Lemma 14.34.2. In Situation 14.34.1 the system $X = (X_ n, d^ n_ j, s^ n_ j)$ is a simplicial object of $\text{Fun}(\mathcal{A}, \mathcal{A})$ and $\epsilon _0$ defines an augmentation $\epsilon $ from $X$ to the constant simplicial object with value $X_{-1} = \text{id}_\mathcal {A}$.
Proof. Consider $Y = U \circ V : \mathcal{A} \to \mathcal{A}$. We already have the transformation $d : Y = U \circ V \to \text{id}_\mathcal {A}$. Let us denote
This places us in the situation of Example 14.33.1. It is immediate from the formulas that the $X, d^ n_ i, s^ n_ i$ constructed above and the $X, s^ n_ i, s^ n_ i$ constructed from $Y, d, s$ in Example 14.33.1 agree. Thus, according to Lemma 14.33.2 it suffices to prove that
The first equal sign translates into the equality
which holds if we have $1_ U = (d \star 1_ U) \circ (1_ U \star \eta )$ which in turn holds by (14.34.1.1). Similarly for the second equal sign. For the last equation we need to prove
For this it suffices to prove $(\eta \star 1_ V \star 1_ U) \circ \eta = (1_ V \star 1_ U \star \eta ) \circ \eta $ which is true because both sides are the same as $\eta \star \eta $. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: