+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Homotopy theory +--{: .hide} [[!include homotopy - contents]] =-- #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea By _1-groupoid_ one means -- for emphasis -- a _[[groupoid]]_ regarded as an _[[∞-groupoid]]_. A quick way to make this precise is to says that a 1-groupoid is a [[Kan complex]] which is [[equivalence|equivalent]] ([[homotopy equivalence|homotopy equivalent]]) to the [[nerve]] of a groupoid: a [[simplicial coskeleton|2-coskeletal]] Kan complex. More abstractly this is a [[truncated object of an (∞,1)-category|1-truncated]] [[∞-groupoid]]. More generally and more vaguely: Fix a meaning of $\infty$-[[infinity-groupoid|groupoid]], however weak or strict you wish. Then a __$1$-groupoid__ is an $\infty$-groupoid such that all [[parallel pairs]] of $j$-morphisms are [[equivalence|equivalent]] for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent. If you rephrase equivalence of $1$-morphisms as [[equality]], which gives the same result up to [[equivalence of categories|equivalence]], then all that is left in this definition is a [[groupoid]]. Thus one may also say that a __$1$-groupoid__ is simply a groupoid. The point of all this is simply to fill in the general concept of $n$-[[n-groupoid|groupoid]]; nobody thinks of $1$-groupoids as a concept in their own right except simply as groupoids. Compare $1$-[[1-category|category]] and $1$-[[1-poset|poset]], which are defined on the same basis. ## Related concepts [[!include homotopy n-types - table]] [[!redirects 1-groupoids]]