Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.1.4.11 (Nonunital Algebras). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $A$ be an object of $\operatorname{\mathcal{C}}$. A nonunital algebra structure on $A$ is a map $m: A \otimes A \rightarrow A$ for which the diagram

\[ \xymatrix { & A \otimes (A \otimes A) \ar [rr]_-{\alpha _{A,A,A}} \ar [dl]_{ \operatorname{id}\otimes m} & & (A \otimes A) \otimes A \ar [dr]^{m \otimes \operatorname{id}} & \\ A \otimes A \ar [drr]^{m} & & & & A \otimes A \ar [dll]_{m} \\ & & A & & } \]

is commutative. A nonunital algebra object of $\operatorname{\mathcal{C}}$ is a pair $(A,m)$, where $A$ is an object of $\operatorname{\mathcal{C}}$ and $m$ is a nonunital algebra structure on $A$. If $(A,m)$ and $(A', m')$ are nonunital algebra objects of $\operatorname{\mathcal{C}}$, then we say that a morphism $f: A \rightarrow A'$ is a nonunital algebra homomorphism if the diagram

\[ \xymatrix { A \otimes A \ar [r]^-{m} \ar [d]^{ f \otimes f} & A \ar [d]^{f} \\ A' \otimes A' \ar [r]^-{m'} & A' } \]

is commutative. We let $\operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ denote the category whose objects are nonunital algebra objects of $\operatorname{\mathcal{C}}$ and whose morphisms are nonunital algebra homomorphisms.

Let $\{ e\} $ denote the trivial monoid, regarded as a (strict) monoidal category having only identity morphisms (Example 2.1.1.3). Then we can identify objects $A \in \operatorname{\mathcal{C}}$ with functors $F: \{ e\} \rightarrow \operatorname{\mathcal{C}}$ (by means of the formula $A = F(e)$). Unwinding the definitions, we see that nonunital lax monoidal structures on the functor $F$ (in the sense of Definition 2.1.4.3) can be identified with nonunital algebra structures on the object $A = F(e)$. Under this identification, nonunital monoidal natural transformations correspond to homomorphisms of nonunital algebras. We therefore have an isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \{ e \} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$.