homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
The Cosmic Cube of higher category theory is the name for a diagram whose vertices correspond to special types of n-categories. The cube looks like this:
We may take $n = \infty$ here as well, and we may also consider a version for (n,r)-categories. The three axes correspond to:
making $n$-categories ‘groupoidal’ — that is, making morphisms invertible, thus passing from general $n$-categories to n-groupoids;
making $n$-categories strict, thus passing from general $n$-categories to strict $n$-categories;
making $n$-categories symmetric monoidal or ‘stable’, thus passing from general $n$-categories to symmetric monoidal $n$-categories.
Each vertex of the cube can also be understood as corresponding to a version of homotopy theory:
$\infty$-groupoids yield ordinary homotopy theory, symmetric monoidal and groupal $\infty$-groupoids correspond to stable homotopy theory, strictly abelian strict $\infty$-groupoids correspond to homological algebra. $\infty$-Categories that are not $\infty$-groupoids correspond to directed homotopy theory.
Here we list the 8 vertices of the cube in the case of $\infty$-categories.
(…)
A strict ∞-groupoid is modeled by a crossed complex. Under ω-nerve it embeds into all ∞-groupoids, modeled as Kan complexes.
A strictly stable strict ∞-groupoid is modeled by a bounded-below chain complex of abelian groups. Under the embedding $\Theta$ of complexes into crossed complexes it embeds into strict ∞-groupoids.
For the definition of $\Theta$ see Nonabelian Algebraic Topology , section Crossed complexes from chain complexes.
Combining the above inclusions
yields in total the map $ChnCplx \to sAb$ from chain complexes to simplicial abelian groups (followed by the forgetful $sAb \to KanCpx$) of the Dold-Kan correspondence.
A strictly stable strict ∞-groupoid is modeled by a bounded-below chain complex of abelian groups. Under ω-nerve it embeds into all (connective) spectras, modeled as spectrum objects in Kan complexes.
A strictly stable ∞-groupoid is modeled by a connective spectrum. The forgetful functor to ∞-groupoids is also called $\Omega^\infty$ or the “zeroth-space functor.”
Some blog discussion