Definition
Category
A category $\mathcal{C}$ consists of the following data:
- A collection of objects, denoted $\operatorname{Ob}(\mathcal{C})$.
- For every pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$, a set $$ \operatorname{Hom}_{\mathcal{C}}(X,Y) $$ whose elements are called morphisms (or arrows) from $X$ to $Y$.
- For every triple of objects $X,Y,Z$, a composition map $$ \operatorname{Hom}_{\mathcal{C}}(Y,Z) \times \operatorname{Hom}_{\mathcal{C}}(X,Y) \to \operatorname{Hom}_{\mathcal{C}}(X,Z), $$ written $$ (g,f) \mapsto g \circ f . $$
- For every object $X$, an identity morphism $$ \operatorname{id}_X \in \operatorname{Hom}_{\mathcal{C}}(X,X). $$
- Associativity: For morphisms $$ X \xrightarrow{f} Y \xrightarrow{g} Z \xrightarrow{h} W $$ we have $$ h \circ (g \circ f) = (h \circ g) \circ f . $$
- Identity: For every morphism $f : X \to Y$, $$ f \circ \operatorname{id}_X = f \qquadand\qquad \operatorname{id}_Y \circ f = f . $$
Used in
Dependency Graph
flowchart LR
classDef current fill:#6366f1,color:#fff,stroke:#4f46e5
n45ea5662["Uniqueness of Inverses"]
nf2c6bdf5["Category"]:::current
n136199b7["Isomorphism"]
nd14b7e1d["Notation for Morphisms"]
nf3e709f9["Duality Principle"]
na02613f0["Subcategory"]
n39e872fb["Functor"]
n310ab4af["Opposite Category"]
n30fdee11["Corollary"]
nf2c6bdf5 --> nd14b7e1d
nf2c6bdf5 --> n136199b7
nf2c6bdf5 --> n45ea5662
nf2c6bdf5 --> n30fdee11
nf2c6bdf5 --> na02613f0
nf2c6bdf5 --> n310ab4af
nf2c6bdf5 --> nf3e709f9
nf2c6bdf5 --> n39e872fb
click n45ea5662 "../objects/45ea5662.html" "_self"
click n136199b7 "../objects/136199b7.html" "_self"
click nd14b7e1d "../objects/d14b7e1d.html" "_self"
click nf3e709f9 "../objects/f3e709f9.html" "_self"
click na02613f0 "../objects/a02613f0.html" "_self"
click n39e872fb "../objects/39e872fb.html" "_self"
click n310ab4af "../objects/310ab4af.html" "_self"
click n30fdee11 "../objects/30fdee11.html" "_self"