Let $\Delta_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta_{+} \hookrightarrow \Delta$ for the inclusion.

Now assume we have a semi-simplicial diagram $X \colon \Delta_{+}^{\text{op}} \to M$. We can then form the left Kan extension $i_! X \colon \Delta^{\text{op}} \to M$, which adds the degeneracies to the semi-simplicial diagram $X$. Now here's my question:

Is $$\text{hocolim}_{\Delta_{+}^{\text{op}}}X = \text{hocolim}_{\Delta^{\text{op}}}i_{!}X,$$ or are they at least equivalent? What about if I take a homotopy left Kan extension instead?

I've got another question, but I'm afraid this might be too specific. A Segal groupoid in a model category $M$ is a simplicial object $X \colon \Delta^{\text{op}} \to M$ which satisfies the Segal condition, i.e. $$X_n \to X_1 \times ^h _{X_0} \dots \times ^h _{X_0} X_1$$ is a weak equivalence, and also $$d_0 \times d_1 \colon X_2 \to X_1 \times ^h _{d_0,X_0, d_0} X_1 $$ is a weak equivalence. The second condition says that every horn of type $b \leftarrow a \to c$ can be filled. Since both conditions only involve face maps, they make perfect sense for a semi-simplicial object too. So I guess it makes sense to ask, given a semisimplicial object $Y \colon \Delta_{+}^{\text{op}} \to M$ satisfying the two conditions above, is the (homotopy) left Kan extension $i_! Y$ a Segal groupoid in $M$?