2 category

In the mathematical field of category theory , 2 categories are the simplest examples of higher categories .

definition

A 2-category consists of a class of objects , a class of morphisms between objects and a class of morphisms between morphisms. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle Ob({\mathcal {C}})}$ ${\displaystyle Mor({\mathcal {C}})}$ ${\displaystyle Mor_{2}({\mathcal {C}})}$

That means both

${\displaystyle (Ob({\mathcal {C}}),Mor({\mathcal {C}}))}$

as well as

${\displaystyle (Mor({\mathcal {C}}),Mor_{2}({\mathcal {C}}))}$

each form a category .

Examples

• Let be the category of groups and group homomorphisms . This becomes a 2-category with the conjugation maps as 2-isomorphisms ${\displaystyle {{\mathcal {G}}rp}}$

${\displaystyle Mor_{2}(f_{0},f_{1})=\left\{h\in H\colon f_{1}(g)=hf_{0}(g)h^{-1}\ \forall \ g\in G\right\}}$

for everyone . ${\displaystyle f_{0},f_{1}\in Mor(G,H)}$

• Let be the category of topological spaces and continuous mappings . This becomes a 2-category with the homotopias as 2-homomorphisms ${\displaystyle {{\mathcal {T}}op}}$

${\displaystyle Mor_{2}(f_{0},f_{1})=\left\{H\colon X\times \left[0,1\right]\to Y\colon H(x,i)=f_{i}(x)\ \forall \ x\in X,i\in \left\{0,1\right\}\right\}}$

for everyone . ${\displaystyle f_{0},f_{1}\in Mor(X,Y)}$

• Be the category of categories and functors . This becomes a 2-category with the natural transformations as 2-morphisms ${\displaystyle {{\mathcal {C}}at}}$

${\displaystyle Mor_{2}({\mathcal {F}}_{0},{\mathcal {F}}_{1})=\left\{\left\{\alpha _{C}\colon {\mathcal {F}}_{0}(C)\to {\mathcal {F}}_{1}(C)\right\}_{C\in {\mathcal {C}}}\colon \alpha _{C^{\prime }}\circ F(f)=G(f)\circ \alpha _{C}\ \forall C,C^{\prime }\in Ob({\mathcal {C}}),f\in Mor(C,C^{\prime })\right\}}$

for everyone . ${\displaystyle {\mathcal {F}}_{0},{\mathcal {F}}_{1}\in Mor({\mathcal {C}},{\mathcal {D}})}$

literature

• Jacob Lurie : Higher Topos Theory (Section 1.1), online (pdf)

Web links

• 2-category (nlab)