Category Theory

A category $\mathcal{C}$ consists of the following data: • A collection of objects, denoted $\operatorname{Ob}(\mathcal{C})$. • For every pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$, a set $$ \opera…

Let $\mathcal{C}$ be a category as in Definition~. If $f \in \Hom_{\mathcal{C}}(X,Y)$ we write $$ f : X \to Y. $$ The object $X$ is called the domain of $f$ and $Y$ the codomain. Composition of morphisms is written…

Let $\mathcal{C}$ be a category (Definition~). A morphism $$ f : X \to Y $$ is called an isomorphism if there exists a morphism $$ g : Y \to X $$ such that $$ g \circ f = \operatorname{id}_X \qquad f \circ g = \oper…

Let $\mathcal{C}$ be a category (Definition~). If a morphism $f : X \to Y$ is an isomorphism (Definition~), then its inverse is unique.

Let $\mathcal{C}$ be a category (Definition~). The relation $$ X \cong Y $$ defined via isomorphisms (Definition~) is an equivalence relation on $\operatorname{Ob}(\mathcal{C})$.

Let $\mathcal{C}$ be a category (Definition~). A subcategory $\mathcal{D}$ of $\mathcal{C}$ consists of • a collection of objects $$ \operatorname{Ob}(\mathcal{D}) \subseteq \operatorname{Ob}(\mathcal{C}) $$ • fo…

Let $\mathcal{C}$ be a category (Definition~). The opposite category $\mathcal{C}^{op}$ is defined as follows. • Objects: $$ \operatorname{Ob}(\mathcal{C}^{op}) = \operatorname{Ob}(\mathcal{C}) $$ • Morphisms: $$…

Let $\mathcal{C}$ be a category (Definition~). Any statement about $\mathcal{C}$ has a dual statement obtained by replacing • morphisms $f : X \to Y$ by $f : Y \to X$ • compositions $g \circ f$ by $f \circ g$ …

A functor $F$ is a map of categories that preserves commutative diagrams. In particular, given two categories, $\mathcal{C}$, $\mathcal{D}$, a functor $F : \mathcal{C} \to \mathcal{D}$ satisfies the following prope…

Let $$ F : \mathcal{C} \to \mathcal{D} $$ be a functor (Definition~). If $$ f : X \to Y $$ is an isomorphism in $\mathcal{C}$ (Definition~), then $$ F(f) : F(X) \to F(Y) $$ is an isomorphism in $\mathcal{D}$. Refe…

Let $$ F : \mathcal{C} \to \mathcal{D}, \qquad G : \mathcal{D} \to \mathcal{E} $$ be functors (Definition~). The composition of functors $$ G \circ F : \mathcal{C} \to \mathcal{E} $$ is defined as follows. On objects…

The composition of functors is a functor. References: Definition~, Definition~.

Let $\mathcal{C}$ be a category. The identity functor $$ \operatorname{Id}_{\mathcal{C}} : \mathcal{C} \to \mathcal{C} $$ is defined by Objects $$ \operatorname{Id}_{\mathcal{C}}(X) = X $$ Morphisms $$ \operator…

The identity functor is a functor. References: Definition~, Definition~.

Functor composition is associative. References: Definition~.

Let $$ \mathbf{Grp} $$ be the category of groups and group homomorphisms, and $$ \mathbf{Set} $$ the category of sets. Define a functor $$ U : \mathbf{Grp} \to \mathbf{Set} $$ as follows. Objects $$ U(G) = the…

Let $$ F,G : \mathcal{C} \to \mathcal{D} $$ be functors (Definition~). A natural transformation $$ \eta : F \Rightarrow G $$ consists of morphisms $$ \eta_X : F(X) \to G(X) $$ for every object $X$ of $\mathcal{C…