The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

14.26 Homotopies

Consider the simplicial sets $\Delta [0]$ and $\Delta [1]$. Recall that there are two morphisms

\[ e_0, e_1 : \Delta [0] \longrightarrow \Delta [1], \]

coming from the morphisms $[0] \to [1]$ mapping $0$ to an element of $[1] = \{ 0, 1\} $. Recall also that each set $\Delta [1]_ k$ is finite. Hence, if the category $\mathcal{C}$ has finite coproducts, then we can form the product

\[ U \times \Delta [1] \]

for any simplicial object $U$ of $\mathcal{C}$, see Definition 14.13.1. Note that $\Delta [0]$ has the property that $\Delta [0]_ k = \{ *\} $ is a singleton for all $k \geq 0$. Hence $U \times \Delta [0] = U$. Thus $e_0, e_1$ above gives rise to morphisms

\[ e_0, e_1 : U \to U \times \Delta [1]. \]

Definition 14.26.1. Let $\mathcal{C}$ be a category having finite coproducts. Suppose that $U$ and $V$ are two simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be two morphisms.

  1. We say a morphism

    \[ h : U \times \Delta [1] \longrightarrow V \]

    is a homotopy connecting $a$ to $b$ if $a = h \circ e_0$ and $b = h \circ e_1$.

  2. We say morphisms $a$ and $b$ are homotopic if there exists a homotopy connecting $a$ to $b$ or a homotopy connecting $b$ to $a$.

Warning: Being homotopic is not an equivalence relation on the set of all morphisms from $U$ to $V$! The relation “there exists a homotopy from $a$ to $b$” is not symmetric.

It turns out we can define homotopies between pairs of maps of simplicial objects in any category. To do this you just work out what it means to have the morphisms $h_ n : (U \times \Delta [1])_ n \to V_ n$ in terms of the mapping property of coproducts.

Let $\mathcal{C}$ be a category with finite coproducts. Let $U$, $V$ be simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms. Further, suppose that $h : U \times \Delta [1] \to V$ is a homotopy connecting $a$ to $b$. For every $n \geq 0$ let us write

\[ \Delta [1]_ n = \{ \alpha ^ n_0, \ldots , \alpha ^ n_{n + 1}\} \]

where $\alpha ^ n_ i : [n] \to [1]$ is the map such that

\[ \alpha ^ n_ i(j) = \left\{ \begin{matrix} 0 & \text{if} & j < i \\ 1 & \text{if} & j \geq i \end{matrix} \right. \]

Thus

\[ h_ n : (U \times \Delta [1])_ n = \coprod U_ n \cdot \alpha ^ n_ i \longrightarrow V_ n \]

has a component $h_{n, i} : U_ n \to V_ n$ which is the restriction to the summand corresponding to $\alpha ^ n_ i$ for all $i = 0, \ldots , n + 1$.

Lemma 14.26.2. In the situation above, we have the following relations:

  1. We have $h_{n, 0} = b_ n$ and $h_{n, n + 1} = a_ n$.

  2. We have $d^ n_ j \circ h_{n, i} = h_{n - 1, i - 1} \circ d^ n_ j$ for $i > j$.

  3. We have $d^ n_ j \circ h_{n, i} = h_{n - 1, i} \circ d^ n_ j$ for $i \leq j$.

  4. We have $s^ n_ j \circ h_{n, i} = h_{n + 1, i + 1} \circ s^ n_ j$ for $i > j$.

  5. We have $s^ n_ j \circ h_{n, i} = h_{n + 1, i} \circ s^ n_ j$ for $i \leq j$.

Conversely, given a system of maps $h_{n, i}$ satisfying the properties listed above, then these define a morphism $h$ which is a homotopy between $a$ and $b$.

Proof. Omitted. You can prove the last statement using the fact, see Lemma 14.2.4 that to give a morphism of simplicial objects is the same as giving a sequence of morphisms $h_ n$ commuting with all $d^ n_ j$ and $s^ n_ j$. $\square$

Example 14.26.3. Suppose in the situation above $a = b$. Then there is a trivial homotopy between $a$ and $b$, namely the one with $h_{n, i} = a_ n = b_ n$.

Remark 14.26.4. Let $\mathcal{C}$ be any category (no assumptions whatsoever). We say that a pair of morphisms $a, b : U \to V$ of simplicial objects are homotopic if there exist morphisms1 $h_{n, i} : U_ n \to V_ n$, for $n \geq 0$, $i = 0, \ldots , n + 1$ satisfying the relations of Lemma 14.26.2 (potentially with the roles of $a$ and $b$ switched). This is a “better” definition, because it applies to any category. Also it has the following property: if $F : \mathcal{C} \to \mathcal{C}'$ is any functor then $a$ homotopic to $b$ implies trivially that $F(a)$ is homotopic to $F(b)$. Since the lemma says that the newer notion is the same as the old one in case finite coproduct exist, we deduce in particular that functors preserve the old notion whenever both categories have finite coproducts.

Remark 14.26.5. Let $\mathcal{C}$ be any category. Suppose two morphisms $a, a' : U \to V$ of simplicial objects are homotopic. Then for any morphism $b : V \to W$ the two maps $b \circ a, b \circ a' : U \to W$ are homotopic. Similarly, for any morphism $c : X \to U$ the two maps $a \circ c, a' \circ c : X \to V$ are homotopic. In fact the maps $b \circ a \circ c, b \circ a' \circ c : X \to W$ are homotopic. Namely, if the maps $h_{n, i} : U \to U$ define a homotopy between $a$ and $a'$ then the maps $b \circ h_{n, i} \circ c$ define a homotopy between $b \circ a \circ c$ and $b \circ a' \circ c$.

Definition 14.26.6. Let $U$ and $V$ be two simplicial objects of a category $\mathcal{C}$. We say a morphism $a : U \to V$ is a homotopy equivalence if there exists a morphism $b : V \to U$ such that $a \circ b$ is homotopic to $\text{id}_ V$ and $b \circ a$ is homotopic to $\text{id}_ U$. If there exists such a morphism between $U$ and $V$, then we say that $U$ and $V$ are homotopy equivalent2.

Example 14.26.7. The simplicial set $\Delta [m]$ is homotopy equivalent to $\Delta [0]$. Namely, there is a unique morphism $f : \Delta [m] \to \Delta [0]$ and we take $g : \Delta [0] \to \Delta [m]$ to be given by the inclusion of the last $0$-simplex of $\Delta [m]$. We have $f \circ g = \text{id}$ and we will give a homotopy $h : \Delta [m] \times \Delta [1] \to \Delta [m]$ between $\text{id}_{\Delta [m]}$ and $g \circ f$. Namely $h$ given by the maps

\[ \mathop{Mor}\nolimits _\Delta ([n], [m]) \times \mathop{Mor}\nolimits _\Delta ([n], [1]) \to \mathop{Mor}\nolimits _\Delta ([n], [m]) \]

which send $(\varphi , \alpha )$ to

\[ k \mapsto \left\{ \begin{matrix} \varphi (k) & \text{if} & \alpha (k) = 0 \\ m & \text{if} & \alpha (k) = 1 \end{matrix} \right. \]

Note that this only works because we took $g$ to be the inclusion of the last $0$-simplex. If we took $g$ to be the inclusion of the first $0$-simplex we could find a homotopy from $g \circ f$ to $\text{id}_{\Delta [m]}$. This is an illustration of the asymmetry inherent in homotopies in the category of simplicial sets.

The following lemma says that $U \times \Delta [1]$ is homotopy equivalent to $U$.

Lemma 14.26.8. Let $\mathcal{C}$ be a category with finite coproducts. Let $U$ be a simplicial object of $\mathcal{C}$. Consider the maps $e_1, e_0 : U \to U \times \Delta [1]$, and $\pi : U \times \Delta [1] \to U$, see Lemma 14.13.3.

  1. We have $\pi \circ e_1 = \pi \circ e_0 = \text{id}_ U$, and

  2. The morphisms $\text{id}_{U \times \Delta [1]}$, and $e_0 \circ \pi $ are homotopic.

  3. The morphisms $\text{id}_{U \times \Delta [1]}$, and $e_1 \circ \pi $ are homotopic.

Proof. The first assertion is trivial. For the second, consider the map of simplicial sets $\Delta [1] \times \Delta [1] \longrightarrow \Delta [1]$ which in degree $n$ assigns to a pair $(\beta _1, \beta _2)$, $\beta _ i : [n] \to [1]$ the morphism $\beta : [n] \to [1]$ defined by the rule

\[ \beta (i) = \max \{ \beta _1(i), \beta _2(i)\} . \]

It is a morphism of simplicial sets, because the action $\Delta [1](\varphi ) : \Delta [1]_ n \to \Delta [1]_ m$ of $\varphi : [m] \to [n]$ is by precomposing. Clearly, using notation from Section 14.26, we have $\beta = \beta _1$ if $\beta _2 = \alpha ^ n_0$ and $\beta = \alpha ^ n_{n + 1}$ if $\beta _2 = \alpha ^ n_{n + 1}$. This implies easily that the induced morphism

\[ U \times \Delta [1] \times \Delta [1] \longrightarrow U \times \Delta [1] \]

of Lemma 14.13.3 is a homotopy between $\text{id}_{U \times \Delta [1]}$ and $e_0 \circ \pi $. Similarly for $e_1 \circ \pi $ (use minimum instead of maximum). $\square$

Lemma 14.26.9. Let $f : Y \to X$ be a morphism of a category $\mathcal{C}$ with fibre products. Assume $f$ has a section $s$. Consider the simplicial object $U$ constructed in Example 14.3.5 starting with $f$. The morphism $U \to U$ which in each degree is the self map $(s \circ f)^{n + 1}$ of $Y \times _ X \ldots \times _ X Y$ given by $s \circ f$ on each factor is homotopic to the identity on $U$. In particular, $U$ is homotopy equivalent to the constant simplicial object $X$.

Proof. Set $g^0 = \text{id}_ Y$ and $g^1 = s \circ f$. We use the morphisms

\begin{eqnarray*} Y \times _ X \ldots \times _ X Y \times \mathop{Mor}\nolimits ([n], [1]) & \to & Y \times _ X \ldots \times _ X Y \\ (y_0, \ldots , y_ n) \times \alpha & \mapsto & (g^{\alpha (0)}(y_0), \ldots , g^{\alpha (n)}(y_ n)) \end{eqnarray*}

where we use the functor of points point of view to define the maps. Another way to say this is to say that $h_{n, 0} = \text{id}$, $h_{n, n + 1} = (s \circ f)^{n + 1}$ and $h_{n, i} = \text{id}_ Y^{i + 1} \times (s \circ f)^{n + 1 - i}$. We leave it to the reader to show that these satisfy the relations of Lemma 14.26.2. Hence they define the desired homotopy. See also Remark 14.26.4 which shows that we do not need to assume anything else on the category $\mathcal{C}$. $\square$

Lemma 14.26.10. Let $\mathcal{C}$ be a category.

  1. If $a_ t, b_ t : X_ t \to Y_ t$, $t \in T$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, then $\prod a_ t, \prod b_ t : \prod X_ t \to \prod Y_ t$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

  2. If $(X_ t, Y_ t)$, $t \in T$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, then $\prod X_ t$ and $\prod Y_ t$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

Proof. If $h_ t = (h_{t, n , i})$ are homotopies connecting $a_ t$ and $b_ t$ (see Remark 14.26.4), then $h = (\prod _ t h_{t, n, i})$ is a homotopy connecting $\prod a_ t$ and $\prod b_ t$. This proves (1). Part (2) follows from part (1) and the definitions. $\square$

[1] In the literature, often the maps $h_{n + 1, i} \circ s_ i : U_ n \to V_{n + 1}$ are used instead of the maps $h_{n, i}$. Of course the relations these maps satisfy are different from the ones in Lemma 14.26.2.
[2] Warning: This notion is not an equivalence relation on objects in general.

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