Category

See category theory, first.
A category $\mathcal{C}$ consists of:

1. A class $\text{ob}(\mathcal{C})$, whose elements are called objects.

2. A class $\text{mor}(\mathcal{C})$, whose elements are called morphisms (or arrows, maps). Each morphism $f$ has:

   • A source object $a\in \text{ob}(\mathcal{C})$
   • A target object $b\in \text{ob}(\mathcal{C})$

   We write $f:a\to b$ to indicate that $f$ is a morphism from $a$ to $b$.

   For any pair of objects $a,b\in \text{ob}(\mathcal{C})$, we denote by ${\text{hom}}_{\mathcal{C}}(a,b)$ (or simply $\text{hom}(a,b)$) the hom-class of all morphisms from $a$ to $b$. Note that $\text{mor}(\mathcal{C})$ is the union of all such hom-classes.

3. A binary operation $\circ$ called composition:
   For any three objects $a,b,c\in \text{ob}(\mathcal{C})$, we have

   $$\circ :\text{hom}(b,c)\times \text{hom}(a,b)\to \text{hom}(a,c)$$

   The composition of $f:a\to b$ and $g:b\to c$ is written as $g\circ f$ (or simply $gf$).

4. Two axioms governing composition:

   • Associativity: For any $f:a\to b$, $g:b\to c$, and $h:c\to d$,$$h\circ (g\circ f)=(h\circ g)\circ f$$
   • Identity: For every object $x\in \text{ob}(\mathcal{C})$, there exists a morphism $1_x:x\to x$ called the identity morphism for $x$, such that for every morphism $f:a\to b$,$$1_b\circ f=f=f\circ 1_a$$

   From these axioms, it follows that the identity morphism for each object is unique.

The "processes" or "maps" between categories that preserve the structure given above are called functors.

In practice, most working mathematicians impose additional size restrictions:

(a) Locally Small Category (Most Common Usage)

A category $\mathcal{C}$ is called locally small if for every pair of objects $a,b\in \text{ob}(\mathcal{C})$, the hom-class $\text{hom}(a,b)$ is actually a set (not a proper class).

Why this matters: Most important constructions in category theory (the Yoneda’s lemma, functor categories, limits/colimits) require local smallness. When mathematicians say "category" without qualification, they usually mean "locally small category."

(b) Small Category

A category $\mathcal{C}$ is called small if both:

• $\text{ob}(\mathcal{C})$ is a set
• $\text{mor}(\mathcal{C})$ is a set

Small categories are particularly manageable and form the objects of the important category Cat.

The "Category of All Categories"

This definition clarifies why there is ambiguity about whether categories form a category:

• Under the general definition above: Categories do form a category called CAT, where:

  • Objects: All categories
  • Morphisms: All functors between categories
  • Composition: Ordinary composition of functors

• Under the locally small restriction: Categories do not form a category, because for two large categories $\mathcal{C}$ and $\mathcal{D}$, the collection of all functors $\mathcal{C}\to \mathcal{D}$ may be a proper class, violating the "hom-classes must be sets" requirement.

Examples

Basic examples of categories:

• groups.
• category of sets
• powerset category
• topos theory
  Two important construction in a category $C$ are product in categories and coproduct in categories.

An important strategy in category theory is to use universal propertys.