(1,0)-category

**category theory**
## Concepts
* category
* functor
* natural transformation
* Cat
## Universal constructions
* universal construction
* representable functor
* adjoint functor
* limit/colimit
* weighted limit
* end/coend
* Kan extension
## Theorems
* Yoneda lemma
* Isbell duality
* Grothendieck construction
* adjoint functor theorem
* monadicity theorem
* adjoint lifting theorem
* Tannaka duality
* Gabriel-Ulmer duality
* small object argument
* Freyd-Mitchell embedding theorem
* relation between type theory and category theory
## Extensions
* sheaf and topos theory
* enriched category theory
* higher category theory
## Applications
* applications of (higher) category theory

By the general rules of $(n,r)$-categories, a **$(1,0)$-category** is an $\infty$-category such that

- any $j$-morphism is an equivalence, for $j \gt 0$;
- any two parallel $j$-morphisms are equivalent, for $j \gt 1$.

You can start from any notion of $\infty$-category, strict or weak; up to equivalence, the result is always the same as a groupoid.

Revised on September 15, 2009 18:57:45
by Urs Schreiber
(80.187.149.88)