Multimorphism operad

Example 12.2.7 (Multimorphism operad). Let \(C\) be a symmetric monoidal category. We define a colored operad \(\Mm _C\), called the multimorphism operad¹ of \(C\), as follows:

• The colors of \(\Mm _C\) are the objects of \(C\).
• For objects \(x,y_1, \dots , y_n\) of \(C\), the multimorphisms in \(\Mm _C\) are given by maps from the tensor product: \[ \Mm _C((y_1, \dots ,y_n);x) \quad := \quad \Hom _C(y_1 \otimes \dots \otimes y_n;x); \]
• The identity operations are given by the identity maps \(\id _x \in \Hom _C(x,x)\);
• The composition of operations is given by the composition in \(C\);
• The permutation operations are induced by the isomorphisms \(y_1 \otimes \dots \otimes y_n \simeq y_{\sigma (1)} \otimes \dots \otimes y_{\sigma (n)}\) coming from the symmetry isomorphisms in \(C\).

Note that for an object \(A\) of \(C\), the non-colored operad \((\Mm _C)_A\) obtained by applying Example 12.2.5 is precisely the endomorphism operad \(\oEnd _C(A)\) introduced in Example 12.1.9.