III.8 Categories, Eugenia Cheng, Class Field Theory, Cohomology: Pages 165-166.

Eugenia Cheng authored the section on mathematical categories.  The important maps between groups are group homomorphisms and between vector spaces are the linear maps.  These maps preserve structure.  A group homomorphism f from group G to group H preserves multiplication in the sense that f(g1g2) = f(g1) f(g2) for any pair of elements g1 and g2 in G.  Similarly linear maps of vector fields preserve addition and scalar multiplication. 

The above structure preserving map can be generalized to mathematical structures, A, B and C, and f and g structure preserving maps from A to B and B to C respectively.  Then the composition map gëf is structure preserving (if the range of one equals the domain of the other).  Structure preserving maps can be composed.  A and B are said to be isomorphic if there is a structure preserving map from A to B and an inverse that also preserves structure. 

A category allows us to discuss the above in the abstract.  A category consists of a collection of objects, and a collection of morphisms between these objects.  There is also a notion of composition of morphisms.  Compositions must be associative.  There must also be an identity morphism.

I addition to the group and vector space examples of categories there are also others, such as:

i.  Natural numbers as objects, morphisms from n to m are real n x m matrices, composition is matrix multiplication, this is an unusual example of a map between natural numbers m and n.

ii.  Objects may be the elements of a set, morphisms from x to y are “x = y”, and ordered set can use instead the inequalities, “x    y”, as the morphisms.

iii.  Any group can be the object and the morphisms from the group to itself are the elements of the group, group multiplication defines the composition of two morphisms.

iv.  Objects can be topological spaces, morphisms are continuous functions or the morphisms could be homotopy classes of continuous functions.

Morphisms are also called maps.  The maps can also be called arrows.  Arrows are used to emphasise the abstraction and in pictorial representations.

Shape refers to morphisms and equations the morphisms satisfy.

A morphism f from a to b is written pictorially as: a  f™ b.  The f should be above the arrow but I couldn’t do that with the font I am using.  The composition of two morphisms f and g may be written: a  f™ g™ c.  g ë f = t ës is an equality between morphisms.  It may be pictorially displayed as a square with top and bottom sides a  f™ b  and d  t™ c.  And sides down-pointing arrows of a  s™ d and b  g™ c and asserting that the diagram commutes in the sense that either path from a to c yields the same composite, as in the equality. 

Commutative diagrams, generalizations of this simple square, can be used to describe free groups, free rings, free algebras, quotients, products, disjoint unions, function spaces, direct and inverse limits, completion, compactification, and geometric realization, among others.

A pictorial diagram that looks like an arrowhead may be used to describe a dijoint union of sets.  The universal property of a disjoint union reflects the “most free” way of having two sets may into another set.  Universal properties are central to how category theory describes canonical structures. 

An isomorphism is a morphism with a two-sided inverse.

Categories form a category – look out for a Russell type paradox.  Functors are the structure preserving maps for categories.  A functor F from a category X to a category Y tales the objects of X to the objects of Y and the morphisms of X to the morphisms of Y so that the identity of a is taken to the identity of Fa and the composite of f and g is taken to the composite of Ff anf Fg.  An important example of this is a functor that takes a “marked point” topological space S to its fundamental group p1(S,s).  And a continuous map between two topological (marked point) spaces gives rise to a homomorphism between their fundamental groups.

There is a concept of “natural transformation” a morphism between functors.  This is analogous to a homotopy between maps of topological spaces.  The analogy includes the topological idea of holes.  Avoidance of holes by paths is reflected in the commutativity of certain commuting squares.  This is known a s the naturality condition.

One example is that every infinite dimensional vector space is cononically isomorphic to its double dual.  this is not rue for finite dimensional vector spaces.  In the presence of natural transformations categories actually form a 2-category – a two dimensional generalization of a category.  An n-category has morphisms for each dimension up to n.

Category theory can be thought of as foundational and may replace set membership as the foundation of mathematics.

“Class Field Theory” is discussed under “From Quadratic Reciprocity to Class Field Theory”.

“Cohomology” is discussed under “Homology and Cohomology”.

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