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Tag 0DBB

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 [SGA4, Exposé V, Subsection 7.3.0]. 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. $\square$

    The code snippet corresponding to this tag is a part of the file hypercovering.tex and is located in lines 103–284 (see updates for more information).

    \section{Semi-representable objects}
    \label{section-semi-representable}
    
    \noindent
    In order to start we make the following definition.
    The letters ``SR'' stand for Semi-Representable.
    
    \begin{definition}
    \label{definition-SR}
    Let $\mathcal{C}$ be a category. We denote $\text{SR}(\mathcal{C})$
    the category of {\it semi-representable objects} defined as follows
    \begin{enumerate}
    \item objects are families of objects $\{U_i\}_{i \in I}$, and
    \item 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 $\mathcal{C}$.
    \end{enumerate}
    Let $X \in \Ob(\mathcal{C})$ be an object of $\mathcal{C}$.
    The category of {\it semi-representable objects over $X$}
    is the category
    $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$.
    \end{definition}
    
    \noindent
    This definition is essentially equivalent to
    \cite[Expos\'e 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 $\text{SR}(\mathcal{C}, X)$:
    \begin{enumerate}
    \item objects are families of morphisms
    $\{U_i \to X\}_{i \in I}$, and
    \item 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 $X$.
    \end{enumerate}
    There is a forgetful functor
    $\text{SR}(\mathcal{C}, X) \to \text{SR}(\mathcal{C})$.
    
    \begin{definition}
    \label{definition-SR-F}
    Let $\mathcal{C}$ be a category.
    We denote $F$ the functor {\it 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 $U$.
    \end{definition}
    
    \noindent
    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.
    
    \begin{lemma}
    \label{lemma-coprod-prod-SR}
    Let $\mathcal{C}$ be a category.
    \begin{enumerate}
    \item the category $\text{SR}(\mathcal{C})$ has coproducts
    and $F$ commutes with them,
    \item the functor $F : \text{SR}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$
    commutes with limits,
    \item if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C})$
    has fibre products,
    \item if $\mathcal{C}$ has products of pairs, then
    $\text{SR}(\mathcal{C})$ has products of pairs,
    \item if $\mathcal{C}$ has equalizers, so does $\text{SR}(\mathcal{C})$, and
    \item if $\mathcal{C}$ has a final object, so does $\text{SR}(\mathcal{C})$.
    \end{enumerate}
    Let $X \in \Ob(\mathcal{C})$.
    \begin{enumerate}
    \item the category $\text{SR}(\mathcal{C}, X)$ has coproducts
    and $F$ commutes with them,
    \item 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.
    \end{enumerate}
    \end{lemma}
    
    \begin{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.
    
    \medskip\noindent
    Proof of (2). For $U$ in $\Ob(\mathcal{C})$ consider the object $\{U\}$ of
    $\text{SR}(\mathcal{C})$. It is clear that
    $\Mor_{\text{SR}(\mathcal{C})}(\{U\}, K)) = F(K)(U)$
    for $K \in \Ob(\text{SR}(\mathcal{C}))$. Since limits of presheaves
    are computed at the level of sections
    (Sites, Section \ref{sites-section-limits-colimits-PSh})
    we conclude that $F$ commutes with limits.
    
    \medskip\noindent
    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.
    
    \medskip\noindent
    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.
    
    \medskip\noindent
    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.
    
    \medskip\noindent
    Proof of (6). If $X$ is a final object of $\mathcal{C}$, then
    $\{X\}$ is a final object of $\text{SR}(\mathcal{C})$.
    
    \medskip\noindent
    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 \ref{sites-lemma-essential-image-j-shriek} 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 \ref{categories-lemma-finite-limits-exist}.
    We omit verifying the statements on the functor $F$ in this case.
    \end{proof}

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