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*}

6.15 Algebraic structures

In this section we mildly formalize the notions we have encountered in the sections above.

Definition 6.15.1. A type of algebraic structure is given by a category $\mathcal{C}$ and a functor $F : \mathcal{C} \to \textit{Sets}$ with the following properties

  1. $F$ is faithful,

  2. $\mathcal{C}$ has limits and $F$ commutes with limits,

  3. $\mathcal{C}$ has filtered colimits and $F$ commutes with them, and

  4. $F$ reflects isomorphisms.

We make this definition to point out the properties we will use in a number of arguments below. But we will not actually study this notion in any great detail, since we are prohibited from studying “big” categories by convention, except for those listed in Categories, Remark 4.2.2. Among those the following have the required properties.

Lemma 6.15.2. The following categories, endowed with the obvious forgetful functor, define types of algebraic structures:

  1. The category of pointed sets.

  2. The category of abelian groups.

  3. The category of groups.

  4. The category of monoids.

  5. The category of rings.

  6. The category of $R$-modules for a fixed ring $R$.

  7. The category of Lie algebras over a fixed field.

Proof. Omitted. $\square$

From now on we will think of a (pre)sheaf of algebraic structures and their stalks, in terms of the underlying (pre)sheaf of sets. This is allowable by Lemmas 6.9.2 and 6.13.1.

In the rest of this section we point out some results on algebraic structures that will be useful in the future.

Lemma 6.15.3. Let $(\mathcal{C}, F)$ be a type of algebraic structure.

  1. $\mathcal{C}$ has a final object $0$ and $F(0) = \{ * \} $.

  2. $\mathcal{C}$ has products and $F(\prod A_ i) = \prod F(A_ i)$.

  3. $\mathcal{C}$ has fibre products and $F(A \times _ B C) = F(A)\times _{F(B)}F(C)$.

  4. $\mathcal{C}$ has equalizers, and if $E \to A$ is the equalizer of $a, b : A \to B$, then $F(E) \to F(A)$ is the equalizer of $F(a), F(b) : F(A) \to F(B)$.

  5. $A \to B$ is a monomorphism if and only if $F(A) \to F(B)$ is injective.

  6. if $F(a) : F(A) \to F(B)$ is surjective, then $a$ is an epimorphism.

  7. given $A_1 \to A_2 \to A_3 \to \ldots $, then $\mathop{\mathrm{colim}}\nolimits A_ i$ exists and $F(\mathop{\mathrm{colim}}\nolimits A_ i) = \mathop{\mathrm{colim}}\nolimits F(A_ i)$, and more generally for any filtered colimit.

Proof. Omitted. The only interesting statement is (5) which follows because $A \to B$ is a monomorphism if and only if $A \to A \times _ B A$ is an isomorphism, and then applying the fact that $F$ reflects isomorphisms. $\square$

Lemma 6.15.4. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $A, B, C \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $f : A \to B$ and $g : C \to B$ be morphisms of $\mathcal{C}$. If $F(g)$ is injective, and $\mathop{\mathrm{Im}}(F(f)) \subset \mathop{\mathrm{Im}}(F(g))$, then $f$ factors as $f = g \circ t$ for some morphism $t : A \to C$.

Proof. Consider $A \times _ B C$. The assumptions imply that $F(A \times _ B C) = F(A) \times _{F(B)} F(C) = F(A)$. Hence $A = A \times _ B C$ because $F$ reflects isomorphisms. The result follows. $\square$

Example 6.15.5. The lemma will be applied often to the following situation. Suppose that we have a diagram

\[ \xymatrix{ A \ar[r] & B \ar[d] \\ C \ar[r] & D } \]

in $\mathcal{C}$. Suppose $C \to D$ is injective on underlying sets, and suppose that the composition $A \to B \to D$ has image on underlying sets in the image of $C \to D$. Then we get a commutative diagram

\[ \xymatrix{ A \ar[r] \ar[d] & B \ar[d] \\ C \ar[r] & D } \]

in $\mathcal{C}$.

Example 6.15.6. Let $F : \mathcal{C} \to \textit{Sets}$ be a type of algebraic structures. Let $X$ be a topological space. Suppose that for every $x \in X$ we are given an object $A_ x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Consider the presheaf $\Pi $ with values in $\mathcal{C}$ on $X$ defined by the rule $\Pi (U) = \prod _{x \in U} A_ x$ (with obvious restriction mappings). Note that the associated presheaf of sets $U \mapsto F(\Pi (U)) = \prod _{x \in U} F(A_ x)$ is a sheaf by Example 6.7.5. Hence $\Pi $ is a sheaf of algebraic structures of type $(\mathcal{C} , F)$. This gives many examples of sheaves of abelian groups, groups, rings, etc.


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