Proposition 11.3 (05VP) · ScalingStacks

(1)

Given $f\in\map(x,y)$ and $g\in\map(y,z)$ , and $g\circ f$ the result of a composition, then $(g\circ f)_{*}\sim g_{*}\circ f_{*}$ and $(g\circ f)^{*}\sim f^{*}\circ g^{*}$ .
(2)

Given $x\in{\operatorname{ob}}W$ then $(\id_{x})_{*}\sim(\id_{x})^{*}\sim\id_{\map(x,x)}$ .

Proof. To prove (1), let $k\in\map(x,y,z)$ be a composition of $f$ and $g$ which results in a composite $g\circ f$ . To show that $(g\circ f)_{*}\sim g_{*}\circ f_{*}$ , it suffices to show that both sides of the equation are equal (in the homotopy category of spaces) to the zig-zag

\map(w,x)\xrightarrow{\{k\}\times 1}\map(x,y,z)\times\map(w,x)\xleftarrow{\sim}\map(w,x,y,z)\rightarrow\map(w,z).

The proof that $(g\circ f)^{*}\sim f^{*}\circ g^{*}$ is similar.

The proof of (2) is straightforward. ∎