Categories and their Algebra James A. Overton B.Math Honours Thesis - - PowerPoint PPT Presentation

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)
• Edge-sets connecting

each pair of vertices G(v,w)

• Graph concepts
• path
• loop
• (strongly) connected
• component

v x w y g e b d c

G

a f

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 1b : b → b ∈ Arr(C) satisfying the compositions f1b = f and 1bg = g (with f and g as above).

Basic Categories

1a 1a 1b 1a 1b 1c 1a 1b f

1 2 3

a a b a b c a b f g h f g

1c0

ab

1c1

ba

c0 c1 a, aba b, bab ab=(ab)2, ba=(ba)2

Examples

Abbr. Name Objects Arrows Empty Category None None 1 One Object a 1a (identity arrows are assumed) 2 Two Objects a, b f : a → b 3 Triangle a, b, c f : a → b, g : b → c, fg : a → c ↓↓ Parallel a, b f, g : a → b ∆ Simplicial Category all finite ordinals all order preserving functions Set Sets all small sets all functions between them Set∗ Pointed Sets small sets with a selected base point all base-point preserving functions EnsV Ensembles over a set V all sets within V all functions between them Cat Category of Categories all small categories all functors Mon Monoids all small monoids all morphisms of monoids Grp Groups all small groups all morphisms of groups Ab Abelian Groups all small (additive) Abelian groups all morphisms between them Rng Rings all small rings all morphisms of rings CRng Commutative Rings all small commutative rings all morphisms of rings

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 1M
• A category with one object is a monoid. For

this reason, categories can be seen as a way to extend monoids.

Functors

• A Functor is a

morphism of categories.

• Examples:
• Power Set P
• Group of Units *
• f a Ring
• General Linear

Group GLn of a Ring

T : C → D T : Obj(C) → Obj(D) T : C(a, b) → D(aT, bT) A functor consists of an object function and hom-set functions where the latter satisfy

1cT = 1cT (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

τ : S → T

a natural transformation

a ∈ Obj(C) → aτ ∈ D(aS, aT)

is a function from

• bjects in C to arrows in D

a aS aτ

✲ aT

b f

❄

bS fS

❄

bτ

✲ bT

fT

❄

such that for all arrows in C, this diagram commutes:

Examples

a aS aτ

✲ aT

b f

✲

bS bτ f S

✲ ✲ bT

f T

✲

c h

❄ ✛

g cS hS

❄

cτ

✲ ✛

g S cT

❄ ✛

g T

GLn(K) detK ✲ K∗ GLn(J) fGLn

❄

detJ ✲ J∗ f∗

❄

The determinate is an example of a natural transformation

Division of Categories

• Division of Monoids and Categories
• The Derived Category Theorem
• Congruences and Generators
• Locally T

rivial Categories

Division

∀m, m ∈ M, mϕmϕ ⊆ (mm)ϕ 1N ∈ 1Mϕ ∀m, mϕ = ∅

A relational morphism is a set-valued function such that A division is a relation morphism where

∀m, m ∈ M, m = m ⇒ mϕ ∩ mϕ = ∅

There are divisions of monoids and categories.

ϕ : M N ϕ : M ≺ N

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 ∈ mϕ}

Obj(Dϕ) = N Arr(Dϕ) = {[n1, (m, n)] : n1 → n2 | (m, n) ∈ ϕ#, n1n = n2} identity for n ∈ N is [n, (1M, 1N)] Dϕ(n1, n2) = {[n1, (m, n)] | (m, n) ∈ ϕ#, n1n = n2}

[n, (m0, n0)] : nϕ−1 → nn0ϕ−1 m[n, (m0, n0)] = mm0, m ∈ nϕ−1 [n, (m0, n0)][nn0, (m1, n1)] = [n, (m0m1, nn1)]

Wreath Product

× (f, n)(g, n0) = (f + ng, nn0)

• 1

f n 1 g n0

• =
• 1

f + ng nn0

• N × V N → V N, (n, f) → nf, n0(nf) = (n0n)f

n(f+g) = nf+ng, n(n0f) = nn0f, 1f = f, nf0 = f0

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 Dϕ ≺ 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 Dϕ ≺ V N.

Congruences and Generators

• A graph equivalence relation is a family of

set equivalence relations for each edge-

• set. W

e can form a quotient graph where the edge-sets are the equivalence classes.

• A congruence on a category is a graph

equivalence relation such that:

• The quotient of a congruence with a

category is a category with composition

• f equivalence classes.
• The free category G* of a graph G has

every path as an arrow.

• If G*=C then G generates C.

∀b, b coterminal and a, c b ≡ b ⇒ abc ≡ abc

G/≡ C/≡ [a][b]=[ab]

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 VC

• A variety V of monoids is “local” if VC equals

the variety of local categories.

• Question: How do we know when a category C

belongs to a variety VC?

Laws

• Birkhoff showed that varieties of monoids are

defined 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 defined by laws.

p

L1 L2

q p q r

(L1; pq = qp) (L2; prq = qrp)

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 defined by a finite

number of path equations.

• Every variety of categories is defined by path

equations, and non-trivial categories by strongly connected path equations.

The Strongly Connected Component Theorem

Let V be a non-trivial variety of categories. Then C ∈ V iff the strongly connected compo- nents of C belong to V.

W e now have the answer our question: How do we know when a category C belongs to a variety VC?

References

[1] G. M. Bergman, An invitation to general algebra and universal constructions, Henry Helson, Berkeley CA, 1998. [2] S. Eilenberg, Automata, languages, and machines, vol. B, Academic Press, New York, 1976. [3] S. Eilenberg and S. Mac Lane, General theory of natural equivalences, Transactions of the Americal Mathematical Society 58 (1945), no. 2, 231–294. [4] L. Kaloujnine and M. Krasner, Produit complete des groupes de permutations et le probl` eme d’extension des group III, Acta Scientiarum Mathematicarum (Szeged) 14 (1951), 69–82. [5] J. Lambek and P.J. Scott, Introduction to higher order categorical logic, Cambridge Studies in Advanced Mathematics, no. 7, Cambridge University Press, Cambridge, 1986. [6] S. Mac Lane, Categories for the working mathematician, second ed., Springer, New York, 1998. [7] B. Steinberg and B. Tilson, Categories as algebra, II, International Journal of Algebra and Computation 13 (2003), 627–703. [8] D. Th´ erien and A. Weiss, Graph congruences and wreath products, Journal of Pure and Applied Algebra 36 (1985), 205–215. [9] B. Tilson, Categories as algebra: An essential ingredient in the theory of monoids, Journal of Pure and Applied Algebra 48 (1987), 83–198. [10] D. B. West, Introduction to graph theory, second ed., Prentice Hall, Upper Saddle River, NJ, 2001.