Category

Definition

A category $\mathcal{C}$ is made up of:

• A type of objects $|\mathcal{C}|:\mathrm{Set}$
• A type of hom-sets ($\mathcal{C}[\_,\_]:|\mathcal{C}|\to |\mathcal{C}|\to \mathrm{Set}$)
• $\mathrm{id}:\forall a\to \mathcal{C}[a,a]$ identity morphism
• $\circ :\mathcal{C}[b,c]\to \mathcal{C}[a,b]\to \mathcal{C}[a,c]$ composition operator Such that,
• ${\mathrm{id}}_a$ is a left- and right-inverse of $\circ$.
• $\circ$ is associative.

Remarks

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

Categories are typlically 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.

I will use the notation $x:\mathcal{C}$ as shorthand for saying that $x$ is an element of the category.

Examples

Category of Sets

Vect (Category)

Category of vector spaces and linear transformations.

Grp (Category)

Ab (Category)

Category of Abelian group

Top (Category)

Topological spaces and continuous maps.

Toph (Category)

Topological spaces and homotopy classes of maps.

Diff (Category)

Category of smooth manifolds and smooth maps

Rng (Category)

Category of rings without identity.

Ring (Category)

CRing (Category)

Category of commutative rings.

Ord (Category)

Category of Ordinal Numbers.

Simplex Category

[[$R$-Mod]]

Category of left $R$ modules.

[[Mod-$R$]]

Category of right $R$ modules.

[[$K$-Mod]]

Category of modules over a commutative ring $K$.

Graph (Category)

Category of (undirected) graphs an graph homomorphisms.

DirGraph (Category)

Category of directed graphs and homomorphisms.

Man (Category)

Category of smooth manifolds and smooth maps. Also called Diff.

Meas (Category)

Category of Measurable Space and measurable functions.

Poset (Category)

Category of partial orders and monotone maps.

Chain Complexes of R-Modules

See riehl2016-category-theory

Quiver (Category Theory)

Special cases

Discrete Categories

For any set $S$, the discrete category of $S$ is a category.

Preorders

Every preorder is also a thin category.

Monoids

All monoids are also categories.

Groupoid (Category Theory)

Algebras

Call the category of algebras $\mathrm{Alg}(\mathcal{C},F)$. Given a Functor $F:\mathcal{C}\Rightarrow \mathcal{C}$ A $F$-algebra is $(X:\mathcal{C},\alpha :FX\to X)$ $X$ is called the carrier.

Pointed Algebra

Let $\mathcal{C}$ be a category with a terminal object and finite Coproduct (Category Theory). A pointed Algebra (Category Theory) extends an algebra by adding a point $x:1\to X$, making $(X,\alpha :FX\to X,x:1\to X)$ which is isomorphic to,

$$(X,\beta :FX+1\to X)$$

If $\mathcal{D}$ is a category of algebras then the category of pointed algebras on $\mathcal{D}$ is denoted ${\mathcal{D}}_{\bullet }$.

Small Category

A category is small when its collection of all of its hom-sets forms a set, i.e. has a cardinality.

Locally Small Category

A category is locally small when all of its hom-sets form a set, i.e. have a cardinality.