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In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

If a category is equivalent to the opposite (or dual) of another category then one speaks of a duality of categories, and says that the two categories are dually equivalent.

An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be naturally isomorphic to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.

Definition

Formally, given two categories C and D, an equivalence of categories consists of a functor F : C → D, a functor G : D → C, and two natural isomorphisms ε: FG→I_D and η : I_C→GF. Here FG: D→D and GF: C→C, denote the respective compositions of F and G, and I_C: C→C and I_D: D→D denote the identity functors on C and D, assigning each object and morphism to itself. If F and G are contravariant functors one speaks of a duality of categories instead.

One often does not specify all the above data. For instance, we say that the categories C and D are equivalent (respectively dually equivalent) if there exists an equivalence (respectively duality) between them. Furthermore, we say that F "is" an equivalence of categories if an inverse functor G and natural isomorphisms as above exist. Note however that knowledge of F is usually not enough to reconstruct G and the natural isomorphisms: there may be many choices (see example below).

Equivalent characterizations

One can show that a functor F : C → D yields an equivalence of categories if and only if it is:

full, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is surjective;
faithful, i.e. for any two objects c₁ and c₂ of C, the map Hom_C(c₁,c₂) → Hom_D(Fc₁,Fc₂) induced by F is injective; and
essentially surjective (dense), i.e. each object d in D is isomorphic to an object of the form Fc, for c in C.

This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" G and the natural isomorphisms between FG, GF and the identity functors. On the other hand, though the above properties guarantee the existence of a categorical equivalence (given a sufficiently strong version of the axiom of choice in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible. Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories (unfortunately this conflicts with terminology from homotopy theory).

There is also a close relation to the concept of adjoint functors. The following statements are equivalent for functors F : C → D and G : D → C:

There are natural isomorphisms from FG to I_D and I_C to GF.
F is a left adjoint of G and both functors are full and faithful.
F is a right adjoint of G and both functors are full and faithful.

One may therefore view an adjointness relation between two functors as a "very weak form of equivalence". Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.

Examples

Consider the category $C$ having a single object $c$ and a single morphism $1_{c}$ , and the category $D$ with two objects $d_{1}$ , $d_{2}$ and four morphisms: two identity morphisms $1_{d_{1}}$ , $1_{d_{2}}$ and two isomorphisms $α : d_{1} \to d_{2}$ and $β : d_{2} \to d_{1}$ . The categories $C$ and $D$ are equivalent; we can (for example) have $F$ map $c$ to $d_{1}$ and $G$ map both objects of $D$ to $c$ and all morphisms to $1_{c}$ .

By contrast, the category $C$ with a single object and a single morphism is not equivalent to the category $E$ with two objects and only two identity morphisms as the two objects therein are not isomorphic.

Consider a category $C$ with one object $c$ , and two morphisms $1_{c}, f : c \to c$ . Let $1_{c}$ be the identity morphism on $c$ and set $f \circ f = 1$ . Of course, $C$ is equivalent to itself, which can be shown by taking $1_{c}$ in place of the required natural isomorphisms between the functor $I_{C}$ and itself. However, it is also true that $f$ yields a natural isomorphism from $I_{C}$ to itself. Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.

Consider the category $C$ of finite-dimensional real vector spaces, and the category $D = M a t (R)$ of all real matrices (the latter category is explained in the article on additive categories). Then $C$ and $D$ are equivalent: The functor $G : D \to C$ which maps the object $A_{n}$ of $D$ to the vector space $R^{n}$ and the matrices in $D$ to the corresponding linear maps is full, faithful and essentially surjective.

One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings. The functor $G$ associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. Its adjoint $F$ associates to every affine scheme its ring of global sections.

In functional analysis the category of commutative C*-algebras with identity is contravariantly equivalent to the category of compact Hausdorff spaces. Under this duality, every compact Hausdorff space $X$ is associated with the algebra of continuous complex-valued functions on $X$ , and every commutative C*-algebra is associated with the space of its maximal ideals. This is the Gelfand representation.

In lattice theory, there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of topological spaces. Probably the most well-known theorem of this kind is Stone's representation theorem for Boolean algebras, which is a special instance within the general scheme of Stone duality. Each Boolean algebra $B$ is mapped to a specific topology on the set of ultrafilters of $B$ . Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra. One obtains a duality between the category of Boolean algebras (with their homomorphisms) and Stone spaces (with continuous mappings). Another case of Stone duality is Birkhoff's representation theorem stating a duality between finite partial orders and finite distributive lattices.

In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.

Any category is equivalent to its skeleton.

Properties

As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true:

the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
the morphism α in C is a monomorphism (or epimorphism, or isomorphism), if and only if Fα is a monomorphism (or epimorphism, or isomorphism) in D.
the functor H : I → C has limit (or colimit) l if and only if the functor FH : I → D has limit (or colimit) Fl. This can be applied to equalizers, products and coproducts among others. Applying it to kernels and cokernels, we see that the equivalence F is an exact functor.
C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos).

Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.

If F : C → D is an equivalence of categories, and G₁ and G₂ are two inverses of F, then G₁ and G₂ are naturally isomorphic.

If F : C → D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)

An auto-equivalence of a category C is an equivalence F : C → C. The auto-equivalences of C form a group under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of C. (One caveat: if C is not a small category, then the auto-equivalences of C may form a proper class rather than a set.)

References

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nl:Equivalentie (categorietheorie) zh-yue:範疇等價性 zh:范畴的等价

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-{{Technical|date=August 2011}}
+In [[category theory]], an abstract branch of [[mathematics]], an '''equivalence of categories''' is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
-{{Development economics sidebar}}
-'''Endogenous growth theory''' holds that [[economic growth]] is primarily the result of [[endogeny|endogenous]] and not external forces.<ref>{{cite journal |journal = [[The Journal of Economic Perspectives]] |volume= 8 |issue= 1 |year= 1994 |url= http://links.jstor.org/sici?sici=0895-3309%28199424%298%3A1%3C3%3ATOOEG%3E2.0.CO%3B2-H |first= P. M. |last= Romer |title= The Origins of Endogenous Growth |authorlink= Paul Romer |page = [http://www.iset.ge/old/upload/Romer%201994.pdf 3]  }}</ref> Endogenous growth theory holds that investment in [[human capital]], [[innovation]], and knowledge are significant contributors to economic growth. The theory also focuses on [[positive externalities]] and [[spillover effects]] of a knowledge-based economy which will lead to economic development. The endogenous growth theory also holds that policy measures can have an impact on the long-run growth rate of an economy. For example, [[subsidies]] for [[research and development]] or [[education]] increase the growth rate in some endogenous growth models by increasing the incentive for innovation.
-==Models in Endogenous Growth==
+If a category is equivalent to the [[dual (category theory)|opposite (or dual)]] of another category then one speaks of
+a '''duality of categories''', and says that the two categories are '''dually equivalent'''.
-In the mid-1980s, a group of growth theorists became increasingly dissatisfied with common accounts of [[exogenous]] factors determining long-run growth. They favored a model that replaced the exogenous growth variable (unexplained technical progress) with a model in which the key determinants of growth were explicit in the model. The initial research was based on the work of [[Kenneth Arrow]] (1962), [[Hirofumi Uzawa]] (1965), and [[Miguel Sidrauski]] (1967).<ref>{{cite web |url= http://www.newschool.edu/nssr/het/essays/growth/moneygrowth.htm |title=Monetary Growth Theory   |work=newschool.edu |year=2011 [last update] |accessdate=11 October 2011}}</ref> [[Paul Romer]] (1986), [[Robert Emerson Lucas, Jr. |Lucas]] (1988),<ref>{{cite journal |url= http://www.fordham.edu/economics/mcleod/LucasMechanicsEconomicGrowth.pdf |title= On the mechanics of Economic Development |first= R. E. |last= Lucas |authorlink= Robert Emerson Lucas, Jr. |journal = [[Journal of Monetary Economics]] |year=1988 |volume = 22  }}</ref> and Rebelo (1991)<ref>{{cite journal |url=http://www.nber.org/papers/w3325 |title= Long-Run Policy Analysis and Long-Run Growth |first= Sergio |last= Rebelo  |journal = [[Journal of Political Economy]] |year=1991 |volume = 99 |issue= 3 |page= 500 }}</ref><ref>{{cite web |url= http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/Growth/RebeloAK.pdf |title= The Rebelo AK Growth Model  |first= C.|last= Carroll |work=econ2.jhu.edu |year=2011 [last update] |accessdate=11 October 2011 |quote= the steady-state growth rate in a Rebelo economy is directly proportional to the saving rate.}}</ref> omitted technological change. Instead, growth in these models was due to indefinite investment in [[human capital]] which had [[spillover effect]] on economy and reduces the diminishing return to [[capital accumulation]].<ref name= "BX">{{cite book |first1= R. J. |last1= Barro |first2= Xavier |authorlink2= Xavier Sala-i-Martin |last2= Sala-i-Martin |title= Economic Growth |isbn= 978-0-262-02459-4 |date= 1998-11-20 |url= http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf  }}</ref>
+An equivalence of categories consists of a [[functor]] between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for [[isomorphism]]s in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be ''[[natural transformation|naturally isomorphic]]'' to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of [[isomorphism of categories]] where a strict form of inverse functor is required, but this is of much less practical use than the ''equivalence'' concept.
-The [[AK model]], which is the simplest endogenous model, gives a constant-saving-rate of endogenous growth. It assumes a constant, exogenous saving rate and fixed level of the technology. It shows elimination of diminishing returns leading to endogenous growth. However, the endogenous growth theory is further supported with models in which agents optimally determined the consumption and saving, optimizing the resources allocation to research and development leading to technological progress. Romer (1987, 1990) and signiﬁcant contributions by Aghion and Howitt (1992) and Grossman and Helpman (1991), incorporated [[imperfect market]]s and R&D to the growth model.<ref name= "BX"/>
+==Definition==
+Formally, given two categories ''C'' and ''D'', an ''equivalence of categories'' consists of a functor ''F'' : ''C'' → ''D'', a functor ''G'' : ''D'' → ''C'', and two natural isomorphisms ε: ''FG''→'''I'''<sub>''D''</sub> and η : '''I'''<sub>''C''</sub>→''GF''. Here ''FG'': ''D''→''D'' and ''GF'': ''C''→''C'', denote the respective compositions of ''F'' and ''G'', and '''I'''<sub>''C''</sub>: ''C''→''C'' and '''I'''<sub>''D''</sub>: ''D''→''D'' denote the ''identity functors'' on ''C'' and ''D'', assigning each object and morphism to itself. If ''F'' and ''G'' are contravariant functors one speaks of a ''duality of categories'' instead.
-== The AK Model ==
+One often does not specify all the above data. For instance, we say that the categories ''C'' and ''D'' are ''equivalent'' (respectively ''dually equivalent'') if there exists an equivalence (respectively duality) between them. Furthermore, we say that ''F'' "is" an equivalence of categories if an inverse functor ''G'' and natural isomorphisms as above exist. Note however that knowledge of ''F'' is usually not enough to reconstruct ''G'' and the natural isomorphisms: there may be many choices (see example below).
-{{main|AK model}}
-The model works on the property of absence of diminishing returns to capital. The simplest form of production function with diminishing return is:
+==Equivalent characterizations==
-[[File:Ak model.png|thumb|figure 1.1]]
+One can show that a functor ''F'' : ''C'' → ''D'' yields an equivalence of categories if and only if it is:
-:<math>Y = AK\,</math>
+* [[full functor|full]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[surjective]];
+* [[faithful functor|faithful]], i.e. for any two objects ''c''<sub>1</sub> and ''c''<sub>2</sub> of ''C'', the map Hom<sub>''C''</sub>(''c''<sub>1</sub>,''c''<sub>2</sub>) → Hom<sub>''D''</sub>(''Fc''<sub>1</sub>,''Fc''<sub>2</sub>) induced by ''F'' is [[injective]]; and
+* [[essentially surjective functor|essentially surjective (dense)]], i.e. each object ''d'' in ''D'' is isomorphic to an object of the form ''Fc'', for ''c'' in ''C''.
+This is a quite useful and commonly applied criterion, because one does not have to explicitly construct the "inverse" ''G'' and the natural isomorphisms between ''FG'', ''GF'' and the identity functors. On the other hand, though the above properties guarantee the ''existence'' of a categorical equivalence (given a sufficiently strong version of the [[axiom of choice]] in the underlying set theory), the missing data is not completely specified, and often there are many choices. It is a good idea to specify the missing constructions explicitly whenever possible.
+Due to this circumstance, a functor with these properties is sometimes called a '''weak equivalence of categories''' (unfortunately this conflicts with terminology from homotopy theory).
-where
+There is also a close relation to the concept of [[adjoint functors]]. The following statements are equivalent for functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'':
-:<math> A\,</math> , is a positive constant that reflects the level of the technology.
+* There are natural isomorphisms from ''FG'' to '''I'''<sub>''D''</sub> and '''I'''<sub>''C''</sub> to ''GF''.
+* ''F'' is a left adjoint of ''G'' and both functors are full and faithful.
+* ''F'' is a right adjoint of ''G'' and both functors are full and faithful.
+One may therefore view an adjointness relation between two functors as a "very weak form of equivalence". Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed. The key property that one has to prove here is that the ''counit'' of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
-:<math> K \,</math> capital (broad sense to include human capital)
+==Examples==
+* Consider the category <math>C</math> having a single object <math>c</math> and a single morphism <math>1_{c}</math>, and the category <math>D</math> with two objects <math>d_{1}</math>, <math>d_{2}</math> and four morphisms: two identity morphisms <math>1_{d_{1}}</math>, <math>1_{d_{2}}</math> and two isomorphisms <math>\alpha \colon d_{1} \to d_{2}</math> and <math>\beta \colon d_{2} \to d_{1}</math>.  The categories <math>C</math> and <math>D</math> are equivalent; we can (for example) have <math>F</math> map <math>c</math> to <math>d_{1}</math> and <math>G</math> map both objects of <math>D</math> to <math>c</math> and all morphisms to <math>1_{c}</math>.
-:<math>y = AK\,</math> , output per capita and the average and marginal product are constant at the level <math>A>0\,</math>
+* By contrast, the category <math>C</math> with a single object and a single morphism is ''not'' equivalent to the category <math>E</math> with two objects and only two identity morphisms as the two objects therein are ''not'' isomorphic.
-If we substitute <math>\frac{f(k)}{k}=A \,</math> in equation of transitional Dynamics of Solow-Swan model ([[Exogenous growth model]]) which shows how an economy’s per capita incomes converges toward its own steady-state value and to the per capita incomes of other nations.
+* Consider a category <math>C</math> with one object <math>c</math>, and two morphisms <math>1_{c}, f \colon c \to c</math>.  Let <math>1_{c}</math> be the identity morphism on <math>c</math> and set <math>f \circ f = 1</math>.  Of course, <math>C</math> is equivalent to itself, which can be shown by taking <math>1_{c}</math> in place of the required natural isomorphisms between the functor <math>\mathbf{I}_{C}</math> and itself.  However, it is also true that <math>f</math> yields a natural isomorphism from <math>\mathbf{I}_{C}</math> to itself.  Hence, given the information that the identity functors form an equivalence of categories, in this example one still can choose between two natural isomorphisms for each direction.
-Transitional Dynamics equation, where Growth rate on <math> k\,</math> is given by,
+* Consider the category <math>C</math> of finite-[[dimension of a vector space|dimensional]] [[real number|real]] [[vector space]]s, and the category <math>D = \mathrm{Mat}(\mathbf{R})</math> of all real [[matrix (mathematics)|matrices]] (the latter category is explained in the article on [[additive category|additive categories]]).  Then <math>C</math> and <math>D</math> are equivalent: The functor <math>G \colon D \to C</math> which maps the object <math>A_{n}</math> of <math>D</math> to the vector space <math>\mathbf{R}^{n}</math> and the matrices in <math>D</math> to the corresponding linear maps is full, faithful and essentially surjective.
-:<math>\gamma_K=\dot{k}/k = s.f(k)/ k - (n+\delta)\ ,</math>
+* One of the central themes of [[algebraic geometry]] is the duality of the category of [[affine scheme]]s and the category of [[commutative ring]]s.  The functor <math>G</math> associates to every commutative ring its [[spectrum of a ring|spectrum]], the scheme defined by the [[prime ideal]]s of the ring.  Its adjoint <math>F</math> associates to every affine scheme its ring of global sections.
-on substituting <math> A\,</math>, we get,
-:<math>\gamma_K= sA -(n+\delta)\ ,</math>
-We return here to the case of zero technological progress, <math> x=0\,</math>, because we want to show that per capita growth can now occur in the long-run even without exogenous technological change. The figure 1.1 explains the perpetual growth, with exogenous technical progress.  The vertical distance between the two line, <math> sA\,</math>and n+&delta; gives the<math>\gamma_K\,</math>
+* In [[functional analysis]] the category of commutative [[C*-algebra]]s with identity is contravariantly equivalent to the category of [[compact space|compact]] [[Hausdorff space]]s.  Under this duality, every compact Hausdorff space <math>X</math> is associated with the algebra of continuous complex-valued functions on <math>X</math>, and every commutative C*-algebra is associated with the space of its [[maximal ideal]]s.  This is the [[Gelfand representation]].
-As, <math> sA>\, </math>n+&delta;, so that<math>\gamma_K > 0\,</math>. Since the two line are parallel, <math>\gamma_K\,</math>is constant; in particular, it is independent of <math>K\,</math>. In other words,<math>K\,</math> always grows at steady states rate,<math>\gamma_K^*= sA -(n+\delta)\ ,</math>.
+* In [[lattice theory]], there are a number of dualities, based on representation theorems that connect certain classes of lattices to classes of [[topology|topological spaces]].  Probably the most well-known theorem of this kind is ''[[Stone's representation theorem for Boolean algebras]]'', which is a special instance within the general scheme of ''[[Stone duality]]''.  Each [[Boolean algebra (structure)|Boolean algebra]] <math>B</math> is mapped to a specific topology on the set of [[lattice theory|ultrafilters]] of <math>B</math>.  Conversely, for any topology the clopen (i.e. closed and open) subsets yield a Boolean algebra.  One obtains a duality between the category of Boolean algebras (with their homomorphisms) and [[Stone space]]s (with continuous mappings). Another case of Stone duality is [[Birkhoff's representation theorem]] stating a duality between finite partial orders and finite distributive lattices.
-Since
+* In [[pointless topology]] the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
-:<math>y = AK\,</math>,<math>\gamma_K\,</math> equals <math>\gamma_K^*\,</math>
-at every point of time. In addition, since
+* Any category is equivalent to its [[skeleton (category theory)|skeleton]].
-:<math>c= (1-s) y\,</math>,
-the growth rate of
+==Properties==
-:<math>c\,</math> equals <math>\gamma_K^*\,</math>.
+As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If ''F'' : ''C'' → ''D'' is an equivalence, then the following statements are all true:
+* the object ''c'' of ''C'' is an [[initial object]] (or [[terminal object]], or [[zero object]]), [[if and only if]] ''Fc'' is an [[initial object]] (or [[terminal object]], or [[zero object]]) of ''D''
+* the morphism α in ''C'' is a [[monomorphism]] (or [[epimorphism]], or [[isomorphism]]), if and only if ''Fα'' is a monomorphism (or epimorphism, or isomorphism) in ''D''.
+* the functor ''H'' : ''I'' → ''C'' has [[limit (category theory)|limit]] (or colimit) ''l'' if and only if the functor ''FH'' : ''I'' → ''D'' has limit (or colimit) ''Fl''. This can be applied to [[equaliser (mathematics)|equalizers]], [[product (category theory)|product]]s and [[coproduct]]s among others. Applying it to [[kernel (category theory)|kernel]]s and [[cokernel]]s, we see that the equivalence ''F'' is an [[Regular_category#Exact_sequences_and_regular_functors|exact functor]].
+* ''C'' is a [[cartesian closed category]] (or a [[topos]]) if and only if ''D'' is cartesian closed (or a topos).
-Hence, the entire per capita variable in the model grows at same rate, given by
+Dualities "turn all concepts around": they turn initial objects into terminal objects, monomorphisms into epimorphisms, kernels into cokernels, limits into colimits etc.
-:<math>\gamma^*= sA -(n+\delta)\ ,</math>
-However, we can observe that<math>y = AK\,</math> technology displays a positive long-run per capita growth without any exogenous technological development. The per capita growth depends on behavioural factors of the model as the saving rate and population. It is unlike neoclassical model, which is higher saving, s, promotes higher long-run per capita growth <math>\gamma^*\,</math>.<ref>Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin</ref>
+If ''F'' : ''C'' → ''D'' is an equivalence of categories, and ''G''<sub>1</sub> and ''G''<sub>2</sub> are two inverses of ''F'', then ''G''<sub>1</sub> and ''G''<sub>2</sub> are naturally isomorphic.
-== Endogenous versus exogenous growth theory ==
+If ''F'' : ''C'' → ''D'' is an equivalence of categories, and if ''C'' is a [[preadditive category]] (or [[additive category]], or [[abelian category]]), then ''D'' may be turned into a preadditive category (or additive category, or abelian category) in such a way that ''F'' becomes an [[additive functor]]. On the other hand, any equivalence between additive categories is necessarily additive. (Note that the latter statement is not true for equivalences between preadditive categories.)
-In neo-classical growth models, the long-run rate of growth is [[exogeny|exogenous]]ly determined by either the savings rate (the [[Harrod–Domar model]]) or the rate of technical progress ([[Solow model]]). However, the savings rate and rate of technological progress remain unexplained. Endogenous growth theory tries to overcome this shortcoming by building macroeconomic models out of [[Microfoundations|microeconomic foundations]]. Households are assumed to maximize utility subject to budget constraints while firms maximize profits. Crucial importance is usually given to the production of new technologies and [[human capital]]. The engine for growth can be as simple as a constant return to scale production function (the AK model) or more complicated set ups with [[Knowledge spillover|spillover]] effects (spillovers are positive externalities, benefits that are attributed to costs from other firms), increasing numbers of goods, increasing qualities, etc.
-Often endogenous growth theory assumes constant marginal product of capital at the aggregate level, or at least that the limit of the marginal product of capital does not tend towards zero. This does not imply that larger firms will be more productive than small ones, because at the firm level the marginal product of capital is still diminishing. Therefore, it is possible to construct endogenous growth models with [[perfect competition]]. However, in many endogenous growth models the assumption of perfect competition is relaxed, and some degree of [[monopoly]] power is thought to exist. Generally monopoly power in these models comes from the holding of patents. These are models with two sectors, producers of final output and an R&D sector. The R&D sector develops ideas that they are granted a monopoly power. R&D firms are assumed to be able to make monopoly profits selling ideas to production firms, but the [[free entry]] condition means that these profits are dissipated on R&D spending.
+An '''auto-equivalence''' of a category ''C'' is an equivalence ''F'' : ''C'' → ''C''. The auto-equivalences of ''C'' form a [[group (mathematics)|group]] under composition if we consider two auto-equivalences that are naturally isomorphic to be identical. This group captures the essential "symmetries" of ''C''. (One caveat: if ''C'' is not a small category, then the auto-equivalences of ''C'' may form a proper [[class (set theory)|class]] rather than a [[Set (mathematics)|set]].)
-==Implications==
+==References==
-An Endogenous growth theory implication is that policies which embrace openness, competition, change and innovation will promote growth.<ref>{{cite journal|last=Fadare|first=Samuel O.|title=Recent Banking Sector Reforms and Economic Growth in Nigeria|journal=Middle Eastern Finance and Economics|issue=Issue 8 (2010)|url=http://www.eurojournals.com/MEFE_8_12.pdf}}</ref>  Conversely, policies which have the effect of restricting or slowing change by protecting or favouring particular existing industries or firms are likely over time to slow growth to the disadvantage of the community. [[Peter Howitt (economist)|Peter Howitt]] has written:
+*{{Springer|id=e036050|title=Equivalence of categories}}
-<blockquote>
+*{{cite book|last=Mac Lane|first=Saunders|title=Categories for the working mathematician|year=1998|publisher=Springer|location=New York|isbn=0-387-98403-8|pages=xii+314}}
-Sustained economic growth is everywhere and always a process of continual transformation. The sort of economic progress that has been enjoyed by the richest nations since the Industrial Revolution would not have been possible if people had not undergone wrenching changes. Economies that cease to transform themselves are destined to fall off the path of economic growth. The countries that most deserve the title of “developing” are not the poorest countries of the world, but the richest. [They] need to engage in the never-ending process of economic development if they are to enjoy continued prosperity. (Conclusion, "Growth and development: a Schumpeterian perspective", 2006 [http://www.cdhowe.org/pdf/commentary_246.pdf]).
-</blockquote>
-== Criticisms ==<!-- This section is linked from [[Economics]] -->
+{{DEFAULTSORT:Equivalence Of Categories}}
+[[Category:Adjoint functors]]
+[[Category:Category theory]]
-One of the main failings of endogenous growth theories is the collective failure to explain conditional convergence reported in the empirical literature.<ref> See {{cite journal |last=Sachs |first=Jeffrey D. |first2=Andrew M. |last2=Warner |year=1997 |title=Fundamental Sources of Long-Run Growth |journal=[[American Economic Review]] |volume=87 |issue=2 |pages=184–188 |doi= |jstor=2950910 }}</ref> Another frequent critique concerns the cornerstone assumption of diminishing returns to capital. Some contend that ''new growth theory'' has proven no more successful than [[exogenous growth model|exogenous growth theory]] in explaining the income divergence between the [[developing nation|developing]] and [[developed nation|developed]] worlds (despite usually being more complex).<ref>See for instance, Professor Stephen Parente's 2001 review, ''The Failure of Endogenous Growth'' ([https://netfiles.uiuc.edu/parente/The%20Failure%20of%20Endogenous%20Growth.pdf Online] at the [[University of Illinois at Urbana-Champaign]]). (Published in [http://www.metapress.com/(gqdg4dzadovmv5fnfj5jki2x)/home/main.mpx Knowledge Technology & Policy] Volume XIII, Number 4.)</ref>
+[[nl:Equivalentie (categorietheorie)]]
+[[zh-yue:範疇等價性]]
-== See also ==
+[[zh:范畴的等价]]
-* [[Economic growth]]
-* [[Human capital]]
-* [[Paul Romer]]
-* [[Neoclassical growth model|Exogenous growth model]]
-* [[Mahalanobis model]]
-* [[Ramsey–Cass–Koopmans model]]
-==Notes==
-{{reflist}}
-== External links ==
-* [http://www.stanford.edu/~promer/EconomicGrowth.pdf Economic Growth] by [[Paul Romer]].
-* [http://www.eda.gov/ImageCache/EDAPublic/documents/pdfdocs/1g3lr_5f7_5fcortright_2epdf/v1/1g3lr_5f7_5fcortright.pdf New Growth Theory, Technology and Learning: A Practitioner's Guide], [[Economic Development Administration|U.S. Economic Development Administration]].
-* [http://tcdc.undp.org/CoopSouth/1998_2/cop9829.pdf Technological Implications of New Growth Theory for the South], [[United Nations Development Programme]].
-*[http://mitpress.mit.edu/books/chapters/0262025531chap1.pdf The AK Model] by Economic Growth, 2nd Edition Robert J. Barro and Xavier Sala-i-Martin
-*The Origins of Endogenous Growth, Romer.M Paul,The Journal of Economic Perspectives, Vol. 8, No. 1. (Winter, 1994), pp. 3-22.
-[[Category:Economic theories]]
-[[Category:Economic growth]]
-[[ca:Desenvolupament endogen]]
-[[de:Endogene Wachstumstheorie]]
-[[fr:Théorie de la croissance endogène]]
-[[it:Teoria della crescita endogena]]
-[[lo:Endogenous growth model]]
-[[nl:Endogene groeitheorie]]
-[[pl:Endogeniczny model wzrostu gospodarczego]]
-[[fi:Endogeenisen kasvun teoria]]

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