# Computing weak operadic colimits as colimits

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.