What is Category Theory?

Author: (1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility F.2 Basic Categories This part helps with understanding the concepts underpinning Definition 13. It also develops the theory behind Figure 1 in Section 9 and draws links with other items in this appendix. Category theory uses formal diagrams to study underlying mathematical relations. A Category is a labelled directed graph whose nodes are called Objects and labelled directed edges are called morphisms or Arrows. Categories have two basic properties: arrows can be composed associatively and for every object there is an identity map. Vakil [18 October, 2017] has an excellent chapter expositing category theory, with a view to algebraic geometry. Leinster [2014] is a general introduction. F.2.1 Discussion and Examples Categories represent abstractions of other mathematical concepts. Many familiar concepts and structures can be viewed as categories. It simplifies the process of stating and proving theorems and helps with understanding similarities between seemingly disparate areas of mathematics. Perhaps the most basic example of a category is the category of sets, where the objects are sets and the arrows represents functions between these sets. The strength of the framework is that any system, that can be built from associative mappings, can be analyzed in this way. Arrows are often said to represent Structure-Preserving Transformations between objects. This is most apparent with the family of concrete categories, informally, those that can be viewed as part of the category of sets but with additional restrictions on the arrows. First is Top, whose objects are topological spaces with continuous functions as morphisms. From basic analysis and topology these preserve compactness, connectedness and separability properties. In differential topology the category Smooth has smooth manifolds as objects and maps that preserve this property (smooth maps) as arrows. In Homotopy maps preserve homotopy between pointed topological spaces. The insight is more general. The class Grp consists of groups and homomorphisms that retain the group structure of the object in the target. Categories have their own structure-preserving processes called Functor. A functor associates to every object in one category an object in another and every morphism one in the other. Categories allow for significant abstraction, for instance, it is possible to define the category of Categories and Functors with categories as objects and functors as arrows. Finally, Natural Transformations are mappings between functors that obey certain commutativity conditions. Functors and natural transformations underpin the "diagram chasing" method, exemplified in Figure 1 in the text. F.2.2 Categories and Morphisms A category C comprises the following three mathematical entities • A family ob(C), whose elements are called objects. • A class hom(C), whose elements are known as morphisms, maps or arrows. f : a → b denotes a morphism from source a to target b. • hom(a, b) denotes the hom-class of all morphisms from a to b, for example, all the smooth functions between sets a and b in Smooth. • A binary operation ◦, called composition of morphisms, such that for any three objects a, b and c ◦ : hom(a, b) × hom(b, c) → hom(a, c). The composition of f : a → b with g : b → c rendered g ◦ f or gf is governed by two familiar axioms i Associativity: If f : a → b, g : b → c and h : c → d then h◦(g◦f) = (h◦g)◦f ii Identity: For every object x, there exists a morphism 1x : x → x, called the identity morphism, such that for every morphism f : a → b, it is the case that 1b ◦ f = f ◦ 1a Intuitively, the first condition is essential to define paths around diagrams. It is common to notions of multiplication and addition. The second, is a requirement that no matter how layered the algebraic structure, one can drill down to the primitive level typically points, sets or maps. Relations between morphisms, like fg = h, are often depicted in commutative diagrams, with vertices representing objects and arrows representing morphism. A morphism f : a → b can possess the following properties: • Monomorphism (or monic) if f ◦ g1 = f ◦ g2 implies g1 = g2, for all morphisms g1, g2 : x → a • Epimorphism (or epic) if g1 ◦ f = g2 ◦ f for any g1, g2 : b → x • Bimorphism if f is both monic and epic • An Isomorphism if there is another morphism g : b → a where f ◦ g = 1b and g ◦ f = 1a A category is Abelian if it has the subsequent attributes: • It is pre-additive, so for any three morphisms f, g and h and the group operator + f ◦ (g + h) = f ◦ g + f ◦ h (f + g) ◦ h = f ◦ h + g ◦ h Thus every set Hom(a, b) has an abelian group structure and function composition is distributive over the group operator. • It has a zero object I, such that for any other object Z, the object is initial, so there is exactly one morphism I → Z and final, in the sense that there is only one mapping Z → I. • It has all kernels and cokernels. • All morphisms and epimorphisms are normal.This means the category behaves as though there is a set which is the destination for all mappings. • An object F(x) in D for any object x in C and • A morphism F(f) : F(x) → F(y) for each morphism f : x → y in C with the properties that • all morphisms compose associatively so f : x → y and g : y → z imply F(g ◦ f) = F(g) ◦ F(f) A Contravariant functor F : C → D behaves like a covariant functor, except with all the morphisms turned around. More specifically, every morphism f : x → y in C is assigned to a morphism F(f) : F(y) → F(x) in D. Equivalently, a contravariant functor acts as a covariant functor from the opposite category C op to D, where the opposite category C op is C with all its arrows reversed Formally, a Natural Transformation is a relation between two functors. Functors often describe so called natural constructions and natural transformations represent natural homomorphisms between these constructions. When two different constructions yield the same result an isomorphism arises between the two functors. If F and G are covariant functors between the categories C and D, then a natural transformation η from F to G associates with every object x in C a morphism ηx : F(x) → G(x) in D, such that for every morphism f : x → y, the relationship ηy ◦ F(f) = G(f) ◦ ηx. Thus, I construct a commuting diagram below. The two functors F and G are referred to as naturally isomorphic because there exists a natural transformation from F to G, such that ηx is an isomorphism for every x in C. Next I can define a Concrete Category as a pair (C, U), such that C is a category, and U : C → Set (the category of sets and functions) is a faithful functor- one which is one-to-one when restricted to morphisms with the same target and source. The functor U is to be thought of as a so called forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function". Thus a concrete category can be viewed as the category of sets but with a restriction on the set of possible morphisms, like differentiability or requiring uniform continuity (often denoted Uni). An Exact Functor is one which preserves short exact sequences (those that end in zero after three or fewer steps). Formally, let P and Q be abelian categories, and let F : P → Q be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if, whenever is a short exact sequence in P, then is a short exact sequence in Q. It is often easier to work with the weaker notion of a Left Exact sequence defined below is a short exact sequence in P, then Finally, I will talk briefly later about Derived Functors. This advanced topic is probably best appreciated by studying Jacobson [1980] or Vakil [18 October, 2017]. I will provide a more heuristic presentation here. Consider a covariant left exact functor F : A → B between two abelian categories A and B. If 0 → A → B → C → 0 is a short exact sequence in A then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and the sequence could continue to the right to form a long exact sequence. [118] A right derived functor RiF : A → B defined for each i ≥ 1 continues the sequence as follows: F.2.3 Applications I supply one mathematical and one economic interpretation, focusing on Figure 1 in the text. The mapping idea of the Lucas critique can be viewed categorically, as the requirement for a natural transformation between the structural model and its econometric approximation in the category of sets or concrete categories, such as categories of local approximations. This links back to the Lucas critique. We can think of the structural mapping as breaking down, when the trivial homology fails. In the next part, we will explore a particular cohomology, which is contravariant to homology, which is central to Econometric Duality. This (Principle 1) represents the deepest connection between micro-foundations, econometrics and the Lucas critique. From this standpoint, category theoretic reasoning is a cornerstone of the paper. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility F.2 Basic Categories This part helps with understanding the concepts underpinning Definition 13. It also develops the theory behind Figure 1 in Section 9 and draws links with other items in this appendix. Category theory uses formal diagrams to study underlying mathematical relations. A Category is a labelled directed graph whose nodes are called Objects and labelled directed edges are called morphisms or Arrows . Categories have two basic properties: arrows can be composed associatively and for every object there is an identity map. Vakil [18 October, 2017] has an excellent chapter expositing category theory, with a view to algebraic geometry. Leinster [2014] is a general introduction. Category Objects Arrows F.2.1 Discussion and Examples F.2.1 Discussion and Examples Categories represent abstractions of other mathematical concepts. Many familiar concepts and structures can be viewed as categories. It simplifies the process of stating and proving theorems and helps with understanding similarities between seemingly disparate areas of mathematics. Perhaps the most basic example of a category is the category of sets, where the objects are sets and the arrows represents functions between these sets. The strength of the framework is that any system, that can be built from associative mappings, can be analyzed in this way. Arrows are often said to represent Structure-Preserving Transformation s between objects. Structure-Preserving Transformation This is most apparent with the family of concrete categories, informally, those that can be viewed as part of the category of sets but with additional restrictions on the arrows. First is Top , whose objects are topological spaces with continuous functions as morphisms. From basic analysis and topology these preserve compactness, connectedness and separability properties. In differential topology the category Smooth has smooth manifolds as objects and maps that preserve this property (smooth maps) as arrows. In Homotopy maps preserve homotopy between pointed topological spaces. Top Smooth Homotopy The insight is more general. The class Grp consists of groups and homomorphisms that retain the group structure of the object in the target. Categories have their own structure-preserving processes called Functor. Grp Functor. A functor associates to every object in one category an object in another and every morphism one in the other. Categories allow for significant abstraction, for instance, it is possible to define the category of Categories and Functors with categories as objects and functors as arrows. Finally, Natural Transformations are mappings between functors that obey certain commutativity conditions. Functors and natural transformations underpin the "diagram chasing" method, exemplified in Figure 1 in the text. Categories and Functors Natural Transformations F.2.2 Categories and Morphisms F.2.2 Categories and Morphisms A category C comprises the following three mathematical entities • A family ob(C), whose elements are called objects. • A class hom(C), whose elements are known as morphisms, maps or arrows. f : a → b denotes a morphism from source a to target b. • hom(a, b) denotes the hom-class of all morphisms from a to b, for example, all the smooth functions between sets a and b in Smooth. Smooth. • A binary operation ◦, called composition of morphisms, such that for any three objects a, b and c ◦ : hom(a, b) × hom(b, c) → hom(a, c). The composition of f : a → b with g : b → c rendered g ◦ f or gf is governed by two familiar axioms i Associativity: If f : a → b, g : b → c and h : c → d then h◦(g◦f) = (h◦g)◦f ii Identity: For every object x, there exists a morphism 1x : x → x, called the identity morphism, such that for every morphism f : a → b, it is the case that 1b ◦ f = f ◦ 1a Intuitively, the first condition is essential to define paths around diagrams. It is common to notions of multiplication and addition. The second, is a requirement that no matter how layered the algebraic structure, one can drill down to the primitive level typically points, sets or maps. Relations between morphisms, like fg = h, are often depicted in commutative diagrams, with vertices representing objects and arrows representing morphism. A morphism f : a → b can possess the following properties: • Monomorphism (or monic) if f ◦ g1 = f ◦ g2 implies g1 = g2, for all morphisms g1, g2 : x → a Monomorphism • Epimorphism (or epic) if g1 ◦ f = g2 ◦ f for any g1, g2 : b → x Epimorphism • Bimorphism if f is both monic and epic Bimorphism • An Isomorphism if there is another morphism g : b → a where f ◦ g = 1b and g ◦ f = 1a Isomorphism A category is Abelian if it has the subsequent attributes: Abelian • It is pre-additive, so for any three morphisms f, g and h and the group operator + f ◦ (g + h) = f ◦ g + f ◦ h (f + g) ◦ h = f ◦ h + g ◦ h Thus every set Hom(a, b) has an abelian group structure and function composition is distributive over the group operator. • It has a zero object I, such that for any other object Z, the object is initial, so there is exactly one morphism I → Z and final, in the sense that there is only one mapping Z → I. • It has all kernels and cokernels. • All morphisms and epimorphisms are normal.This means the category behaves as though there is a set which is the destination for all mappings. • An object F(x) in D for any object x in C and • A morphism F(f) : F(x) → F(y) for each morphism f : x → y in C with the properties that • all morphisms compose associatively so f : x → y and g : y → z imply F(g ◦ f) = F(g) ◦ F(f) A Contravariant functor F : C → D behaves like a covariant functor, except with all the morphisms turned around. More specifically, every morphism f : x → y in C is assigned to a morphism F(f) : F(y) → F(x) in D. Equivalently, a contravariant functor acts as a covariant functor from the opposite category C op to D, where the opposite category C op is C with all its arrows reversed Contravariant Formally, a Natural Transformation is a relation between two functors. Functors often describe so called natural constructions and natural transformations represent natural homomorphisms between these constructions. When two different constructions yield the same result an isomorphism arises between the two functors. Natural Transformation If F and G are covariant functors between the categories C and D, then a natural transformation η from F to G associates with every object x in C a morphism ηx : F(x) → G(x) in D, such that for every morphism f : x → y, the relationship ηy ◦ F(f) = G(f) ◦ ηx. Thus, I construct a commuting diagram below. The two functors F and G are referred to as naturally isomorphic because there exists a natural transformation from F to G, such that ηx is an isomorphism for every x in C. Next I can define a Concrete Category as a pair (C, U), such that C is a category, and U : C → Set (the category of sets and functions) is a faithful functor- one which is one-to-one when restricted to morphisms with the same target and source. The functor U is to be thought of as a so called forgetful functor, which assigns to every object of C its "underlying set", and to every morphism in C its "underlying function". Thus a concrete category can be viewed as the category of sets but with a restriction on the set of possible morphisms, like differentiability or requiring uniform continuity (often denoted Uni ). Concrete Category Set Uni An Exact Functor is one which preserves short exact sequences (those that end in zero after three or fewer steps). Formally, let P and Q be abelian categories, and let F : P → Q be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if, whenever Exact Functor is a short exact sequence in P, then is a short exact sequence in Q. It is often easier to work with the weaker notion of a Left Exact sequence defined below Left Exact is a short exact sequence in P, then Finally, I will talk briefly later about Derived Functors. This advanced topic is probably best appreciated by studying Jacobson [1980] or Vakil [18 October, 2017]. I will provide a more heuristic presentation here. Consider a covariant left exact functor F : A → B between two abelian categories A and B. If 0 → A → B → C → 0 is a short exact sequence in A then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) and the sequence could continue to the right to form a long exact sequence. [118] A right derived functor RiF : A → B defined for each i ≥ 1 continues the sequence as follows: F.2.3 Applications F.2.3 Applications I supply one mathematical and one economic interpretation, focusing on Figure 1 in the text. The mapping idea of the Lucas critique can be viewed categorically, as the requirement for a natural transformation between the structural model and its econometric approximation in the category of sets or concrete categories, such as categories of local approximations. This links back to the Lucas critique. We can think of the structural mapping as breaking down, when the trivial homology fails. In the next part, we will explore a particular cohomology, which is contravariant to homology, which is central to Econometric Duality. This (Principle 1) represents the deepest connection between micro-foundations, econometrics and the Lucas critique. From this standpoint, category theoretic reasoning is a cornerstone of the paper. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv