Tags: #higher-algebra/infty-cats
- An ∞-groupoid is an infinity categories in which all morphisms are invertible.
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0-groupoid: A set -1-groupoid: An ordinary groupoid, -Play the role analogous to sets in classical category theory.
- Have homs that are again infinity groupoids.
- Pullbacks in \({ \underset{\infty}{ {\mathsf{Grpd}}} }\): limits over morphisms in \({ \underset{\infty}{ {\mathsf{Grpd}}} }\) of \(A_1 \to B \leftarrow A_2\)
- Fibers in \({ \underset{\infty}{ {\mathsf{Grpd}}} }\): for an object \(b\in B \in { \underset{\infty}{ {\mathsf{Grpd}}} }\), fibers are pullbacks over the morphism \(s_b: \one \to B\) that selects the object \(b\in B\)
- See homotopy sum
- Maps of ∞-groupoids with codomain \(\mathsf{B}\) form the objects of a slice category \({ \underset{\infty}{ {\mathsf{Grpd}}} }_{/B}\)
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A morphism of ∞-groupoids \(X \to B\) can be interpreted as a family of ∞-groupoids parametrised by \(B\), namely the fibres \(X_b\).
- Equivalently, a presheaf \(B\to { \underset{\infty}{ {\mathsf{Grpd}}} }\)
- operads.
- Simplicial \({ \underset{\infty}{ {\mathsf{Grpd}}} }= {\mathsf{Fun}}(\Delta^{\operatorname{op}}, { \underset{\infty}{ {\mathsf{Grpd}}} })\)
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∞-groupoids form a (large) ∞-category denoted \({ \underset{\infty}{ {\mathsf{Grpd}}} }\)
- It can be described explicitly as the Kan complex.