symmetric monoidal (∞,1)-category of spectra
The unitalization of a non-unital algebra is a unital algebra with a “unit” (an identity element) freely adjoined.
This is a free functor: Rng Ring.
For a commutative ring write for the category of nonassociative algebras with unit over and unit-preserving homomorphisms, and write for nonunital nonassociative -algebras. Note that is a subcategory of , as we use both ‘non-unital’ and ‘non-associative’ in accordance with the red herring principle.
The inclusion functor has a left adjoint . For we say is the unitalization of .
Explicitly, as an -module, with product given by
or in general
We often write as or , which makes the above formulas obvious.
If is an associative algebra, then will also be associative; if is a commutative algebra, then will also be commutative.
See
Unitisation in the generality of Ek-algebra – hence for nonunital Ek-algebras – unitalization is the content of (Lurie, prop. 5.2.3.13).
Last revised on May 31, 2017 at 05:44:48. See the history of this page for a list of all contributions to it.