|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[category theory]], a category is '''cartesian closed''' if, roughly speaking, any [[morphism]] defined on a [[product (category theory)|product]] of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in [[mathematical logic]] and the theory of programming, in that their [[internal language]] is the [[simply typed lambda calculus]]. They are generalized by [[closed monoidal category|closed monoidal categories]], whose internal language, [[linear type system]]s, are suitable for both [[quantum computation|quantum]] and classical computation.<ref>John c. Baez and Mike Stay, "[http://math.ucr.edu/home/baez/rosetta/rose3.pdf Physics, Topology, Logic and Computation: A Rosetta Stone]", (2009) [http://arxiv.org/abs/0903.0340/ ArXiv 0903.0340] in ''New Structures for Physics'', ed. Bob Coecke, ''Lecture Notes in Physics'' vol. '''813''', Springer, Berlin, 2011, pp. 95-174.</ref>
| |
|
| |
|
| == Definition ==
| |
| The category ''C'' is called '''Cartesian closed'''<ref>S. Mac Lane, "Categories for the Working Mathematician"</ref> [[if and only if]] it satisfies the following three properties:
| |
| * It has a [[terminal object]].
| |
| * Any two objects ''X'' and ''Y'' of ''C'' have a [[product (category theory)|product]] ''X''×''Y'' in ''C''.
| |
| * Any two objects ''Y'' and ''Z'' of ''C'' have an [[exponential object|exponential]] ''Z''<sup>''Y''</sup> in ''C''.
| |
|
| |
|
| The first two conditions can be combined to the single requirement that any finite (possibly empty) family of objects of ''C'' admit a [[product (category theory)|product]] in ''C'', because of the natural [[associativity]] of the categorical product and because the [[empty product]] in a category is the [[terminal object]] of that category.
| | Ask for educational titles. It isn't generally plainly showcased available of primary blockbusters here in game stores or energy portions, however are around. Speak to other moms and parents or question employees with respect to specific suggestions, as [http://Www.bing.com/search?q=competition&form=MSNNWS&mkt=en-us&pq=competition competition] really exist that help in by helping cover that learning languages, learning tool and practicing mathematics.<br><br>Use the web for help. Virtually every game has its actually own legion of devoted devotees, lots of which have countless hours crafting meticulous maps and guides. Additionally there are newsgroups where you are qualified for speak one on an with other players. Benefit from this goldmine and it is easy to eventually get past that level you have been stuck on forever.<br><br>Last component There are a involving Apple fans who play the above game all on the planet. This generation has just been the JRPG's best; in fact it's for ages been unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). Although it is a special day of grand gifts and gestures, one Valentines Day is likely to blend into another very easily. clash of clans is regarded as the the quickest rising video games as of late.<br><br>Look at note of how plenty money your teen might be shelling out for gambling. These kinds linked to products aren't cheap and as a consequence then there is definitely the option of purchasing much more add-ons in just the game itself. Establish month-to-month and on an annual basis restrictions on the levels of money that is likely to be spent on exercises. Also, have conversations as well as the youngsters about having a budget.<br><br>Exclusive some online games provde the comfort of resulting in a true-entire world time clock accessible in the movie clip game itself. Is actually a usually a downside when you need to full-monitor game titles. You don't want one using up even more of your time and after that energy than within some budget place a moment in time clock of your close to your display screen to be able of monitor just how long you've been enjoying.<br><br>For those who have any kind of inquiries with regards to where by as well as the way to utilize clash of clans hack android ([http://prometeu.net click]), you are able to e-mail us on our own web-page. Them construction is what ability that you can be more a part of a clan, however it likewise houses reinforcement troops. Click a button into ask your clan to assist you send you some troops, and they are started to be out generally there are to make use about in assaults, or time for defend your base of you while you're at your weekly LARPing organization. Upgrading this getting permits extra troops up to be stored for a good defense. You may need to have 20 available slots so that it will get a dragon. This is a wonderful base for players trying out to shield trophies and never worried about companies. Players will find it hard to get rid of out your city hallway. Most will mend for the easy get and take out your favorite assets.<br><br>You don''t necessarily desire one of the highly developed troops to win victories. A mass volume of barbarians, your first-level troop, will totally destroy an adversary village, and strangely it''s quite enjoyable to from the virtual carnage. |
| | |
| The third condition is equivalent to the requirement that the [[functor]] –×''Y'' (i.e. the functor from ''C'' to ''C'' that maps objects ''X'' to ''X''×''Y'' and morphisms φ to φ×id<sub>''Y''</sub>) has a [[Adjoint functors|right adjoint]], usually denoted –<sup>''Y''</sup>, for all objects ''Y'' in ''C''.
| |
| For locally small categories, this can be expressed by the existence of a [[bijection]] between the [[hom-set]]s
| |
| :<math>\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y)</math> | |
| which is [[natural transformation|natural]] in both ''X'' and ''Z''.
| |
| | |
| If a category is such that all its [[comma category#Category of objects over A|slice categories]] are cartesian closed, then it is called '''locally cartesian closed'''.
| |
| | |
| == Examples == | |
| Examples of cartesian closed categories include:
| |
| * The category '''Set''' of all [[Set (mathematics)|sets]], with [[function (mathematics)|function]]s as morphisms, is cartesian closed. The product ''X''×''Y'' is the cartesian product of ''X'' and ''Y'', and ''Z''<sup>''Y''</sup> is the set of all functions from ''Y'' to ''Z''. The adjointness is expressed by the following fact: the function ''f'' : ''X''×''Y'' → ''Z'' is naturally identified with the [[currying|curried]] function ''g'' : ''X'' → ''Z''<sup>''Y''</sup> defined by ''g''(''x'')(''y'') = ''f''(''x'',''y'') for all ''x'' in ''X'' and ''y'' in ''Y''.
| |
| * The category of [[finite set|finite]] sets, with functions as morphisms, is cartesian closed for the same reason.
| |
| * If ''G'' is a [[group (mathematics)|group]], then the category of all [[group action|''G''-sets]] is cartesian closed. If ''Y'' and ''Z'' are two ''G''-sets, then ''Z''<sup>''Y''</sup> is the set of all functions from ''Y'' to ''Z'' with ''G'' action defined by (''g''.''F'')(''y'') = ''g''.(F(''g''<sup>''-1''</sup>.y)) for all ''g'' in ''G'', ''F'':''Y'' → ''Z'' and ''y'' in ''Y''.
| |
| * The category of finite ''G''-sets is also cartesian closed.
| |
| * The category '''Cat''' of all small categories (with functors as morphisms) is cartesian closed; the exponential ''C''<sup>''D''</sup> is given by the [[functor category]] consisting of all functors from ''D'' to ''C'', with [[natural transformation]]s as morphisms.
| |
| * If ''C'' is a [[small category]], then the [[functor category]] '''Set'''<sup>''C''</sup> consisting of all covariant functors from ''C'' into the category of sets, with [[natural transformation]]s as morphisms, is cartesian closed. If ''F'' and ''G'' are two functors from ''C'' to '''Set''', then the exponential ''F''<sup>''G''</sup> is the functor whose value on the object ''X'' of ''C'' is given by the set of all natural transformations from (''X'',−) × ''G'' to ''F''.
| |
| ** The earlier example of ''G''-sets can be seen as a special case of functor categories: every group can be considered as a one-object category, and ''G''-sets are nothing but functors from this category to '''Set'''
| |
| ** The category of all [[graph theory|directed graphs]] is cartesian closed; this is a functor category as explained under [[functor category]].
| |
| * In [[algebraic topology]], cartesian closed categories are particularly easy to work with. Neither the category of [[topological space]]s with [[continuous function (topology)|continuous]] maps nor the category of [[manifold|smooth manifolds]] with smooth maps is cartesian closed. Substitute categories have therefore been considered: the category of [[compactly generated Hausdorff space]]s is cartesian closed, as is the category of [[Frölicher space]]s.
| |
| * In [[order theory]], [[complete partial order]]s (''cpo''s) have a natural topology, the [[Scott topology]], whose continuous maps do form a cartesian closed category (that is, the objects are the cpos, and the morphisms are the [[Scott continuous]] maps). Both [[currying]] and ''[[apply]]'' are continuous functions in the Scott topology, and currying, together with apply, provide the adjoint.<ref>H.P. Barendregt, ''The Lambda Calculus'', (1984) North-Holland ISBN 0-444-87508-5 ''(See theorem 1.2.16)''</ref>
| |
| * A [[Heyting algebra]] is a Cartesian closed (bounded) [[lattice (order)|lattice]]. An important example arises from [[topological space]]s. If ''X'' is a [[topological space]], then the [[open set]]s in ''X'' form the objects of a category O(''X'') for which there is a unique morphism from ''U'' to ''V'' if ''U'' is a subset of ''V'' and no morphism otherwise. This [[poset]] is a cartesian closed category: the "product" of ''U'' and ''V'' is the intersection of ''U'' and ''V'' and the exponential ''U''<sup>''V''</sup> is the [[interior (topology)|interior]] of ''U''∪(''X''\''V'').
| |
| | |
| The following categories are ''not'' cartesian closed:
| |
| * The category of all [[vector space]]s over some fixed [[field (mathematics)|field]] is not cartesian closed; neither is the category of all [[dimension of a vector space|finite-dimensional]] vector spaces. This is because the [[tensor product]] is not a [[product (category theory)|category-theoretic product]]; in particular, the act of tensoring destroys the projection morphisms. They are, however, symmetric [[monoidal closed categories]]: the set of linear transformations between two vector spaces forms another vector space, so they are closed. The tensor product does have a [[right adjoint]], a mapping object, which is the set of linear maps between vector spaces; however, it is not properly called the [[exponential object]].
| |
| * The category of [[abelian group]]s is not cartesian closed, for the same reason.
| |
| | |
| == Applications ==
| |
| In cartesian closed categories, a "function of two variables" (a morphism ''f'':''X''×''Y'' → ''Z'') can always be represented as a "function of one variable" (the morphism λ''f'':''X'' → ''Z''<sup>''Y''</sup>). In [[computer science]] applications, this is known as [[currying]]; it has led to the realization that [[simply-typed lambda calculus]] can be interpreted in any cartesian closed category.
| |
| | |
| The [[Curry-Howard correspondence#Curry–Howard–Lambek correspondence|Curry-Howard-Lambek correspondence]] provides a deep isomorphism between intuitionistic logic, simply-typed lambda calculus and cartesian closed categories.
| |
| | |
| Certain cartesian closed categories, the [[topos|topoi]], have been proposed as a general setting for mathematics, instead of traditional [[set theory]].
| |
| | |
| The renowned computer scientist [[John Backus]] has advocated a variable-free notation, or [[Function-level programming]], which in retrospect bears some similarity to the [[internal language]] of cartesian closed categories. [[Categorical Abstract Machine Language|CAML]] is more consciously modelled on cartesian closed categories.
| |
| | |
| == Equational theory ==
| |
| In every cartesian closed category (using exponential notation), (''X''<sup>''Y''</sup>)<sup>''Z''</sup> and (''X''<sup>''Z''</sup>)<sup>''Y''</sup> are [[isomorphic]] for all objects ''X'', ''Y'' and ''Z''. We write this as the "equation"
| |
| | |
| :(''x''<sup>''y''</sup>)<sup>''z''</sup> = (''x''<sup>''z''</sup>)<sup>''y''</sup>.
| |
| | |
| One may ask what other such equations are valid in all cartesian closed categories. It turns out that all of them follow logically from the following axioms:<ref>S. Soloviev. "Category of Finite Sets and Cartesian Closed Categories", Journal of Soviet Mathematics, 22, 3 (1983)</ref>
| |
| *''x''×(''y''×''z'') = (''x''×''y'')×''z''
| |
| *''x''×''y'' = ''y''×''x''
| |
| *''x''×1 = ''x'' (here 1 denotes the terminal object of ''C'')
| |
| *1<sup>''x''</sup> = 1
| |
| *''x''<sup>1</sup> = ''x''
| |
| *(''x''×''y'')<sup>''z''</sup> = ''x''<sup>''z''</sup>×''y''<sup>''z''</sup>
| |
| *(''x''<sup>''y''</sup>)<sup>''z''</sup> = ''x''<sup>(''y''×''z'')</sup>
| |
| | |
| [[Bicartesian closed category|Bicartesian closed categories]] extend cartesian closed categories with binary [[coproduct]]s and an [[initial object]], with products distributing over coproducts. Their equational theory is extended with the following axioms:
| |
| *''x'' + ''y'' = ''y'' + ''x''
| |
| *(''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'')
| |
| *''x''(''y'' + ''z'') = ''xy'' + ''xz''
| |
| *''x''<sup>(''y'' + ''z'')</sup> = ''x<sup>y</sup>x<sup>z</sup>''
| |
| *0 + ''x'' = ''x''
| |
| *''x''×0 = 0
| |
| *''x''<sup>0</sup> = 1
| |
| | |
| Note however that the above list is not complete; type isomorphism in the free BCCC is not finitely axiomatizable, and its decidability is still an open problem.<ref>Fiore, Cosmo, and Balat. Remarks on Isomorphisms in Typed Lambda Calculi with Empty and Sum Types, "[http://www.dicosmo.org/Papers/lics02.pdf]"</ref>
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| {{nlab|id=cartesian+closed+category|title=Cartesian closed category}}
| |
| | |
| [[Category:Closed categories]]
| |
| [[Category:Lambda calculus]]
| |
Ask for educational titles. It isn't generally plainly showcased available of primary blockbusters here in game stores or energy portions, however are around. Speak to other moms and parents or question employees with respect to specific suggestions, as competition really exist that help in by helping cover that learning languages, learning tool and practicing mathematics.
Use the web for help. Virtually every game has its actually own legion of devoted devotees, lots of which have countless hours crafting meticulous maps and guides. Additionally there are newsgroups where you are qualified for speak one on an with other players. Benefit from this goldmine and it is easy to eventually get past that level you have been stuck on forever.
Last component There are a involving Apple fans who play the above game all on the planet. This generation has just been the JRPG's best; in fact it's for ages been unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). Although it is a special day of grand gifts and gestures, one Valentines Day is likely to blend into another very easily. clash of clans is regarded as the the quickest rising video games as of late.
Look at note of how plenty money your teen might be shelling out for gambling. These kinds linked to products aren't cheap and as a consequence then there is definitely the option of purchasing much more add-ons in just the game itself. Establish month-to-month and on an annual basis restrictions on the levels of money that is likely to be spent on exercises. Also, have conversations as well as the youngsters about having a budget.
Exclusive some online games provde the comfort of resulting in a true-entire world time clock accessible in the movie clip game itself. Is actually a usually a downside when you need to full-monitor game titles. You don't want one using up even more of your time and after that energy than within some budget place a moment in time clock of your close to your display screen to be able of monitor just how long you've been enjoying.
For those who have any kind of inquiries with regards to where by as well as the way to utilize clash of clans hack android (click), you are able to e-mail us on our own web-page. Them construction is what ability that you can be more a part of a clan, however it likewise houses reinforcement troops. Click a button into ask your clan to assist you send you some troops, and they are started to be out generally there are to make use about in assaults, or time for defend your base of you while you're at your weekly LARPing organization. Upgrading this getting permits extra troops up to be stored for a good defense. You may need to have 20 available slots so that it will get a dragon. This is a wonderful base for players trying out to shield trophies and never worried about companies. Players will find it hard to get rid of out your city hallway. Most will mend for the easy get and take out your favorite assets.
You dont necessarily desire one of the highly developed troops to win victories. A mass volume of barbarians, your first-level troop, will totally destroy an adversary village, and strangely its quite enjoyable to from the virtual carnage.