Grothendieck category
Let
be a category. Moreover, let
be a family of objects
of
. The family
is said to be a family of generators of the category
if for any object
of
and any subobject
of
, distinct from
, there is at least an index
, and a morphism,
, that cannot be factorized through the canonical injection
. Then, an object
of
is said to be a generator of the category
provided that
belongs to the family of generators
of
([4]).
By duality, that is, by simply reversing all arrows in the above definition one obtains the notion of a
family of cogenerators
of the same category
, and also the notion of cogenerator
of
, if all of the required, reverse arrows exist. Notably, in a groupoid- regarded as a small category
with all its morphisms invertible- this is always possible, and thus a groupoid
can always be cogenerated via duality. Moreover, any generator in the dual category
is a cogenerator of
.
- (Ab3). Let us recall that an Abelian category
is cocomplete
(or an
-category) if it has arbitrary direct sums.
- (Ab5). A cocomplete Abelian category
is said to be an
-category if for any directed family
of subobjects of
, and for any subobject
of
, the following equation holds
- One notes that the condition Ab3 is equivalent to the existence of arbitrary direct limits.
- Furthermore, Ab5 is equivalent to the following proposition:
there exist inductive limits and the inductive limits over directed families of indices are exact,
that is, if
is a directed set and
is an exact
sequence for any
, then
is also an exact sequence.
- By duality, one readily obtains conditions Ab3* and Ab5* simply
by reversing the arrows in the above conditions defining Ab3 and Ab5.
Definition 0.1
A
Grothendieck category is an

category
with a generator.
As an example consider the category
of Abelian groups
such that if
is a family of abelian groups, then
a direct product
is defined by the Cartesian product
with addition defined by the rule:
.
One then defines a projection
given by
. A direct sum is obtained by taking the appropriate subgroup
consisting of all elements
such that
for all but a finite number of indices
. Then one also defines a structural injection , and it is straightforward
to prove that
is an
and
category. (viz. p 61 in ref. [4]).
Definition 0.2
A
co-Grothendieck category is an

category that has a set of cogenerators,
i.e., a category whose dual is a Grothendieck category.
- Let
be an abelian category
and
a small category.
One defines then a functor
as follows: for any
,
is the
constant functor which is associated to
. Then
is an
category (respectively,
), if and only if for any directed set
, as above, the functor
has an exact left (or respectively, right) adjoint.
- With
,
,
, and
one can construct categories of (pre) additive functors.
- A preabelian category is an additive category
with the additional (
) condition that for any morphism
in the category there exist also both
and
;
- An Abelian category can be then also defined as a preabelian category in which for any morphism
, the morphism
is an isomorphism (the
condition).
-
- 1
-
Alexander Grothendieck et al. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1
(or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results.),
AG4 is freely available
in French;
also available here is an extensive
Abstract in English.
- 2
-
Alexander Grothendieck, 1984. ``Esquisse d'un Programme'', (1984 manuscript), finally published in ``Geometric Galois Actions'', L. Schneps, P. Lochak, eds.,
London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.5-48;
English transl., ibid., pp. 243-283. MR 99c:14034 .
- 3
-
Alexander Grothendieck, ``La longue marche in á travers la théorie de Galois''
= ``The Long March Towards/Across the Theory of Galois'', 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire.
- 4
-
Nicolae Popescu. Abelian Categories with Applications to Rings and Modules.,
Academic Press: New York and London, 1973 and 1976 edns., (English translation by I. C. Baianu.)
- 5
-
Leila Schneps. 1994.
The Grothendieck Theory of Dessins d'Enfants.
(London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
- 6
-
David Harbater and Leila Schneps. 2000.
Fundamental groups of moduli and the Grothendieck-Teichmüller group, Trans. Amer. Math. Soc. 352 (2000), 3117-3148.
MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.
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