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In mathematics, a '''Grothendieck category''' is a certain kind of [[abelian category]], introduced by [[Alexander Grothendieck]] in 1957<ref>{{citation|first=A.|last=Grothendieck|authorlink=Alexander Grothendieck|title=Sur quelques points d’algèbre homologique|journal=[[Tôhoku Mathematical Journal]]|volume=9|series=(2)|pages=119–221|year=1957|mr=0102537|url=http://projecteuclid.org/euclid.tmj/1178244839}}. [ftp://ftp.math.mcgill.ca/barr/pdffiles/gk.pdf English translation].</ref> in order to develop the machinery of [[homological algebra]] for [[module (mathematics)|module]]s and for [[sheaf (mathematics)|sheaves]] in a unified manner.
Baron Haussmann got a Parisian thoroughfare named after himself in the mid-nineteenth century by reconfiguring the city's circulatory road systems into wide, grand avenues.<br><br>The idea was to widen streets and prevent blockades and riots. It didn't entirely work, but was a noble effort.<br>What did it take for Chanel to get their own street? Well, if you pump vast amounts of cash into building a set that's a photographic facsimile of a Parisian rue, you can call it whatever you damn well want. So, the models at Karl Lagerfeld's spring/summer 2015 Chanel show [http://Www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=sauntered sauntered] in groups down the centre of "Boulevard Chanel," easily chatting amongst themselves.<br><br>A few carried boomboxes packed into chic Chanel handbags, the tinny sound reverberating until the thumping main soundtrack took  [http://tinyurl.com/kecvhhb discount ugg boots] over. "I'm every woman," warbled Whitney Houston, at one point, "It's all in me."<br>She could have been singing about the collection because, in typical Lagerfeld style, it was all in there. Wide trousers, tunics, print, plain, military, pinstripes, sweaters, Gisele in a striped cardigan and something suspiciously close to a pair of Chanel Uggs. All present and correct, on flat shoes, women striding meaningfully off to work.<br><br>Which, in the fashion world, means the end of the (catwalk) road and back. There were also, obviously, plenty of bags, printed with faux-politicised slogans like "F�ministe mais Feminine," or "Votez Coco." Chanel Spring/Summer 2015 show by German designer Karl Lagerfeld during the Paris Fashion Week <br>There's been an undercurrent of women dressing women this week in Paris - Lagerfeld isn't a woman, obviously. But he does operate under the mantle of Coco Chanel, who tied her fashion to the feminist cause, even if she didn't realise it. Clothes, she stated, must be logical.<br><br>A button should have a buttonhole, and should work. Pockets should be real.<br>Chanel glorified function over form, or at least the ideal twinning together of the two. Her clothes addressed her own reality, the demands of the clothes she wanted to wear, which found resonance in women in the wider world. It was as simple as the chain handle on her 2.55 handbag, leaving the hands free to get on with something else, or the fact said bag was originally lined in burgundy, to make it easier to find your stuff inside.<br><br>Gisele Bundchen on the Chanel spring/summer 2015 catwalk<br>Chanel's bag was a form  [http://tinyurl.com/kecvhhb ugg boots sale] of protest against the impracticality of everyday fashion in the mid fifties - she loathed the corsets and petticoats of the New Look, for  [http://tinyurl.com/kecvhhb ugg boots sale] instance, and following her 1954 comeback railed against it everyday she could. Likewise, Lagerfeld's faux-sloganeering on his pouches and pochettes were portents of protest to come.<br><br>After the show finished, models returned to the stage (or rather, the street) brandishing placards daubed with pronouncements like "Make fashion not war" (fair enough), or "Tweed is better than tweet" (a bit of pointless punnery). Their clarion call, via megaphone? "What do we want? Tweed!" Okay.<br>Gisele Bundchen leads the unusual finale at Chanel show <br><br>A few of the models had the good grace to look embarrassed; most seemed to think it was a bit of a laugh. Which also summarised the audience's reaction. Maybe Lagerfeld was cynically poking fun at the whole idea of fashion commenting on culture at large, intentionally reducing its protests to facile fashion commandments rather than an attempt at genuine change.<br><br>But the co-opting of protest polemic as a tool instigating you to buy, as opposed to question why, struck a bum note. Was tweed all we should read into this collection? Should a fashion show just make you want to go out and charge something, rather than change something?<br><br>That's not a judgement on the clothes, which were fine. They were playful, colourful, vibrant. The clusters of models walking together had a bounce and an energy, as well as a reflection of the everyday world. There was something compelling about our odd transportation into a unreal "everyday" Paris street - which ended up feeling more like a scene from Paris When It Sizzles, a choreographed and entirely false occurrence for the benefit of a few thousand onlookers, than a real slice of Paris life.<br><br>Cara Delevingne presents a creation for Chanel during the 2015 Spring/Summer ready-to-wear collection fashion show in Paris <br>Perhaps Chanel's riot was inspired those real-life scenes last season, when the fashion press tore apart the Supermarket the label erected in the middle of the Grand Palais - the whole thing was cordoned off with riot barriers this time, just in case. But why riot if you've got nothing to say?<br><br>Chanel wasn't making a political statement, they were just attracting attention by making a great deal of noise and fuss. It was the artifice of  [http://tinyurl.com/kecvhhb discount ugg boots] anarchy. A joke, sure, but not an especially funny one.<br>And it overshadowed the real message of the clothes on show, which in their diversity and lack of fashion diktat, did have something of a feminist, Chanel-for-all message running through them, like the flecks of bright colour through those tweeds.<br><br>Paris Fashion Week spring/summer 2015<br>Looking through the ranks of the audience, and seeing  [http://tinyurl.com/kecvhhb ugg boots sale] the Chanel jacket crop up again and again, from front-row couture client to standing fashion students, you were arrested by the continuing universality of that style Coco Chanel herself originated. It's a great jacket, but it's not worth rioting over.<br><br>Haven't we all - fashion designers included - got something more interesting to say [http://tinyurl.com/kecvhhb http://tinyurl.com/kecvhhb] than that?<br>In short? Nice show. Shame it all had to end like that.
 
To every [[algebraic variety]] ''V'' one can associate a Grothendieck category qcoh(''V''), consisting of the [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V''. This category encodes all the relevant information about ''V'', and ''V'' can be recovered from qcoh(''V''). This example gives rise to one approach to [[noncommutative algebraic geometry]]: the study of "non-commutative varieties" is then nothing but the study of Grothendieck categories.<ref>{{cite web|title=Quantum Ruled Surfaces|author=Izuru Mori|year=2007|url=http://mathsoc.jp/section/algebra/algsymp_past/algsymp07_files/mouri.pdf}}</ref>
 
==Definition==
By definition, a Grothendieck category ''A'' is an [[AB5 category]] with a [[Generator (category theory)|generator]]. Spelled out, this means that
* ''A'' is an [[abelian category]];
* every (possibly infinite) family of objects in ''A'' has a [[coproduct]] (a.k.a. direct sum) in ''A'';
* [[direct limit]]s (a.k.a. filtered colimits) of [[exact sequence]]s are exact; this means that if a direct system of [[short exact sequence]]s in ''A'' is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always [[exact functor|right-exact]]; the important point here is that we require them to be [[exact functor|left-exact]] as well.)
* ''A'' possesses a generator, i.e. there is an object ''G'' in ''A'' such that Hom(''G'',&nbsp;–) is a [[faithful functor]] from ''A'' to the [[category of sets]]. (In our situation, this is equivalent to saying that every object ''X'' of ''A'' admits an [[epimorphism]] ''G''<sup>(''I'')</sup>→''X'', where ''G''<sup>(''I'')</sup> denotes a direct sum of copies of ''G'', one for each element of the (possibly infinite) set ''I''.)
 
==Examples==
* The prototypical example of a Grothendieck category is the [[category of abelian groups]]; the group '''Z''' of integers can serve as a generator.
* More generally, given any [[ring (mathematics)|ring]] ''R'' (associative, with 1, but not necessarily commutative), the category Mod-''R'' of all left (or alternatively: right) [[module (mathematics)|module]]s over ''R'' is a Grothendieck category; ''R'' itself can serve as a generator.
* Given a [[topological space]] ''X'', the category of all [[sheaf (mathematics)|sheaves]] of abelian groups on ''X'' is a Grothendieck category. (More generally: the category of all sheaves of left ''R''-modules on ''X'' is a Grothendieck category for any ring ''R''.) 
* Given a [[ringed space]] (''X'',''O<sub>X</sub>''), the category of [[Sheaf_(mathematics)#Sheaves_of_modules|sheaves of ''O<sub>X</sub>''-modules]] is a Grothendieck category.
* Any category that's [[equivalence of categories|equivalent]] to a Grothendieck category is itself a Grothendieck category.
* Given a [[Category_(mathematics)#Small_and_large_categories|small category]] ''C'' and a Grothendieck category ''A'', the [[functor category]] Funct(''C'',''A'') is a Grothendieck category; if ''C'' is [[preadditive category|preadditive]], then the functor category Add(''C'',''A'') of all additive functors from ''C'' to ''A'' is a Grothendieck category as well.
* Given an (affine or projective) [[algebraic variety]] ''V'' (or more generally: a [[quasi-compact]] [[Glossary_of_scheme_theory#Separated_and_proper_morphisms|quasi-separated]] [[scheme (mathematics)|scheme]]), the category of [[quasi-coherent sheaf|quasi-coherent sheaves]] on ''V'' is a Grothendieck category.
 
==Properties==
Every object in a Grothendieck category ''A'' has an [[injective hull]] in ''A''. This allows to construct [[injective resolution]]s and thereby the use of the tools of [[homological algebra]] in ''A''. Further, ''A'' contains an [[injective cogenerator]].
 
In a Grothendieck category, any family of [[subobject]]s (''U''<sub>''i''</sub>) of a given object ''X'' has a [[supremum]] Σ<sub>''i''</sub>''U''<sub>''i''</sub> which is again a subobject of ''X''. Further, if the family (''U''<sub>''i''</sub>) is directed (i.e. for any two objects in the family, the family has a third object that contains the two), and ''V'' is another subobject of ''X'', we have
:<math>\sum_{i}(U_i\cap V) = \left(\sum_{i}U_i\right) \cap V.</math>
 
In a Grothendieck category, arbitrary [[limit (category theory)|limit]]s (and in particular [[product (category theory)|product]]s) exist. It follows directly from the definition that arbitrary [[limit (category theory)|colimits]] and [[coproduct]]s (direct sums) exist as well. We can thus say that every Grothendieck category is [[complete category|complete and co-complete]]. 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.
 
If ''A'' is a Grothendieck category and ''T'' is a [[localizing subcategory]] of ''A'' (i.e. ''T'' is a full subcategory of ''A'', closed under ''A''-colimits, and for any [[short exact sequence]] <math>0\rarr A_1\rarr A_2\rarr A_3\rarr 0</math> in ''A'', ''A<sub>2</sub>'' is in ''T'' if and only if ''A<sub>1</sub>'' and ''A<sub>3</sub>'' are in ''T''), we can form the [[Serre quotient category]] ''A''/''T''. This quotient is again a Grothendieck category.
 
The [[Gabriel–Popesco theorem]] states that any Grothendieck category ''A'' is equivalent to a [[full subcategory]] of the category Mod-''R'' of right modules over some unital ring ''R'' (which can be taken to be the [[endomorphism ring]] of a generator of ''A''), and ''A'' can be obtained as a Serre quotient of Mod-''R'' by some localizing subcategory.<ref>{{cite journal |authors=N. Popesco, P. Gabriel|title=Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes|journal=Comptes Rendus de l'Académie des Sciences|volume=258|year=1964|pages=4188–4190}}</ref>
 
==References==
{{reflist}}
*{{cite book|author=N. Popescu|title=Abelian categories with applications to rings and modules|publisher=Academic Press|year=1973}}
*{{citation|first1=Pascual|last1=Jara|first2=Alain|last2=Verschoren|first3=Conchi|last3=Vidal|title=Localization and sheaves: a relative point of view|series=Pitman Research Notes in Mathematics Series|volume=339|publisher=Longman, Harlow|year=1995}}.
 
==External links==
*{{SpringerEOM
| title=Grothendieck category
| id=Grothendieck_category
| last=Tsalenko
| first=M.Sh.
| author-link=
| last2=
| first2=
  | author2-link=
}}
*[http://therisingsea.org/notes/AbelianCategories.pdf Abelian Categories], notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.
 
[[Category:Homological algebra]]
[[Category:Category theory]]

Revision as of 04:29, 7 February 2014

Baron Haussmann got a Parisian thoroughfare named after himself in the mid-nineteenth century by reconfiguring the city's circulatory road systems into wide, grand avenues.

The idea was to widen streets and prevent blockades and riots. It didn't entirely work, but was a noble effort.
What did it take for Chanel to get their own street? Well, if you pump vast amounts of cash into building a set that's a photographic facsimile of a Parisian rue, you can call it whatever you damn well want. So, the models at Karl Lagerfeld's spring/summer 2015 Chanel show sauntered in groups down the centre of "Boulevard Chanel," easily chatting amongst themselves.

A few carried boomboxes packed into chic Chanel handbags, the tinny sound reverberating until the thumping main soundtrack took discount ugg boots over. "I'm every woman," warbled Whitney Houston, at one point, "It's all in me."
She could have been singing about the collection because, in typical Lagerfeld style, it was all in there. Wide trousers, tunics, print, plain, military, pinstripes, sweaters, Gisele in a striped cardigan and something suspiciously close to a pair of Chanel Uggs. All present and correct, on flat shoes, women striding meaningfully off to work.

Which, in the fashion world, means the end of the (catwalk) road and back. There were also, obviously, plenty of bags, printed with faux-politicised slogans like "F�ministe mais Feminine," or "Votez Coco." Chanel Spring/Summer 2015 show by German designer Karl Lagerfeld during the Paris Fashion Week
There's been an undercurrent of women dressing women this week in Paris - Lagerfeld isn't a woman, obviously. But he does operate under the mantle of Coco Chanel, who tied her fashion to the feminist cause, even if she didn't realise it. Clothes, she stated, must be logical.

A button should have a buttonhole, and should work. Pockets should be real.
Chanel glorified function over form, or at least the ideal twinning together of the two. Her clothes addressed her own reality, the demands of the clothes she wanted to wear, which found resonance in women in the wider world. It was as simple as the chain handle on her 2.55 handbag, leaving the hands free to get on with something else, or the fact said bag was originally lined in burgundy, to make it easier to find your stuff inside.

Gisele Bundchen on the Chanel spring/summer 2015 catwalk
Chanel's bag was a form ugg boots sale of protest against the impracticality of everyday fashion in the mid fifties - she loathed the corsets and petticoats of the New Look, for ugg boots sale instance, and following her 1954 comeback railed against it everyday she could. Likewise, Lagerfeld's faux-sloganeering on his pouches and pochettes were portents of protest to come.

After the show finished, models returned to the stage (or rather, the street) brandishing placards daubed with pronouncements like "Make fashion not war" (fair enough), or "Tweed is better than tweet" (a bit of pointless punnery). Their clarion call, via megaphone? "What do we want? Tweed!" Okay.
Gisele Bundchen leads the unusual finale at Chanel show

A few of the models had the good grace to look embarrassed; most seemed to think it was a bit of a laugh. Which also summarised the audience's reaction. Maybe Lagerfeld was cynically poking fun at the whole idea of fashion commenting on culture at large, intentionally reducing its protests to facile fashion commandments rather than an attempt at genuine change.

But the co-opting of protest polemic as a tool instigating you to buy, as opposed to question why, struck a bum note. Was tweed all we should read into this collection? Should a fashion show just make you want to go out and charge something, rather than change something?

That's not a judgement on the clothes, which were fine. They were playful, colourful, vibrant. The clusters of models walking together had a bounce and an energy, as well as a reflection of the everyday world. There was something compelling about our odd transportation into a unreal "everyday" Paris street - which ended up feeling more like a scene from Paris When It Sizzles, a choreographed and entirely false occurrence for the benefit of a few thousand onlookers, than a real slice of Paris life.

Cara Delevingne presents a creation for Chanel during the 2015 Spring/Summer ready-to-wear collection fashion show in Paris
Perhaps Chanel's riot was inspired those real-life scenes last season, when the fashion press tore apart the Supermarket the label erected in the middle of the Grand Palais - the whole thing was cordoned off with riot barriers this time, just in case. But why riot if you've got nothing to say?

Chanel wasn't making a political statement, they were just attracting attention by making a great deal of noise and fuss. It was the artifice of discount ugg boots anarchy. A joke, sure, but not an especially funny one.
And it overshadowed the real message of the clothes on show, which in their diversity and lack of fashion diktat, did have something of a feminist, Chanel-for-all message running through them, like the flecks of bright colour through those tweeds.

Paris Fashion Week spring/summer 2015
Looking through the ranks of the audience, and seeing ugg boots sale the Chanel jacket crop up again and again, from front-row couture client to standing fashion students, you were arrested by the continuing universality of that style Coco Chanel herself originated. It's a great jacket, but it's not worth rioting over.

Haven't we all - fashion designers included - got something more interesting to say http://tinyurl.com/kecvhhb than that?
In short? Nice show. Shame it all had to end like that.

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