In order to start we make the following definition. The letters “SR” stand for Semi-Representable.
Definition 25.2.1. Let $\mathcal{C}$ be a category. We denote $\text{SR}(\mathcal{C})$ the category of semi-representable objects defined as follows
objects are families of objects $\{ U_ i\} _{i \in I}$, and
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 $\text{SR}(\mathcal{C}, X)$:
objects are families of morphisms $\{ U_ i \to X\} _{i \in I}$, and
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 25.2.2. Let $\mathcal{C}$ be a category. We denote $F$ the functor which associates a presheaf to a semi-representable object. In a formula where $h_ U$ denotes the representable presheaf associated to the object $U$.
\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*}
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:
Next we discuss the existence of limits in the category of semi-representable objects.
Lemma 25.2.3. Let $\mathcal{C}$ be a category.
the category $\text{SR}(\mathcal{C})$ has coproducts and $F$ commutes with them,
the functor $F : \text{SR}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ commutes with limits,
if $\mathcal{C}$ has fibre products, then $\text{SR}(\mathcal{C})$ has fibre products,
if $\mathcal{C}$ has products of pairs, then $\text{SR}(\mathcal{C})$ has products of pairs,
if $\mathcal{C}$ has equalizers, so does $\text{SR}(\mathcal{C})$, and
if $\mathcal{C}$ has a final object, so does $\text{SR}(\mathcal{C})$.
Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.
the category $\text{SR}(\mathcal{C}, X)$ has coproducts and $F$ commutes with them,
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 $\mathop{\mathrm{Mor}}\nolimits _{\text{SR}(\mathcal{C})}(\{ U\} , K)) = F(K)(U)$ 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
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
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 7.25.4 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
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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