Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Peter Gabriel's seminal thèse in 1962.

To every algebraic variety $V$ one can associate a Grothendieck category $\operatorname {Qcoh} (V)$, consisting of the quasi-coherent sheaves on $V$. This category encodes all the relevant geometric information about $V$, and $V$ can be recovered from $\operatorname {Qcoh} (V)$ (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.

Definition

By definition, a Grothendieck category ${\mathcal {A}}$ is an AB5 category with a generator. Spelled out, this means that

• ${\mathcal {A}}$ is an abelian category;
• every (possibly infinite) family of objects in ${\mathcal {A}}$ has a coproduct (a.k.a. direct sum) in ${\mathcal {A}}$;
• direct limits of exact sequences are exact; this means that if a direct system of short exact sequences in ${\mathcal {A}}$ is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always right-exact; the important point here is that we require them to be left-exact as well.)
• ${\mathcal {A}}$ possesses a generator, i.e. there is an object $G$ in ${\mathcal {A}}$ such that $\operatorname {Hom} (G,-)$ is a faithful functor from ${\mathcal {A}}$ to the category of sets. (In our situation, this is equivalent to saying that every object $X$ of ${\mathcal {A}}$ admits an epimorphism $G^{(I)}\rightarrow X$, where $G^{(I)}$ denotes a direct sum of copies of $G$, one for each element of the (possibly infinite) set $I$.)

The name "Grothendieck category" neither appeared in Grothendieck's Tôhoku paper nor in Gabriel's thesis; it came into use in the second half of the 1960s by authors including J.-E. Roos, B. Stenström, U. Oberst, and B. Pareigis. (Some authors use a different definition in that they don't require the existence of a generator.)

Examples

• The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group $\mathbb {Z}$ of integers can serve as a generator.
• More generally, given any ring $R$ (associative, with $1$, but not necessarily commutative), the category $\operatorname {Mod} (R)$ of all right (or alternatively: left) modules over $R$ is a Grothendieck category; $R$ itself can serve as a generator.
• Given a topological space $X$, the category of all sheaves of abelian groups on $X$ is a Grothendieck category. (More generally: the category of all sheaves of right $R$-modules on $X$ is a Grothendieck category for any ring $R$.)
• Given a ringed space $(X,{\mathcal {O}}_{X})$, the category of sheaves of O_X-modules is a Grothendieck category.
• Given an (affine or projective) algebraic variety $V$ (or more generally: a quasi-compact quasi-separated scheme), the category $\operatorname {Qcoh} (V)$ of quasi-coherent sheaves on $V$ is a Grothendieck category.
• Given a small site (C, J) (i.e. a small category C together with a Grothendieck topology J), the category of all sheaves of abelian groups on the site is a Grothendieck category.

Constructing further Grothendieck categories

• Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
• Given Grothendieck categories ${\mathcal {A_{1}}},\ldots ,{\mathcal {A_{n}}}$, the product category ${\mathcal {A_{1}}}\times \ldots \times {\mathcal {A_{n}}}$ is a Grothendieck category.
• Given a small category ${\mathcal {C}}$ and a Grothendieck category ${\mathcal {A}}$, the functor category $\operatorname {Funct} ({\mathcal {C}},{\mathcal {A}})$, consisting of all covariant functors from ${\mathcal {C}}$ to ${\mathcal {A}}$, is a Grothendieck category.
• Given a small preadditive category ${\mathcal {C}}$ and a Grothendieck category ${\mathcal {A}}$, the functor category $\operatorname {Add} ({\mathcal {C}},{\mathcal {A}})$ of all additive covariant functors from ${\mathcal {C}}$ to ${\mathcal {A}}$ is a Grothendieck category.
• If ${\mathcal {A}}$ is a Grothendieck category and ${\mathcal {C}}$ is a localizing subcategory of ${\mathcal {A}}$, then both ${\mathcal {C}}$ and the Serre quotient category ${\mathcal {A}}/{\mathcal {C}}$ are Grothendieck categories.

Properties and theorems

Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the quotient group $\mathbb {Q} /\mathbb {Z}$.

Every object in a Grothendieck category ${\mathcal {A}}$ has an injective hull in ${\mathcal {A}}$. This allows to construct injective resolutions and thereby the use of the tools of homological algebra in ${\mathcal {A}}$, such as derived functors. (Note that not all Grothendieck categories allow projective resolutions for all objects; examples are categories of sheaves of abelian groups on many topological spaces, such as on the space of real numbers.)

In a Grothendieck category, any family of subobjects $(U_{i})$ of a given object $X$ has a supremum (or "sum") ${\textstyle \sum _{i}U_{i}}$ as well as an infimum (or "intersection") $\cap _{i}U_{i}$, both of which are again subobjects of $X$. Further, if the family $(U_{i})$ is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and $V$ is another subobject of $X$, we have

$\sum _{i}(U_{i}\cap V)=\left(\sum _{i}U_{i}\right)\cap V.$

Grothendieck categories are well-powered (sometimes called locally small, although that term is also used for a different concept), i.e. the collection of subobjects of any given object forms a set (rather than a proper class).

An object $X$ in a Grothendieck category is called finitely generated if the sum of every directed family of proper subobjects of $X$ is again a proper subobject of $X$. (In the case ${\cal {A}}=\operatorname {Mod} (R)$ of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) A Grothendieck category need not contain any non-zero finitely generated objects. A Grothendieck category is called locally finitely generated if it has a set of finitely generated generators. In such a category, every object is the sum of its finitely generated subobjects.

It is a rather deep result that every Grothendieck category ${\mathcal {A}}$ is complete, i.e. that arbitrary limits (and in particular products) exist in ${\mathcal {A}}$. By contrast, it follows directly from the definition that ${\mathcal {A}}$ is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in ${\mathcal {A}}$. Coproducts in a Grothendieck category are exact (i.e. the coproduct of a family of short exact sequences is again a short exact sequence), but products need not be exact.

A functor $F:{\cal {A}}\to {\cal {X}}$ from a Grothendieck categories ${\mathcal {A}}$ to an arbitrary category ${\mathcal {X}}$ has a left adjoint if and only if it commutes with all limits, and it has a right adjoint if and only if it commutes with all colimits. This follows from Freyd's special adjoint functor theorem and its dual.

The Gabriel–Popescu theorem states that any Grothendieck category ${\mathcal {A}}$ is equivalent to a full subcategory of the category $\operatorname {Mod} (R)$ of right modules over some unital ring $R$ (which can be taken to be the endomorphism ring of a generator of ${\mathcal {A}}$), and ${\mathcal {A}}$ can be obtained as a Serre quotient of $\operatorname {Mod} (R)$ by some localizing subcategory.

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.

Every small abelian category ${\mathcal {C}}$ can be embedded in a Grothendieck category, in the following fashion. The category ${\mathcal {A}}:=\operatorname {Lex} ({\mathcal {C}}^{op},\mathrm {Ab} )$ of left-exact additive (covariant) functors ${\mathcal {C}}^{op}\rightarrow \mathrm {Ab}$ (where $\mathrm {Ab}$ denotes the category of abelian groups) is a Grothendieck category, and the functor $h\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$, with $C\mapsto h_{C}=\operatorname {Hom} (-,C)$, is full, faithful and exact. A generator of ${\mathcal {A}}$ is given by the coproduct of all $h_{C}$, with $C\in {\mathcal {C}}$. The category ${\mathcal {A}}$ is equivalent to the category ${\text{Ind}}({\mathcal {C}})$ of ind-objects of ${\mathcal {C}}$ and the embedding $h$ corresponds to the natural embedding ${\mathcal {C}}\to {\text{Ind}}({\mathcal {C}})$.

References

• N. Popescu (1973). Abelian categories with applications to rings and modules. Academic Press.
• Jara, Pascual; Verschoren, Alain; Vidal, Conchi (1995), Localization and sheaves: a relative point of view, Pitman Research Notes in Mathematics Series, vol. 339, Longman, Harlow.

External links

• Tsalenko, M.Sh. (2001) [1994], "Grothendieck category", Encyclopedia of Mathematics, EMS Press
• Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.