The Stacks project

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14.28 Homotopies and cosimplicial objects

Let $\mathcal{C}$ be a category with finite products. Let $V$ be a cosimplicial object and consider $\mathop{\mathrm{Hom}}\nolimits (\Delta [1], V)$, see Section 14.14. The morphisms $e_0, e_1 : \Delta [0] \to \Delta [1]$ produce two morphisms $e_0, e_1 : \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V) \to V$.

Definition 14.28.1. Let $\mathcal{C}$ be a category having finite products. Suppose that $U$ and $V$ are two cosimplicial objects of $\mathcal{C}$. We say morphisms $a, b : U \to V$ are homotopic if there exists a morphism

\[ h : U \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V) \]

such that $a = e_0 \circ h$ and $b = e_1 \circ h$. In this case $h$ is called a homotopy connecting $a$ and $b$.

This is really exactly the same as the notion we introduced for simplicial objects earlier. In particular, recall that $\Delta [1]_ n$ is a finite set, and that

\[ h_ n = (h_{n, \alpha }) : U \longrightarrow \prod \nolimits _{\alpha \in \Delta [1]_ n} V_ n \]

is given by a collection of maps $h_{n, \alpha } : U_ n \to V_ n$ parametrized by elements of $\Delta [1]_ n = \mathop{Mor}\nolimits _\Delta ([n], [1])$. As in Lemma 14.26.2 these morphisms satisfy some relations. Namely, for every $f : [n] \to [m]$ in $\Delta $ we should have

14.28.1.1
\begin{equation} \label{simplicial-equation-property-homotopy-cosimplicial} h_{m, \alpha } \circ U(f) = V(f) \circ h_{n, \alpha \circ f} \end{equation}

The condition that $a = e_0 \circ h$ means that $a_ n = h_{n, 0 : [n] \to [1]}$ where $0 : [n] \to [1]$ is the constant map with value zero. Similarly, we should have $b_ n = h_{n, 1 : [n] \to [1]}$. In particular we deduce once more that the notion of homotopy can be formulated between cosimplicial objects of any category, i.e., existence of products is not necessary. Here is a precise formulation of why this is dual to the notion of a homotopy between morphisms of simplicial objects.

Lemma 14.28.2. Let $\mathcal{C}$ be a category having finite products. Suppose that $U$ and $V$ are two cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of cosimplicial objects. Recall that $U$, $V$ correspond to simplicial objects $U'$, $V'$ of $\mathcal{C}^{opp}$. Moreover $a, b$ correspond to morphisms $a', b' : V' \to U'$. The following are equivalent

  1. The morphisms $a, b : U \to V$ of cosimplicial objects are homotopic.

  2. The morphisms $a', b' : V' \to U'$ of simplicial objects of $\mathcal{C}^{opp}$ are homotopic.

Proof. If $\mathcal{C}$ has finite products, then $\mathcal{C}^{opp}$ has finite coproducts. And the contravariant functor $(-)' : \mathcal{C} \to \mathcal{C}^{opp}$ transforms products into coproducts. Then it is immediate from the definitions that $(\mathop{\mathrm{Hom}}\nolimits (\Delta [1], V))' = V' \times \Delta [1]$. And so on and so forth. $\square$

slogan

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! $\square$

Lemma 14.28.4. Let $f : X \to Y$ be a morphism of a category $\mathcal{C}$ with pushouts. Assume $f$ has a section $s$. Consider the cosimplicial object $U$ constructed in Example 14.5.5 starting with $f$. The morphism $U \to U$ which in each degree is the self map of $Y \amalg _ X \ldots \amalg _ X Y$ given by $s \circ f$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant cosimplicial object $X$.

Proof. The dual statement which is Lemma 14.26.9. Hence this lemma follows on applying Lemma 14.28.2. $\square$

slogan

Lemma 14.28.5. Let $\mathcal{A}$ be an abelian category. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \to s(V)$, and $Q(a), Q(b) : Q(U) \to Q(V)$ are homotopic maps of cochain complexes.

Proof. Let $(-)' : \mathcal{A} \to \mathcal{A}^{opp}$ be the contravariant functor $A \mapsto A$. By Lemma 14.28.4 the maps $a'$ and $b'$ are homotopic. By Lemma 14.27.1 we see that $s(a')$ and $s(b')$ are homotopic maps of chain complexes. Since $s(a') = (s(a))'$ and $s(b') = (s(b))'$ we conclude that also $s(a)$ and $s(b)$ are homotopic by applying the additive contravariant functor $(-)'' : \mathcal{A}^{opp} \to \mathcal{A}$. The result for the $Q$-complexes follows from the direct sum decomposition of Lemma 14.25.1 for example. $\square$


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