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The Stacks project

Lemma 14.5.2. Let \mathcal{C} be a category.

  1. Given a cosimplicial object U in \mathcal{C} we obtain a sequence of objects U_ n = U([n]) endowed with the morphisms \delta ^ n_ j = U(\delta ^ n_ j) : U_{n - 1} \to U_ n and \sigma ^ n_ j = U(\sigma ^ n_ j) : U_{n + 1} \to U_ n. These morphisms satisfy the relations displayed in Lemma 14.2.3.

  2. Conversely, given a sequence of objects U_ n and morphisms \delta ^ n_ j, \sigma ^ n_ j satisfying these relations there exists a unique cosimplicial object U in \mathcal{C} such that U_ n = U([n]), \delta ^ n_ j = U(\delta ^ n_ j), and \sigma ^ n_ j = U(\sigma ^ n_ j).

  3. A morphism between cosimplicial objects U and U' is given by a family of morphisms U_ n \to U'_ n commuting with the morphisms \delta ^ n_ j and \sigma ^ n_ j.

Proof. This follows from Lemma 14.2.4. \square


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