nLab
cosmic cube

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Higher category theory

higher category theory

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Extra properties and structure

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Contents

Idea

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=n = \infty here as well, and we may also consider a version for (n,r)-categories. The three axes correspond to:

  • making nn-categories ‘groupoidal’ — that is, making morphisms invertible, thus passing from general nn-categories to n-groupoids;

  • making nn-categories strict, thus passing from general nn-categories to strict nn-categories;

  • making nn-categories symmetric monoidal or ‘stable’, thus passing from general nn-categories to symmetric monoidal nn-categories.

In terms of homotopy theory

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.

Vertices of the cube

Here we list the 8 vertices of the cube in the case of \infty-categories.

Strict \infty-categories

Strict \infty-groupoids

Stably monoidal \infty-categories

Stably monoidal \infty-groupoids

Strictly stably monoidal strict \infty-groupoids

Etc.

(…)

Edges of the cube

Strict \infty-groupoids in all \infty-groupoids

A strict ∞-groupoid is modeled by a crossed complex. Under ω-nerve it embeds into all ∞-groupoids, modeled as Kan complexes.

CrsCplx N Δ KanCplx StrGrpd Grpd. \array{ CrsCplx &\stackrel{N^\Delta}{\hookrightarrow}& KanCplx \\ \downarrow && \downarrow \\ Str \infty Grpd &\hookrightarrow& \infty Grpd } \,.

Strictly stable strict \infty-groupoids in strict \infty-groupoids

A strictly stable strict ∞-groupoid is modeled by a bounded-below chain complex of abelian groups. Under the embedding Θ\Theta of complexes into crossed complexes it embeds into strict ∞-groupoids.

ChnCplx Θ CrsCplx StrAbStrGrpd StrGrpd. \array{ ChnCplx &\stackrel{\Theta}{\hookrightarrow}& CrsCplx \\ \downarrow && \downarrow \\ StrAb Str \infty Grpd &\hookrightarrow& Str \infty Grpd } \,.

For the definition of Θ\Theta see Nonabelian Algebraic Topology , section Crossed complexes from chain complexes.

Strictly stable strict \infty-groupoids in all \infty-groupoids

Combining the above inclusions

ChainCplx Θ CrossedCplx N Δ KanCplx StrAbStrGrpd StrGrpd Grpd \array{ ChainCplx &\stackrel{\Theta}{\hookrightarrow}& CrossedCplx &\stackrel{N^\Delta}{\hookrightarrow}& KanCplx \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd }

yields in total the map ChnCplxsAbChnCplx \to sAb from chain complexes to simplicial abelian groups (followed by the forgetful sAbKanCpxsAb \to KanCpx) of the Dold-Kan correspondence.

Strictly stable strict \infty-groupoids in strictly stable \infty-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.

ChnCplx + Σ N ΔΘ Sp(KanCplx) StrAbStrGrpd Sp(Grpd). \array{ ChnCplx^+ &\stackrel{\Sigma^\infty \N^\Delta \Theta}{\hookrightarrow}& Sp(KanCplx) \\ \downarrow && \downarrow \\ StrAb Str \infty Grpd &\hookrightarrow& Sp(\infty Grpd) } \,.

Strictly stable \infty-groupoids in all \infty-groupoids

A strictly stable ∞-groupoid is modeled by a connective spectrum. The forgetful functor to ∞-groupoids is also called Ω \Omega^\infty or the “zeroth-space functor.”

References

Revised on October 6, 2010 22:55:34 by Urs Schreiber (87.212.203.135)