Infinity category

In mathematics , the term infinite category , category, or quasi -category is a generalization of the concept of category . ${\displaystyle \infty }$

While in a category morphisms between objects and in a 2-category also has 2-morphisms between morphisms, there is an infinity in Category -Morphismen between -Morphismen for all ${\displaystyle k}$ ${\displaystyle k-1}$ ${\displaystyle k=1,2,\dotsc }$

definition

An infinite category is a simplicial set that satisfies the weak Kan extension property : ${\displaystyle S_{*}}$

For each simplicial mapping can be continued to a simplicial mapping . ${\displaystyle 0<i<n}$ ${\displaystyle \sigma _{0}\colon \Lambda _{i}^{n}\to S_{*}}$ ${\displaystyle \sigma \colon \Delta ^{n}\to S_{*}}$

Here referred to the dimensional standard simplex and the by omitting from emerging "Horn". ${\displaystyle \Delta ^{n}}$ ${\displaystyle n}$ ${\displaystyle \Lambda _{i}^{n}}$ ${\displaystyle \partial _{i}\Delta ^{n}}$ ${\displaystyle \partial \Delta ^{n}}$

Examples

• Kan complexes are infinite categories in which the desired continuation also exists for and always. ${\displaystyle i=0}$ ${\displaystyle i=n}$
• The nerve of a small category is an infinite category in which the desired continuation is always clear. Conversely, an infinite category with clear continuations is isomorphic to the nerve of a small category.
• The product and co- product (as simplicial sets) of infinite categories is an infinite category.

Objects, morphisms and functors

An object of an infinite category is a 0-simplex . An infinite category morphism is a 1-simplex . Its edges and are called the source and destination of morphism. One then says there is a morphism of to . For each object , the degenerate edge is called the identity morphism . ${\displaystyle x\in S_{0}}$ ${\displaystyle f\in S_{1}}$ ${\displaystyle X=d_{1}f}$ ${\displaystyle Y=d_{0}f}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle s_{0}(X)}$ ${\displaystyle id_{X}}$

A homotopy between two morphisms is a 2-simplex with . ${\displaystyle f,g\colon X\to Y}$ ${\displaystyle \sigma }$ ${\displaystyle d_{0}\sigma =id_{Y},d_{1}\sigma =f,d_{2}\sigma =g}$

A morphism is called the composition of two morphisms and , if there is a 2-simplex with . This morphism always exists, but it is only clearly defined except for homotopy. ${\displaystyle h\colon X\to Z}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon Y\to Z}$ ${\displaystyle \sigma }$ ${\displaystyle d_{0}\sigma =g,d_{1}\sigma =h,d_{2}\sigma =f}$ ${\displaystyle h}$

The homotopy category of an infinite category has as objects the objects of and as morphisms the homotopy classes of morphisms in . The homotopy class of is identity morphism and the well-defined composition of homotopy classes defines the composition of morphisms. ${\displaystyle h{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle id_{X}}$

An isomorphism in the infinity category is a morphism whose homotopy class is an isomorphism in . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle h{\mathcal {C}}}$

A functor of infinite categories is a simplicial mapping . The functors again form an infinite category . ${\displaystyle F\colon {\mathcal {C}}\to {\mathcal {D}}}$ ${\displaystyle Fun({\mathcal {C}},{\mathcal {D}})}$

Web links

• Jacob Lurie : Kerodon