category

Definition

A category $\mathcal{C}$ is the data

• a class of objects denoted $\text{ob}(\mathcal{C})$

• for each pair of objects $X,Y\in \text{ob}(\mathcal{C})$, a set of morphisms from $X$ to $Y$ denoted ${\text{Hom}}_{\mathcal{C}}(X,Y)$. that satisfies the following conditions:

There is an associative composition law

$$\begin{array}{rl}{\text{Hom}}_{\mathcal{C}}(Y,Z)\times {\text{Hom}}_{\mathcal{C}}(X,Y)& ⟶{\text{Hom}}_{\mathcal{C}}(X,Z)\\ (g,f)& ⟼g\circ f\end{array}$$

and for each $X\in \mathcal{C}$, there exists and identity morphism

$${\text{id}}_X\in {\text{Hom}}_{\mathcal{C}}(X,X)$$

such that for every $f\in {\text{Hom}}_{\mathcal{C}}(X,Y)$ and $g\in {\text{Hom}}_{\mathcal{C}}(Y,X)$

$$f\circ {\text{id}}_X=f{\text{id}}_X\circ g=g$$

Examples

Sets

$\text{Set}$

• Objects: Sets

• Morphisms: functions between sets Composition is composition of functions, and ${\text{id}}_X$ is the identity function.

Vector fields

$\text{Vect}$

• Objects: For a field $k$, objects are vector fields over $k$.

• Morphisms: $k$-linear maps (linear transformations). Composition is composition of linear functions (preserves linearity).

Groups

$\text{Grp}$

• Objects: groups

• Morphisms: group homomorphisms

Subcategory

A subcategory $\mathcal{S}\subset \mathcal{C}$ is the data of a subclass $\text{Ob }\mathcal{S}\subset \text{Ob }\mathcal{C}$ and subsets

$${\text{Hom}}_{\mathcal{S}}(X,Y)\subset {\text{Hom}}_{\mathcal{C}}(X,Y)X,Y\in \text{Ob }\mathcal{S}$$

with make $\mathcal{S}$ into a category using the composition rules from $\mathcal{C}$.

Therefore, there is a canonical inclusion functor

$$i:\mathcal{S}⟶\mathcal{C}$$

with is injective on objects and morphisms.