equivalences in/of $(\infty,1)$-categories
The notion of $\infty$-groupoid is the generalization of that of group and groupoids to higher category theory:
an $\infty$-groupoid – equivalently an (∞,0)-category – is an ∞-category in which all k-morphisms for all $k$ are equivalences.
The collection of all $\infty$-groupoids forms the (∞,1)-category ∞Grpd.
Special cases of $\infty$-groupoids include groupoids, 2-groupoids, 3-groupoids, n-groupoids, deloopings of groups, 2-groups, ∞-groups.
There are many ways to present the (∞,1)-category ∞Grpd of all $\infty$-groupoids, or at least obtain its homotopy category.
A simple and very useful incarnation of $\infty$-groupoids is available using a geometric definition of higher categories in the form of simplicial sets that are Kan complexes: the $k$-cells of the underlying simplicial set are the k-morphisms of the $\infty$-groupoid, and the Kan horn-filler conditions encode the fact that adjacent $k$-morphisms have a (non-unique) composite $k$-morphism and that every $k$-morphism is invertible with respect to this composition. See Kan complex for a detailed discussion of how these incarnate $\infty$-groupoids.
The (∞,1)-category of all $\infty$-groupoids is presented along these lines by the Quillen model structure on simplicial sets, whose fibrant-cofibrant objects are precisely the Kan complexes:
One may turn this geometric definition into an algebraic definition of ∞-groupoids by choosing horn-fillers . The resulting notion is that of an algebraic Kan complex that has been shown by Thomas Nikolaus to yield an equivalent (∞,1)-category of $\infty$-groupoids.
There are various model categories which are Quillen equivalent to $sSet_{Quillen}$. For instance the standard model structure on topological spaces, a model structure on marked simplicial sets and many more. All these therefore present ∞Grpd.
Moreover, the corresponding homotopy category of an (∞,1)-category $Ho(\infty Grpd)$, hence a category whose objects are homotopy types of $\infty$-groupoids, is given by the homotopy category of the category of presheaves over any test category. See there for more details.
Every other algebraic definition of omega-categories is supposed to yield an equivalent notion of $\infty$-groupoid when restricted to $\omega$-categories all whose k-morphisms are invertible. This is the statement of the homotopy hypothesis, which is for Kan complexes and algebraic Kan complexes a theorem and for other definitions regarded as a consistency condition.
Notably in Pursuing Stacks and the earlier letter to Larry Breen, Alexander Grothendieck initiated the study of $\infty$-groupoids and the homotopy hypothesis with his original definition of Grothendieck weak infinity-groupoids, that has recently attracted renewed attention.
One may also consider entirely strict $\infty$-groupoids, usually called $\omega$-groupoids or strict ω-groupoids. These are equivalent to crossed complexes of groups and groupoids.
0-connected $\infty$-groupoids are the delooping $\mathbf{B}G$ of ∞-groups (see looping and delooping).
These are presented by simplicial groups. Notably abelian simplicial groups are therefore a model for abelian $\infty$-groupoids. Under the Dold-Kan correspondence these are equivalent to non-negatively graded chain complexes, which therefore also are a model for abelian $\infty$-groupoids. This way much of homological algebra is secretly the study of special $\infty$-groupoids.
homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|
h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |
h-level 1 | (-1)-truncated | (-1)-groupoid/truth value | (0,1)-sheaf | mere proposition, h-proposition | |
h-level 2 | 0-truncated | homotopy 0-type | 0-groupoid/set | sheaf | h-set |
h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid |
h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | (3,1)-sheaf | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
Formulations in homotopy type theory include
See also at category object in an (infinity,1)-category for more along these lines.