homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 ∞-groupoids. These are equivalent to crossed complexes of groups and groupoids.
Pointed, 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 (more precisely, H$\mathbb{Z}$-modules). 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 (structured) $\infty$-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 | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | 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/2-stack | h-2-groupoid |
h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |
h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |
h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |
The term ∞-groupoid is sometimes considered to be too unwieldy, and some alternatives have been suggested or used, but none has gained wide acceptance.
Historically, the word “space” is often (ab)used to mean ∞-groupoid, due to the traditional presentation of $\infty Gpd$ by the model structure on topological spaces. Some have condemned this usage, but others argue that a “homotopy space” is a valid notion of “space”.
The term “space” is also often used to refer to simplicial sets. In particular, Bousfield and Kan in their book “Homotopy Limits, Completions and Localizations” write:
“These notes are written simplicially, i.e. whenever we say space we mean simplicial set.
More recently, Jacob Lurie‘s work continues this usage.
The term “homotopy type” is also quite close in meaning to “∞-groupoid”. Historically, it differed in that morphisms of homotopy types were mere homotopy classes of maps of ∞-groupoids, but more recently (especially with the advent of homotopy type theory some have used “homotopy type” synonymously with “$\infty$-groupoid”.
More radically, Dustin Clausen and Peter Scholze use the term “anima” (plural: anima) as a synonym for “$\infty$-groupoid”, in particular in relation to condensed mathematics. Reference: nCafe. In Purity for flat cohomology by Kęstutis Česnavičius and Peter Scholze they write “If the objects of $C$ are called widgets, then we call those of $Ani(C)$ animated widgets, except that we abbreviate $Ani(Set)$ to $Ani$ and the term ‘animated set’ to anima (plural: anima).”
Formulations in homotopy type theory include
See also at category object in an (infinity,1)-category for more along these lines.
Last revised on May 26, 2022 at 12:11:36. See the history of this page for a list of all contributions to it.