Lemma 3.8 (05U0) · ScalingStacks

Lemma 3.8. Let $C$ be a category. Then there are natural isomorphisms

N([m]\times C)\approx F(m)\times NC\qquad\text{and}\qquad N(C^{I[n]})\approx(NC)^{\Delta[n]}

of simplicial spaces.

[05U1]

Proof. The first isomorphism follows from the fact that $N$ preserves products and that $N([m])\approx F(m)$ . The second isomorphism may be derived from the fact that $\iso(D^{I[n]})\approx(\iso D)^{I[n]}\approx(\iso D)^{[n]}$ for any category $D$ , and thus in particular when $D=C^{[m]}$ . ∎