I am trying to reduce the computation of weak operadic colimits to colimits. Let me introduct some notation. Let $q:C^{\otimes} \to N(Fin_*)$ be a symmetric monoidal category.

Let $p: K \to C^{\otimes}_{act}$ be a diagram. Define $p^{\otimes}: K \to C$ in the following way. For every $k \in K$, there is a cocartesian lifting $s(k): p(k) \to k'$ of the unique active map $qp(k) \to <1>$.

Furthermore, we have an homotopy between $qp: K \to N(Fin_*)^{act}$ and the constant functor $<1>$, because $<1>$ is final in $N(Fin_*)^{act}$.

By a lemma used in the proof of the existence of transition functors, one can lift this homotopy to an homotopy $H: K \times \Delta^1 \to C^{\otimes}_{act}$. In particular, this yield a functor $H | K \times \Delta^{\{1\}} =: p^{\otimes}: K \to C$, the underlying category of $C^{\otimes}$.

Set $C^{act}_{p/}:= C \times_{C^{\otimes}_{act}} (C^{\otimes}_{act})_{p/} $, i.e. cocones with cocone point lying over 1. I want to show that $$ C^{act}_{p/} \simeq C_{p^{\otimes}/} $$

The idea behind is the following: if $K$ was a point, we could lift a map $p^{\otimes}(k) \to c$ to a map $p(k) \to c$ via the cocartesian lift $s(k)$. This realizes the intuitive idea that $Mul_{C^{\otimes}}( (X_1, \ldots, X_n), c) \simeq Hom_C( X_1 \otimes \ldots \otimes X_n, c)$. I believe that this could be generalized to arbitrary diagrams: intuitively, if we have a cone towards an object lying over 1, this factors through a cone lying over 1, made of "pointwise tensor products".

Note that this would yield the following

Corollary. Let $\bar{p}:K^{\triangleright} \to C^{\otimes}_{act}$ a diagram. Then it is a weak operadic q-colimit diagram iff $\bar{p}^{\otimes}$ is a colimit diagram, because equivalences preserve initial objects.

I would be happy also if someone could solve the case $K=\Delta^1$, that would shed light on how to solve the general case simplex by simplex.


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