\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*}

The Stacks project

24.2 Semi-representable objects

In order to start we make the following definition. The letters “SR” stand for Semi-Representable.

Definition 24.2.1. Let $\mathcal{C}$ be a category. We denote $\text{SR}(\mathcal{C})$ the category of semi-representable objects defined as follows

  1. objects are families of objects $\{ U_ i\} _{i \in I}$, and

  2. morphisms $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ are given by a map $\alpha : I \to J$ and for each $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ of $\mathcal{C}$.

Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. The category of semi-representable objects over $X$ is the category $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$.

This definition is essentially equivalent to [Exposé V, Subsection 7.3.0, SGA4]. Note that this is a “big” category. We will later “bound” the size of the index sets $I$ that we need for hypercoverings of $X$. We can then redefine $\text{SR}(\mathcal{C}, X)$ to become a category. Let's spell out the objects and morphisms $\text{SR}(\mathcal{C}, X)$:

  1. objects are families of morphisms $\{ U_ i \to X\} _{i \in I}$, and

  2. morphisms $\{ U_ i \to X\} _{i \in I} \to \{ V_ j \to X\} _{j \in J}$ are given by a map $\alpha : I \to J$ and for each $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ over $X$.

There is a forgetful functor $\text{SR}(\mathcal{C}, X) \to \text{SR}(\mathcal{C})$.

Definition 24.2.2. Let $\mathcal{C}$ be a category. We denote $F$ the functor which associates a presheaf to a semi-representable object. In a formula

\begin{eqnarray*} F : \text{SR}(\mathcal{C}) & \longrightarrow & \textit{PSh}(\mathcal{C}) \\ \{ U_ i\} _{i \in I} & \longmapsto & \amalg _{i\in I} h_{U_ i} \end{eqnarray*}

where $h_ U$ denotes the representable presheaf associated to the object $U$.

Given a morphism $U \to X$ we obtain a morphism $h_ U \to h_ X$ of representable presheaves. Thus we often think of $F$ on $\text{SR}(\mathcal{C}, X)$ as a functor into the category of presheaves of sets over $h_ X$, namely $\textit{PSh}(\mathcal{C})/h_ X$. Here is a picture:

\[ \xymatrix{ \text{SR}(\mathcal{C}, X) \ar[r]_ F \ar[d] & \textit{PSh}(\mathcal{C})/h_ X \ar[d] \\ \text{SR}(\mathcal{C}) \ar[r]^ F & \textit{PSh}(\mathcal{C}) } \]

Next we discuss the existence of limits in the category of semi-representable objects.

Lemma 24.2.3. Let $\mathcal{C}$ be a category.

  1. the category $\text{SR}(\mathcal{C})$ has coproducts and $F$ commutes with them,

  2. the functor $F : \text{SR}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ commutes with limits,

  3. if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C})$ has fibre products,

  4. if $\mathcal{C}$ has products of pairs, then $\text{SR}(\mathcal{C})$ has products of pairs,

  5. if $\mathcal{C}$ has equalizers, so does $\text{SR}(\mathcal{C})$, and

  6. if $\mathcal{C}$ has a final object, so does $\text{SR}(\mathcal{C})$.

Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

  1. the category $\text{SR}(\mathcal{C}, X)$ has coproducts and $F$ commutes with them,

  2. if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C}, X)$ has finite limits and $F : \text{SR}(\mathcal{C}, X) \to \textit{PSh}(\mathcal{C})/h_ X$ commutes with them.

Proof. Proof of the results on $\text{SR}(\mathcal{C})$. Proof of (1). The coproduct of $\{ U_ i\} _{i \in I}$ and $\{ V_ j\} _{j \in J}$ is $\{ U_ i\} _{i \in I} \amalg \{ V_ j\} _{j \in J}$, in other words, the family of objects whose index set is $I \amalg J$ and for an element $k \in I \amalg J$ gives $U_ i$ if $k = i \in I$ and gives $V_ j$ if $k = j \in J$. Similarly for coproducts of families of objects. It is clear that $F$ commutes with these.

Proof of (2). For $U$ in $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ consider the object $\{ U\} $ of $\text{SR}(\mathcal{C})$. It is clear that $\mathop{Mor}\nolimits _{\text{SR}(\mathcal{C})}(\{ U\} , K)) = F(K)(U)$ for $K \in \mathop{\mathrm{Ob}}\nolimits (\text{SR}(\mathcal{C}))$. Since limits of presheaves are computed at the level of sections (Sites, Section 7.4) we conclude that $F$ commutes with limits.

Proof of (3). Suppose given a morphism $(\alpha , f_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ and a morphism $(\beta , g_ k) : \{ W_ k\} _{k \in K} \to \{ V_ j\} _{j \in J}$. The fibred product of these morphisms is given by

\[ \{ U_ i \times _{f_ i, V_ j, g_ k} W_ k\} _{(i, j, k) \in I \times J \times K \text{ such that } j = \alpha (i) = \beta (k)} \]

The fibre products exist if $\mathcal{C}$ has fibre products.

Proof of (4). The product of $\{ U_ i\} _{i \in I}$ and $\{ V_ j\} _{j \in J}$ is $\{ U_ i \times V_ j\} _{i \in I, j \in J}$. The products exist if $\mathcal{C}$ has products.

Proof of (5). The equalizer of two maps $(\alpha , f_ i), (\alpha ', f'_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ is

\[ \{ \text{Eq}(f_ i, f'_ i : U_ i \to V_{\alpha (i)}) \} _{i \in I,\ \alpha (i) = \alpha '(i)} \]

The equalizers exist if $\mathcal{C}$ has equalizers.

Proof of (6). If $X$ is a final object of $\mathcal{C}$, then $\{ X\} $ is a final object of $\text{SR}(\mathcal{C})$.

Proof of the statements about $\text{SR}(\mathcal{C}, X)$. These follow from the results above applied to the category $\mathcal{C}/X$ using that $\text{SR}(\mathcal{C}/X) = \text{SR}(\mathcal{C}, X)$ and that $\textit{PSh}(\mathcal{C}/X) = \textit{PSh}(\mathcal{C})/h_ X$ (Sites, Lemma 7.24.4 applied to $\mathcal{C}$ endowed with the chaotic topology). However we also argue directly as follows. It is clear that the coproduct of $\{ U_ i \to X\} _{i \in I}$ and $\{ V_ j \to X\} _{j \in J}$ is $\{ U_ i \to X\} _{i \in I} \amalg \{ V_ j \to X\} _{j \in J}$ and similarly for coproducts of families of families of morphisms with target $X$. The object $\{ X \to X\} $ is a final object of $\text{SR}(\mathcal{C}, X)$. Suppose given a morphism $(\alpha , f_ i) : \{ U_ i \to X\} _{i \in I} \to \{ V_ j \to X\} _{j \in J}$ and a morphism $(\beta , g_ k) : \{ W_ k \to X\} _{k \in K} \to \{ V_ j \to X\} _{j \in J}$. The fibred product of these morphisms is given by

\[ \{ U_ i \times _{f_ i, V_ j, g_ k} W_ k \to X \} _{(i, j, k) \in I \times J \times K \text{ such that } j = \alpha (i) = \beta (k)} \]

The fibre products exist by the assumption that $\mathcal{C}$ has fibre products. Thus $\text{SR}(\mathcal{C}, X)$ has finite limits, see Categories, Lemma 4.18.4. We omit verifying the statements on the functor $F$ in this case. $\square$


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