Higher category theory
higher category theory
Extra properties and structure
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 here as well, and we may also consider a version for (n,r)-categories. The three axes correspond to:
making -categories ‘groupoidal’ — that is, making morphisms invertible, thus passing from general -categories to n-groupoids;
making -categories strict, thus passing from general -categories to strict -categories;
making -categories symmetric monoidal or ‘stable’, thus passing from general -categories to symmetric monoidal -categories.
In terms of homotopy theory
Each vertex of the cube can also be understood as corresponding to a version of homotopy theory:
-groupoids yield ordinary homotopy theory, symmetric monoidal and groupal -groupoids correspond to stable homotopy theory, strictly abelian strict -groupoids correspond to homological algebra. -Categories that are not -groupoids correspond to directed homotopy theory.
Vertices of the cube
Here we list the 8 vertices of the cube in the case of -categories.
Stably monoidal -categories
Stably monoidal -groupoids
Strictly stably monoidal strict -groupoids
Edges of the cube
Strict -groupoids in all -groupoids
A strict ∞-groupoid is modeled by a crossed complex. Under ω-nerve it embeds into all ∞-groupoids, modeled as Kan complexes.
Strictly stable strict -groupoids in strict -groupoids
A strictly stable strict ∞-groupoid is modeled by a bounded-below chain complex of abelian groups. Under the embedding of complexes into crossed complexes it embeds into strict ∞-groupoids.
For the definition of see Nonabelian Algebraic Topology , section Crossed complexes from chain complexes.
Strictly stable strict -groupoids in all -groupoids
Combining the above inclusions
yields in total the map from chain complexes to simplicial abelian groups (followed by the forgetful ) of the Dold-Kan correspondence.
Strictly stable strict -groupoids in strictly stable -groupoids
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.
Strictly stable -groupoids in all -groupoids
A strictly stable ∞-groupoid is modeled by a connective spectrum. The forgetful functor to ∞-groupoids is also called or the “zeroth-space functor.”
Revised on October 6, 2010 22:55:34
by Urs Schreiber