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Definition Let $\operatorname{\mathcal{C}}$ be a category. A nonunital monoidal structure on $\operatorname{\mathcal{C}}$ consists of the following data:

  • A functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, which we will refer to as the tensor product functor.

  • A collection of isomorphisms $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$, for $X,Y,Z \in \operatorname{\mathcal{C}}$, called the associativity constraints of $\operatorname{\mathcal{C}}$. We demand that the associativity constraints $\alpha _{X,Y,Z}$ depend functorially on $X,Y,Z$ in the following sense: for every triple of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$, and $h: Z \rightarrow Z'$, the diagram

    \[ \xymatrix@C =50pt@R=50pt{ X \otimes (Y \otimes Z) \ar [r]^-{ \alpha _{X,Y,Z } }_{\sim } \ar [d]^{ f \otimes (g \otimes h) } & (X \otimes Y) \otimes Z \ar [d]^{ (f \otimes g) \otimes h} \\ X' \otimes (Y' \otimes Z' ) \ar [r]^-{ \alpha _{X',Y',Z'} }_{\sim } & (X' \otimes Y') \otimes Z' } \]

    is commutative. In other words, we require that $\alpha = \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}}$ can be regarded as a natural isomorphism from the functor

    \[ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\xrightarrow { (X,Y,Z) \mapsto X \otimes (Y \otimes Z) } \operatorname{\mathcal{C}} \]

    to the functor

    \[ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\xrightarrow { (X,Y,Z) \mapsto (X \otimes Y) \otimes Z } \operatorname{\mathcal{C}}. \]

The associativity constraints of $\operatorname{\mathcal{C}}$ are required to satisfy the following additional condition:


For every quadruple of objects $W,X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram of isomorphisms

\[ \xymatrix@C =-30pt@R=30pt{ & W \otimes ((X \otimes Y) \otimes Z) \ar [rr]^-{ \alpha _{W, X \otimes Y, Z} }_{\sim } & & (W \otimes (X \otimes Y)) \otimes Z \ar [dr]^{ \alpha _{W, X, Y} \otimes \operatorname{id}_ Z}_{\sim } & \\ W \otimes (X \otimes (Y \otimes Z) ) \ar [ur]^{ \operatorname{id}_ W \otimes \alpha _{X,Y,Z} }_{\sim } \ar [drr]_{ \alpha _{W,X,Y \otimes Z}}^{\sim } & & & & ((W \otimes X) \otimes Y) \otimes Z \\ & & (W \otimes X) \otimes (Y \otimes Z) \ar [urr]_{\alpha _{ W \otimes X, Y, Z} }^{\sim } & & } \]


A nonunital monoidal category is a triple $(\operatorname{\mathcal{C}}, \otimes , \alpha )$, where $\operatorname{\mathcal{C}}$ is a category and $(\otimes , \alpha )$ is a nonunital monoidal structure on $\operatorname{\mathcal{C}}$.