Timeline for Can one bypass the geometric realization in the definition of algebraic $K$-theory?
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6 events
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| Aug 5, 2023 at 20:43 | comment | added | Dmitri Pavlov | @Stabilo: Using a concrete description of $\def\Exi{\mathop{\sf Ex^∞}}\def\cC{{\cal C}}\Exi \cC$, you can interpret an element in the $n$th homotopy group of $\Exi \cC$ as a diagram in $\cC$ whose indexing category is given by the subdivided $n$-sphere (which is always a poset if you subdivide two or more times). | |
| Aug 5, 2023 at 20:39 | comment | added | Dmitri Pavlov | @Stabilo: Of course: $\def\Exi{\mathop{\sf Ex^∞}}\def\cC{{\cal C}}\Exi \cC$ can be shown to present the ∞-groupoid $\cC[\cC^{-1}]$, i.e., $\cC$ with all of its morphisms inverted up to homotopy. Even if $\cC$ is an ordinary category, $\cC[\cC^{-1}]$ is typically an ∞-groupoid, not a 1-groupoid. The $n$th homotopy group of this ∞-groupoid is precisely the group of isomorphism classes of n-morphisms in $\cC[\cC^{-1}]$ with source and target being identities on the basepoint object and the group structure being given by composition in $\cC[\cC^{-1}]$. | |
| Aug 5, 2023 at 9:07 | comment | added | Stabilo | A curiosity: modeling $\infty$-categories by weak Kan complexes, thanks to the definition you mention, one could define homotopy groups of a $\infty$-category $\mathcal{C}$ (say, pointed by an initial object). Is there an interpretation of these, e.g. when $\mathcal{C}$ is the nerve of an ordinary category? | |
| Aug 5, 2023 at 8:48 | comment | added | Stabilo | Many thanks! This is helpful. I also got me a copy of Goerss-Jardine! | |
| Aug 5, 2023 at 8:46 | vote | accept | Stabilo | ||
| Aug 4, 2023 at 16:25 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |