Functors preserve homotopic morphisms of (co)simplicial objects.

Lemma 14.28.3. Let $\mathcal{C}, \mathcal{C}', \mathcal{D}, \mathcal{D}'$ be categories such that $\mathcal{C}, \mathcal{C}'$ have finite products, and $\mathcal{D}, \mathcal{D}'$ have finite coproducts.

1. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{D}$. Let $F : \mathcal{D} \to \mathcal{D}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \to F(V)$ of simplicial objects.

2. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. Let $F : \mathcal{C} \to \mathcal{C}'$ be a covariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(U) \to F(V)$ of cosimplicial objects.

3. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{D}$. Let $F : \mathcal{D} \to \mathcal{C}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \to F(U)$ of cosimplicial objects.

4. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. Let $F : \mathcal{C} \to \mathcal{D}$ be a contravariant functor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$ are homotopic morphisms $F(V) \to F(U)$ of simplicial objects.

Proof. By Lemma 14.28.2 above, we can turn $F$ into a covariant functor between a pair of categories which have finite coproducts, and we have to show that the functor preserves homotopic pairs of maps. It is explained in Remark 14.26.4 how this is the case. Even if the functor does not commute with coproducts! $\square$

Comment #989 by on

On the first line, there is a comma missing between $\mathcal{D}$ and $\mathcal{D}'$.

Comment #990 by on

Suggested slogan: Functors preserve homotopic morphisms of (co)simplicial objects.

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