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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 $M$?

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It might be helpful to note that the inclusion i (on the level of oo-categories) is cofinal, see Lurie Higher Topos Theory Lemma 6.5.3.7 –  David Ben-Zvi Jun 9 '11 at 3:17
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1 Answer 1

up vote 8 down vote accepted

As I understand it, homotopy left Kan extension along any functor $i:\mathcal C\to\mathcal D$ preserves hocolim.

Left Kan extension $i_!$ is left adjoint to restriction $i^\star$, i.e. composition with $i$. (Of course, if $i$ is not fully faithful, "extension" is a bit of a misnomer: restriction composed with "extension" is not isomorphic to the identity.) Colimit over $\mathcal D$ is left Kan extension along the functor $p:\mathcal D\to \star$. Since $(p\circ i)^\star=i^\star\circ p^\star$, the left adjoints also compose as they should.

Surely the same is true in a derived sense.

Kan extension (as opposed to homotopy Kan extension) will preserve hocolim in cases where it coincides with homotopy Kan extension, i.e. in cases where it preserves weak equivalences. In the case of $i:\Delta_+\to\Delta$, this requires only that coproduct in $\mathcal M$ preserve weak equivalences, which is true for example for spaces and for based simplicial sets but not for based spaces.

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