theory is a powerful framework for understanding mathematical structures and relationships. It focuses on objects and morphisms, providing a unified language for describing diverse mathematical concepts across different fields.
At its core, category theory deals with , identity, and . These fundamental principles allow us to analyze complex systems by breaking them down into simpler components and understanding how they interact.
Fundamental Concepts of Category Theory
Components of categories
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A category C consists of two main components:
Objects denoted as A, B, C, etc. represent the entities or structures within the category (groups, sets, vector spaces)
Morphisms (arrows) denoted as f, g, h, etc. represent the relationships, transformations, or mappings between objects (group homomorphisms, functions, linear transformations)
For every pair of objects A and B, there exists a set HomC(A,B) called the hom-set
Contains all morphisms from A to object B, capturing the possible ways to relate or transform A into B
If f∈HomC(A,B), it is denoted as f:A→B, indicating that f is a from A to B (function from set A to set B)
Notation for category theory
Composition of morphisms: If f:A→B and g:B→C, then there exists a morphism g∘f:A→C
The composition is read from right to left: (g∘f)(a)=g(f(a)) for a∈A, meaning first apply f to a, then apply g to the result
Represents the idea of chaining transformations or mappings together (composing functions, group homomorphisms, or linear transformations)
Commutative diagrams used to represent the equality of compositions of morphisms
If f:A→B, g:B→C, h:A→C, and h=g∘f, then the following diagram commutes:
\begin{CD}
A @>f>> B \\
@VhVV @VVgV \\
C @= C
\end{CD}
Visually captures the idea that following different paths in the diagram yields the same result (going from A to C directly via h is the same as going from A to B via f, then from B to C via g)
Properties of morphisms
Identity morphisms: For every object A, there exists a unique morphism idA:A→A called the
For any morphism f:A→B, the following hold:
f∘idA=f, meaning composing f with the identity on its domain yields f itself (identity function, identity matrix)
idB∘f=f, meaning composing f with the identity on its codomain yields f itself
Captures the idea of a neutral element or transformation that preserves the object
Associativity: For morphisms f:A→B, g:B→C, and h:C→D, the following equality holds:
(h∘g)∘f=h∘(g∘f), meaning the order of composition does not matter when grouping morphisms (associativity of function composition, matrix multiplication)
Allows for unambiguous composition of multiple morphisms without the need for parentheses
Types of morphisms
Isomorphisms: A morphism f:A→B is an if there exists a morphism g:B→A such that:
g∘f=idA and f∘g=idB, meaning f and g are inverses of each other
If an isomorphism exists between A and B, they are considered isomorphic, denoted as A≅B, indicating that A and B are essentially the same object up to relabeling (isomorphic groups, homeomorphic spaces)
Monomorphisms: A morphism f:A→B is a (monic) if for any object C and morphisms g1,g2:C→A, the following implication holds:
If f∘g1=f∘g2, then g1=g2, meaning f is left-cancellative or injective (injective functions, injective group homomorphisms)
Epimorphisms: A morphism f:A→B is an (epic) if for any object C and morphisms h1,h2:B→C, the following implication holds:
If h1∘f=h2∘f, then h1=h2, meaning f is right-cancellative or surjective (surjective functions, surjective group homomorphisms)