En-ring
In mathematics, an -algebra in a symmetric monoidal infinity category C consists of the following data:
- An object for any open subset U of Rn homeomorphic to an n-disk.
- A multiplication map:
- for any disjoint open disks contained in some open disk V
subject to the requirements that the multiplication maps are compatible with composition, and that is an equivalence if . An equivalent definition is that A is an algebra in C over the little n-disks operad.
Examples
- An -algebra in vector spaces over a field is a unital associative algebra if n=1, and a unital commutative associative algebra if n≥2.
- An -algebra in categories is a monoidal category if n=1, a braided monoidal category if n=2, and a symmetric monoidal category if n≥3.
- If Λ is a commutative ring, then defines an -algebra in the infinity category of chain complexes of -modules.
gollark: ++exec```haskellmain = putStr $ (['\n','a','`','`','`','\n']) ++ (take 1900 $ concat $ repeat ":frog: ") ++ (['\n','`','`','`'])```
gollark: ++exec```haskellmain = putStr $ (['\n','a','`','`','`','\n']) ++ (take 1900 $ concat $ repeat ":chips: ") ++ (['\n','`','`','`'])```
gollark: ++exec```python# -*- coding: utf-8 -*-tb = "`" * 3print(" " + tb + "εσολανγς" + tb)```
gollark: ++exec```pythontb = "`" * 3print(" " + tb + "εσολανγς" + tb)```
gollark: ++exec```pythontb = "`" * 3print(" " + tb + "ABC" + tb)```
See also
- Categorical ring
References
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