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 $\to$ Ring.
For $R$ a commutative ring write $R Alg_{\mathrm{u}}$ for the category of nonassociative algebras with unit over $R$ and unit-preserving homomorphisms, and write $R Alg_{\mathrm{nu}}$ for nonunital nonassociative $R$-algebras. Note that $R Alg_{\mathrm{u}}$ is a subcategory of $R Alg_{\mathrm{nu}}$, as we use both ‘non-unital’ and ‘non-associative’ in accordance with the red herring principle.
The inclusion functor $U\colon R Alg_{\mathrm{u}} \to R Alg_{\mathrm{nu}}$ has a left adjoint $(-)^+\colon R Alg_{\mathrm{u}} \to R Alg{\mathrm{u}}$. For $A \in R Alg_{\mathrm{nu}}$ we say $A^+$ is the unitalization of $A$.
Explicitly, $A^+ = A \oplus R$ as an $R$-module, with product given by
or in general
We often write $(a, r)$ as $a + r$ or $a \oplus r$, which makes the above formulas obvious.
If $A$ is an associative algebra, then $A^+$ will also be associative; if $A$ is a commutative algebra, then $A^+$ will also be commutative.
See
The unitalization functor is not a conservative functor (Andruszkiewicz).
However, it does become conservative when restricted to the full subcategory of nonunital algebras over a field $k$ that do not admit a surjective homomorphism to $k$ (Andruszkiewicz).
Unitisation in the generality of Ek-algebra – hence for nonunital Ek-algebras – unitalization is the content of (Lurie, prop. 5.2.3.13).
Jacob Lurie, section 5.2.3 of Higher Algebra
Ryszard R. Andruszkiewicz, Surprise at Adjoining an Identity to an Algebra, Vietnam Journal of Mathematics, doi.
Last revised on April 24, 2021 at 19:27:32. See the history of this page for a list of all contributions to it.