Lemma 115.10.1. Assumptions and notation as in Simplicial, Lemma 14.32.1. There exists a section g : U \to V to the morphism f and the composition g \circ f is homotopy equivalent to the identity on V. In particular, the morphism f is a homotopy equivalence.
115.10 Simplicial methods
Proof. Immediate from Simplicial, Lemmas 14.32.1 and 14.30.8. \square
Lemma 115.10.2. Let \mathcal{C} be a category with finite coproducts and finite limits. Let X be an object of \mathcal{C}. Let k \geq 0. The canonical map
\mathop{\mathrm{Hom}}\nolimits (\Delta [k], X) \longrightarrow \text{cosk}_1 \text{sk}_1 \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X)
is an isomorphism.
Proof. For any simplicial object V we have
\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits (V, \text{cosk}_1 \text{sk}_1 \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X)) & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_1 V, \text{sk}_1 \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X)) \\ & = & \mathop{\mathrm{Mor}}\nolimits (i_{1!} \text{sk}_1 V, \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X)) \\ & = & \mathop{\mathrm{Mor}}\nolimits (i_{1!} \text{sk}_1 V \times \Delta [k], X) \end{eqnarray*}
The first equality by the adjointness of \text{sk} and \text{cosk}, the second equality by the adjointness of i_{1!} and \text{sk}_1, and the first equality by Simplicial, Definition 14.17.1 where the last X denotes the constant simplicial object with value X. By Simplicial, Lemma 14.20.2 an element in this set depends only on the terms of degree 0 and 1 of i_{1!} \text{sk}_1 V \times \Delta [k]. These agree with the degree 0 and 1 terms of V \times \Delta [k], see Simplicial, Lemma 14.21.3. Thus the set above is equal to \mathop{\mathrm{Mor}}\nolimits (V \times \Delta [k], X) = \mathop{\mathrm{Mor}}\nolimits (V, \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X)). \square
Lemma 115.10.3. Let \mathcal{C} be a category. Let X be an object of \mathcal{C} such that the self products X \times \ldots \times X exist. Let k \geq 0 and let C[k] be as in Simplicial, Example 14.5.6. With notation as in Simplicial, Lemma 14.15.2 the canonical map
\mathop{\mathrm{Hom}}\nolimits (C[k], X)_1 \longrightarrow (\text{cosk}_0 \text{sk}_0 \mathop{\mathrm{Hom}}\nolimits (C[k], X))_1
is identified with the map
\prod \nolimits _{\alpha : [k] \to [1]} X \longrightarrow X \times X
which is the projection onto the factors where \alpha is a constant map.
Proof. This is shown in the proof of Hypercoverings, Lemma 25.7.3. \square
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