## 25.2 Semi-representable objects

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

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 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

\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 25.2.3. 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 $\mathop{\mathrm{Mor}}\nolimits _{\text{SR}(\mathcal{C})}(\{ U\} , K)) = F(K)(U)$ 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

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

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 7.25.4 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

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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DBB. Beware of the difference between the letter 'O' and the digit '0'.