Category Theory Notes

A structure-preserving map from one mathematical object to another.

In category theory, obey conditions specific to category theory itself.

A structure-preserving map between two algebraic structures of the same type.

A homomorphism or morphism that admits an inverse. Two objects are isomorphic if an isomorphism exists between them.

An isomorphism from a mathematical object to itself, i.e. an invertible endomorphism.

A morphism (or homomorphism) from a mathematical object to itself.

Also known as an epic morphism, or an epi. A morphism, \(f: X \to Y\) that is right-cancellative i.e. \[ \forall g_1, g_2: Y \to Z, \ g_1 \circ f = g_2 \circ f \implies g_1 = g_2 \]

Epimorphisms are categorical analogues of surjective functions.

"onto" \[ f : X \to Y, \ \forall y \in Y,\ \exists x \in X,\ f(x) = y \]

An injective homomorphism, e.g. \(X \hookrightarrow Y\). In category theory, a monomorphism (a.k.a monic morphism or mono) is a left-cancellative morphism, i.e. \[ \forall g_1, g_2 : Z \to X, \ f : X \to Y\ s.t.\ f \circ g_1 = f \circ g_2 \implies g_1 = g_2 \]

Monomorphisms are a categorical generalization of injective functions

"one-to-one" \[ \forall a, b \in X,\ f(a) = f(b) \implies a = b \] contrapositive: \[ \forall a, b \in X,\ a \neq b \implies f(a) \neq f(b) \]

An algebraic structure comprised of objects linked by arrows, e.g. the category of sets, which links sets with functions. Arrows can be composed associatively and there exists an identity arrow.

A category in which morphisms and objects can be added and in which kernels and cokernels exist and have desireable properties, e.g. the category of abelian groups, Ab.

• has a zero object
• has all binary products and binary coproducts
• has all kernels and cokernels
• all monomorphisms and epimorphisms are normal

The category of abelian groups.

An object that is both intial and terminal. a.k.a. null object.

A category with a zero object.

An initial object for which every morphism into \(I\) is an isomorphism.

\(I \in C, \forall X \in C \exists\) precisely one morphism \(I \to X\) a.k.a. coterminal or universal

\(T\) is terminal if \(\forall X \in C \exists\) a single morphism \(X \to T\). a.k.a terminal element

The "most general" object which admits a morphism to each of the given objects.

a.k.a. categorical sum

An algebraic structure with a single associative binary operation and an identity element. A monoid is a semigroup with an identity.