Category

Definition

A category $\mathcal{C}$ consists of:

• A type of objects $|\mathcal{C}|:\mathrm{Set}$
• A type of hom-sets $\mathcal{C}[\_,\_]:|\mathcal{C}|\to |\mathcal{C}|\to \mathrm{Set}$
• Identity morphisms: $\mathrm{id}:\forall a\to \mathcal{C}[a,a]$
• Composition: $\circ :\mathcal{C}[b,c]\to \mathcal{C}[a,b]\to \mathcal{C}[a,c]$

Such that:

• $\mathrm{id}$ is a left- and right-Isomorphism of $\circ$
• $\circ$ is associative

Notation

For every arrow $f:\mathcal{C}[a,b]$, the domain is $a$ and codomain is $b$.

Composition is sometimes written as $g\cdot f$ or $gf$, particularly when 2-morphisms are present.

Properties

• Categories are typically named by the type of object, though it is frequently the case where two categories share the same object types even if their hom-sets differ.
• A category is small when its collection of all hom-sets forms a set.
• A category is locally small when each individual hom-set forms a set.

Examples

Algebraic Categories

• Category of Sets
• Grp: Category of groups
• Ab (Category): Category of Abelian groups
• Rng (Category): Category of rings without identity
• Ring (Category): Category of rings
• CRing (Category): Category of commutative rings
• R-Mod: Category of left $R$ modules, for a fixed ring $R$
• Mod-R: Category of right $R$ modules
• K-Mod: Category of modules over a commutative ring $K$
• Vect (Category): Category of vector spaces and linear transformations
• Pointed Algebra: Algebras with a distinguished point

Topological and Geometric Categories

• Top (Category): Topological spaces and continuous maps
• Toph (Category): Topological spaces and homotopy classes of maps
• Man (Category): Category of smooth manifolds and smooth maps
• Diff (Category): Category of smooth manifolds and smooth maps

Combinatorial Categories

• Simplex Category: Category of finite ordinals and order-preserving maps
• Graph (Category): Category of (undirected) graphs and graph homomorphisms
• DirGraph (Category): Category of directed graphs and homomorphisms
• Quiver (Category Theory): Category of directed multigraphs

Order-Theoretic Categories

• Ord (Category): Category of Ordinal Numbers
• Poset (Category): Category of partial orders and monotone maps

Other Categories

• Meas (Category): Category of measurable spaces and measurable functions
• Chain Complexes of R-Modules: See riehl2016-category-theory

Special Cases

• Discrete Categories: For any set $S$, the discrete category of $S$ is a category where the only morphisms are identity morphisms.
• Preorders: Every preorder is also a thin category.
• Monoids: All monoids are categories with a single object.
• Groupoids: A Groupoid is a category in which every morphism is an isomorphism.

Related Concepts

• Functor
• Natural Transformation
• Object (Category Theory)
• Morphism
• Homset

References

riehl2016-category-theory