Operadic limits in envelope

Remark 18.1.2. These conditions are direct operadic analogues of the mapping-anima characterizations of ordinary limits and colimits, with unary mapping animae replaced by multimorphism animae. Taking \(n=1\) shows that an operadic limit is a limit in \(\Oo _{\lra {1}}\), and an operadic colimit is a colimit in \(\Oo _{\lra {1}}\). We will therefore usually assume that the relevant (co)limits already exist in \(\Oo _{\lra {1}}\) and only ask whether they are operadic.

For limits, there is also a useful reformulation through the envelope. Every object of \(\Env (\Oo )\) is a tensor \(x_1 \otimes \dots \otimes x_n\) of colors, and by Lemma 17.3.17 the multimorphism animae compute the mapping animae out of such tensors, \(\Oo ((x_1, \dots , x_n); y) \simeq \Hom _{\Env (\Oo )}(x_1 \otimes \dots \otimes x_n, y)\). Testing against all colors \(x_1, \dots , x_n\) therefore shows that a cone in \(\Oo _{\lra {1}}\) is an operadic limit if and only if it becomes a limit cone in \(\Env (\Oo )\) under the inclusion \(\Oo _{\lra {1}} \hookrightarrow \Env (\Oo )\).

For colimits, the corresponding statement is only relative: after tensoring with any fixed colors \(x_2,\dots ,x_n\), the cocone is colimiting when tested by mapping into colors of \(\Oo \). It need not be a colimit cocone in the whole envelope; in particular, an operadic initial object need not be initial there. Compare Example 18.1.7.