Natural T ransformations • A Natural T ransformation is a morphism of functors. Given functors S, T : C → D a natural transformation τ : S → T is a function from a ∈ Obj( C ) → a τ ∈ D ( aS, aT ) objects in C to arrows in D a τ ✲ aT a aS such that for all arrows in C, this f fS fT diagram commutes: b τ ❄ ❄ ❄ ✲ bT b bS
Examples a τ ✲ aT a aS f f f S T ✲ ✲ ✲ b τ ✲ bT h b hS bS S T g g g ✛ ✛ ✛ c τ ❄ ❄ ❄ c cS cT ✲ GL n ( K ) det K ✲ K ∗ The determinate is an example of a natural f GL n f ∗ transformation det J ✲ J ∗ ❄ ❄ GL n ( J )
Division of Categories • Division of Monoids and Categories • The Derived Category Theorem • Congruences and Generators • Locally T rivial Categories
Division ϕ : M � N ∀ m, m ϕ � = ∅ A relational morphism ∀ m, m � ∈ M, m ϕ m � ϕ ⊆ ( mm � ) ϕ is a set - valued function such that 1 N ∈ 1 M ϕ ∀ m, m � ∈ M, ϕ : M ≺ N A division is a relation m � = m � ⇒ m ϕ ∩ m � ϕ = ∅ morphism where There are divisions of monoids and categories.
The Derived Category • The Derived Category of a relational morphism generalizes the kernel of a group homomorphism to monoids. • W e consider the structure which the relational morphism imposes on the domain, as we do with the cosets of a group. • The derived category of a division is trivial.
ϕ : M � N ϕ # = { ( m, n ) | n ∈ m ϕ } [ n, ( m 0 , n 0 )] : n ϕ − 1 → nn 0 ϕ − 1 m [ n, ( m 0 , n 0 )] = mm 0 , m ∈ n ϕ − 1 [ n, ( m 0 , n 0 )][ nn 0 , ( m 1 , n 1 )] = [ n, ( m 0 m 1 , n � n 1 )] Obj( D ϕ ) = N Arr( D ϕ ) = { [ n 1 , ( m, n )] : n 1 → n 2 | ( m, n ) ∈ ϕ # , n 1 n = n 2 } identity for n ∈ N is [ n, (1 M , 1 N )] D ϕ ( n 1 , n 2 ) = { [ n 1 , ( m, n )] | ( m, n ) ∈ ϕ # , n 1 n = n 2 }
Wreath Product N × V N → V N , ( n, f ) → n f, n 0 ( n f ) = ( n 0 n ) f n ( f + g ) = n f + n g, n ( n 0 f ) = nn 0 f, 1 f = f, n f 0 = f 0 � � � � � � 1 0 1 0 1 0 = × f + n g f n g n 0 nn 0 ( f, n )( g, n 0 ) = ( f + n g, nn 0 ) 1 ◦ N ≈ N and V ◦ 1 ≈ V V × N ≺ V ◦ N If V ≺ V � and N ≺ N � , then V ◦ N ≺ V � ◦ N �
Derived Category Theorem (a) Let ϕ : M � N be a relational morphism of monoids, and let V be a monoid satsifying D ϕ ≺ V . Then there is a division of monoids θ : M ≺ V ◦ N satisfying θπ = ϕ (where π is the projection mor- phism π : V ◦ N → N ). (b) Let θ : M ≺ V ◦ N be a division of monoids, and let ϕ = θπ : M � N be the associated rela- tional morphism. Then D ϕ ≺ V N .
Congruences and Generators • A graph equivalence relation is a family of set equivalence relations for each edge - G/ ≡ set. W e can form a quotient graph where the edge - sets are the equivalence classes. ∀ b, b � coterminal and a, c • A congruence on a category is a graph b ≡ b � ⇒ abc ≡ ab � c equivalence relation such that: • The quotient of a congruence with a C/ ≡ category is a category with composition of equivalence classes. [a][b]=[ab] • The free category G* of a graph G has every path as an arrow. • If G*=C then G generates C .
Locally T rivial Categories • The local arrows of a category are those in the hom - sets C(c,c) . W e can call these the local monoids of C . • Hom - sets are trivial if they contain only the identity. • A category is locally trivial if all local hom - sets are trivial. • The strongly connected components of a locally trivial category are also trivial.
V arieties of Categories • V arieties of Monoids and Categories • Laws • Path Equations • The Strongly Connected Component Theorem
V arieties • A variety of monoids or of categories is a collection which is closed under products and division. • A variety of categories is generated by its connected components. A variety of monoids V can generate a variety of categories V C • A variety V of monoids is “ local ” if V C equals the variety of local categories. • Question: How do we know when a category C belongs to a variety V C ?
Laws • Birkho ff showed that varieties of monoids are de fi ned by the equations they satisfy. • Tilson extends the notion of “ equations ” to the category case. • A Law (G; p=q) is a pair of coterminal paths in a graph G . • Products and divisions maintain laws. Every variety is de fi ned by laws. p p L 1 L 2 ( L 1 ; pq = qp ) r ( L 2 ; prq = qrp ) q q
Path Equations • The minimal support for a law (G; p=q) is a graph such that every edge is either in p or in q . Such a law is a path equation, written p=q . When G is strongly connected we have a strongly connected path equation. • Every non - trivial law is de fi ned by a fi nite number of path equations. • Every variety of categories is de fi ned by path equations, and non - trivial categories by strongly connected path equations.
The Strongly Connected Component Theorem W e now have the answer our question: How do we know when a category C belongs to a variety V C ? Let V be a non-trivial variety of categories. Then C ∈ V i ff the strongly connected compo- nents of C belong to V .
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