Higher Groups and Higher Normality
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914901003337728 |
|---|---|
| author | Beardsley, Jonathan Fox, Landon |
| author_facet | Beardsley, Jonathan Fox, Landon |
| contents | In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $\infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $\infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $\infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $\infty$-topoi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21210 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Groups and Higher Normality Beardsley, Jonathan Fox, Landon Algebraic Topology Category Theory 18N60, 55P35, 18N70 In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary $\infty$-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a "normal closure" monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of $\infty$-topoi preserve normal maps or, equivalently, that monoidal functors of $\infty$-topoi preserve the property of "being a fiber" for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in $\infty$-topoi in the "usual way." One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in $\infty$-topoi. |
| title | Higher Groups and Higher Normality |
| topic | Algebraic Topology Category Theory 18N60, 55P35, 18N70 |
| url | https://arxiv.org/abs/2407.21210 |