# Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957[1] 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.[2]

To every algebraic variety ${\displaystyle V}$ one can associate a Grothendieck category ${\displaystyle \operatorname {Qcoh} (V)}$, consisting of the quasi-coherent sheaves on ${\displaystyle V}$. This category encodes all the relevant geometric information about ${\displaystyle V}$, and ${\displaystyle V}$ can be recovered from ${\displaystyle \operatorname {Qcoh} (V)}$ (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.[3]

## Definition

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

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

The name "Grothendieck category" neither appeared in Grothendieck's Tôhoku paper[1] nor in Gabriel's thesis;[2] 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 ${\displaystyle \mathbb {Z} }$ of integers can serve as a generator.
• More generally, given any ring ${\displaystyle R}$ (associative, with ${\displaystyle 1}$, but not necessarily commutative), the category ${\displaystyle \operatorname {Mod} (R)}$ of all right (or alternatively: left) modules over ${\displaystyle R}$ is a Grothendieck category; ${\displaystyle R}$ itself can serve as a generator.
• Given a topological space ${\displaystyle X}$, the category of all sheaves of abelian groups on ${\displaystyle X}$ is a Grothendieck category.[1] (More generally: the category of all sheaves of right ${\displaystyle R}$-modules on ${\displaystyle X}$ is a Grothendieck category for any ring ${\displaystyle R}$.)
• Given a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$, the category of sheaves of OX-modules is a Grothendieck category.[1]
• Given an (affine or projective) algebraic variety ${\displaystyle V}$ (or more generally: a quasi-compact quasi-separated scheme), the category ${\displaystyle \operatorname {Qcoh} (V)}$ of quasi-coherent sheaves on ${\displaystyle V}$ 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 ${\displaystyle {\mathcal {A_{1}}},\ldots ,{\mathcal {A_{n}}}}$, the product category ${\displaystyle {\mathcal {A_{1}}}\times \ldots \times {\mathcal {A_{n}}}}$ is a Grothendieck category.
• Given a small category ${\displaystyle {\mathcal {C}}}$ and a Grothendieck category ${\displaystyle {\mathcal {A}}}$, the functor category ${\displaystyle \operatorname {Funct} ({\mathcal {C}},{\mathcal {A}})}$, consisting of all covariant functors from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {A}}}$, is a Grothendieck category.[1]
• Given a small preadditive category ${\displaystyle {\mathcal {C}}}$ and a Grothendieck category ${\displaystyle {\mathcal {A}}}$, the functor category ${\displaystyle \operatorname {Add} ({\mathcal {C}},{\mathcal {A}})}$ of all additive covariant functors from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {A}}}$ is a Grothendieck category.[4]
• If ${\displaystyle {\mathcal {A}}}$ is a Grothendieck category and ${\displaystyle {\mathcal {C}}}$ is a localizing subcategory of ${\displaystyle {\mathcal {A}}}$, then both ${\displaystyle {\mathcal {C}}}$ and the Serre quotient category ${\displaystyle {\mathcal {A}}/{\mathcal {C}}}$ are Grothendieck categories.[2]

## 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 ${\displaystyle \mathbb {Q} /\mathbb {Z} }$.

Every object in a Grothendieck category ${\displaystyle {\mathcal {A}}}$ has an injective hull in ${\displaystyle {\mathcal {A}}}$.[1][2] This allows to construct injective resolutions and thereby the use of the tools of homological algebra in ${\displaystyle {\mathcal {A}}}$, 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 ${\displaystyle (U_{i})}$ of a given object ${\displaystyle X}$ has a supremum (or "sum") ${\textstyle \sum _{i}U_{i}}$ as well as an infimum (or "intersection") ${\displaystyle \cap _{i}U_{i}}$, both of which are again subobjects of ${\displaystyle X}$. Further, if the family ${\displaystyle (U_{i})}$ is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and ${\displaystyle V}$ is another subobject of ${\displaystyle X}$, we have

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

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).[4]

An object ${\displaystyle X}$ in a Grothendieck category is called finitely generated if the sum of every directed family of proper subobjects of ${\displaystyle X}$ is again a proper subobject of ${\displaystyle X}$. (In the case ${\displaystyle {\cal {A}}=\operatorname {Mod} (R)}$ 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.[4]

It is a rather deep result that every Grothendieck category ${\displaystyle {\mathcal {A}}}$ is complete, i.e. that arbitrary limits (and in particular products) exist in ${\displaystyle {\mathcal {A}}}$. By contrast, it follows directly from the definition that ${\displaystyle {\mathcal {A}}}$ is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in ${\displaystyle {\mathcal {A}}}$. 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 ${\displaystyle F:{\cal {A}}\to {\cal {X}}}$ from a Grothendieck categories ${\displaystyle {\mathcal {A}}}$ to an arbitrary category ${\displaystyle {\mathcal {X}}}$ 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.[5]

The Gabriel–Popescu theorem states that any Grothendieck category ${\displaystyle {\mathcal {A}}}$ is equivalent to a full subcategory of the category ${\displaystyle \operatorname {Mod} (R)}$ of right modules over some unital ring ${\displaystyle R}$ (which can be taken to be the endomorphism ring of a generator of ${\displaystyle {\mathcal {A}}}$), and ${\displaystyle {\mathcal {A}}}$ can be obtained as a Gabriel quotient of ${\displaystyle \operatorname {Mod} (R)}$ by some localizing subcategory.[6]

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.[7]

Every small abelian category ${\displaystyle {\mathcal {C}}}$ can be embedded in a Grothendieck category, in the following fashion. The category ${\displaystyle {\mathcal {A}}:=\operatorname {Lex} ({\mathcal {C}}^{op},\mathrm {Ab} )}$ of left-exact additive (covariant) functors ${\displaystyle {\mathcal {C}}^{op}\rightarrow \mathrm {Ab} }$ (where ${\displaystyle \mathrm {Ab} }$ denotes the category of abelian groups) is a Grothendieck category, and the functor ${\displaystyle h\colon {\mathcal {C}}\rightarrow {\mathcal {A}}}$, with ${\displaystyle C\mapsto h_{C}=\operatorname {Hom} (-,C)}$, is full, faithful and exact. A generator of ${\displaystyle {\mathcal {A}}}$ is given by the coproduct of all ${\displaystyle h_{C}}$, with ${\displaystyle C\in {\mathcal {C}}}$.[2] The category ${\displaystyle {\mathcal {A}}}$ is equivalent to the category ${\displaystyle {\text{Ind}}({\mathcal {C}})}$ of ind-objects of ${\displaystyle {\mathcal {C}}}$ and the embedding ${\displaystyle h}$ corresponds to the natural embedding ${\displaystyle {\mathcal {C}}\to {\text{Ind}}({\mathcal {C}})}$.

## References

1. Grothendieck, A. (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2), 9 (2): 119–221, doi:10.2748/tmj/1178244839, MR 0102537. English translation.
2. Gabriel, P. (1962), "Des catégories abéliennes" (PDF), Bull. Soc. Math. Fr., 90: 323–448
3. ^ Izuru Mori (2007). "Quantum Ruled Surfaces" (PDF).
4. ^ a b c Faith, Carl (1973). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498. ISBN 9783642806346.
5. ^ Lane, Saunders Mac (1978). Categories for the Working Mathematician, 2nd edition. Springer. p. 130.
6. ^ N. Popesco, P. Gabriel (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes Rendus de l'Académie des Sciences. 258: 4188–4190.CS1 maint: Uses authors parameter (link)
7. ^ Šťovíček, Jan (2013-01-01). "Deconstructibility and the Hill Lemma in Grothendieck categories". Forum Mathematicum. 25 (1). arXiv:1005.3251. Bibcode:2010arXiv1005.3251S. doi:10.1515/FORM.2011.113.
• 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, 339, Longman, Harlow.