Category

Definition

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

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

Examples

Quiver (Category Theory)

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

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|Locally Small Category]] A category is locally small when all of its [[Homset|hom-sets]] form a [[set]], i.e. have a cardinality.