DOCSLIB.ORG

On Coalgebras Over Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Coalgebras Over Algebras On coalgebras over algebras Adriana Balan1 Alexander Kurz2 1University Politehnica of Bucharest, Romania 2University of Leicester, UK 10th International Workshop on Coalgebraic Methods in Computer Science A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 1 / 31 Outline 1 Motivation 2 The final coalgebra of a continuous functor 3 Final coalgebra and lifting 4 Commuting pair of endofunctors and their fixed points A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 2 / 31 Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 Motivation Starting data: category C, endofunctor H : C −! C Among fixed points: final coalgebra, initial algebra Categories enriched over complete metric spaces: unique fixed point [Adamek, Reiterman 1994] Categories enriched over cpo: final coalgebra L coincides with initial algebra I [Plotkin, Smyth 1983] Category with no extra structure Set: final coalgebra L is completion of initial algebra I [Barr 1993] Deficit: if H0 = 0, important cases not covered (as A × (−)n, D, Pκ+) Locally finitely presentable categories: Hom(B; L) completion of Hom(B; I ) for all finitely presentable objects B [Adamek 2003] A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 3 / 31 In this talk Category: Alg(M) for a Set-monad M Alg(M)-functor: obtained from lifting A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 4 / 31 Outline 1 Motivation 2 The final coalgebra of a continuous functor 3 Final coalgebra and lifting 4 Commuting pair of endofunctors and their fixed points A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 5 / 31 The limit of the terminal sequence is the final H-coalgebra by cocontinuity ξ = τ −1 : L ' HL Construction of the final coalgebra Assumption 1: functor H : Set −! Set !op-continuous Terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 The limit of the terminal sequence is the final H-coalgebra by cocontinuity ξ = τ −1 : L ' HL Construction of the final coalgebra Assumption 1: functor H : Set −! Set !op-continuous Terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: d O pn L A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 Construction of the final coalgebra Assumption 1: functor H : Set −! Set !op-continuous Terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: ^d O Z pn L Hpn−1 O τ HL The limit of the terminal sequence is the final H-coalgebra by cocontinuity ξ = τ −1 : L ' HL A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 6 / 31 Topology: Discrete topology on Hn1. Initial topology on L, HL and C =) L complete ultrametric space. All maps are continuous. Final coalgebras and anamorphisms ξ For each coalgebra C −!C HC there is a cone over the terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: a [ R Hα0 HC f αn α0 C A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Topology: Discrete topology on Hn1. Initial topology on L, HL and C =) L complete ultrametric space. All maps are continuous. Final coalgebras and anamorphisms ξ For each coalgebra C −!C HC there is a cone over the terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: a [ R Z Hα0 pn HC L f αn O αC α0 C A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Final coalgebras and anamorphisms ξ For each coalgebra C −!C HC there is a cone over the terminal sequence t Hnt 1 o H1 o ::: o Hn1 o ::: a [ R Z Hα0 pn HC L f αn O αC α0 C Topology: Discrete topology on Hn1. Initial topology on L, HL and C =) L complete ultrametric space. All maps are continuous. A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 7 / 31 Outline 1 Motivation 2 The final coalgebra of a continuous functor 3 Final coalgebra and lifting 4 Commuting pair of endofunctors and their fixed points A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 8 / 31 () Distributive law λ : MH −! HM Mλ λM M2H / MHM / HM2 mH Hm λ MH / HM uH H / MH C CC CC λ Hu CC C! HM Lifting of H to Alg(M) H~ Alg(M) / Alg(M) UM UM H Set / Set Lifting functors to algebras over a monad Monad M = (M; m : M2 −! M; u : Id −! M) Adjunction F M a UM : Alg(M) −! Set Initial object M20 −! M0, terminal object M1 −! 1 A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 9 / 31 Lifting functors to algebras over a monad Monad M = (M; m : M2 −! M; u : Id −! M) Adjunction F M a UM : Alg(M) −! Set Initial object M20 −! M0, terminal object M1 −! 1 Lifting of H to Alg(M) () Distributive law λ : MH −! HM λ H~ 2 Mλ M 2 Alg(M) / Alg(M) M H / MHM / HM mH Hm UM UM λ H Set / Set MH / HM uH H / MH C CC CC λ Hu CC C! HM A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 9 / 31 The final coalgebra and the lifting Assumption 2: there is a lifting He of H to Alg(M) ξ γ Then (L; L−!HL) inherits an algebra structure map ML−!L making it the final He-coalgebra. Lemma Mp The cone ML −!n MHn1 −!an Hn1 is induced by the H-coalgebra structure of ML Mp x n M1 o MH1 o ::: o MHn1 o ::: ML Mt MHnt a0 a1 an γ t Hnt 1 o H1 o ::: o Hn1 o ::: L f pn Hence the unique coalgebra map γ : ML −! L is also the anamorphism αML : ML −! L for the coalgebra ML. A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 10 / 31 Topology Discrete topology on both sequences Initial topologies on ML and L Proposition The final H-coalgebra inherits a structure of a topological M-algebra. The final coalgebra and the lifting Diagram in Alg(M) with limiting lower sequence Mp x n M1 o MH1 o ::: o MHn1 o ::: ML Mt MHnt a0 a1 an γ t Hnt 1 o H1 o ::: o Hn1 o ::: L f pn A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 Proposition The final H-coalgebra inherits a structure of a topological M-algebra. The final coalgebra and the lifting Diagram in Alg(M) with limiting lower sequence Mp x n M1 o MH1 o ::: o MHn1 o ::: ML Mt MHnt a0 a1 an γ t Hnt 1 o H1 o ::: o Hn1 o ::: L f pn Topology Discrete topology on both sequences Initial topologies on ML and L A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 The final coalgebra and the lifting Diagram in Alg(M) with limiting lower sequence Mp x n M1 o MH1 o ::: o MHn1 o ::: ML Mt MHnt a0 a1 an γ t Hnt 1 o H1 o ::: o Hn1 o ::: L f pn Topology Discrete topology on both sequences Initial topologies on ML and L Proposition The final H-coalgebra inherits a structure of a topological M-algebra. A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 11 / 31 Assumption 3: M0=1 [Adamek 2003] He has also (non empty) initial algebra I built upon this sequence in Alg(M), with unique M-algebra monomorphism f : I −! L 6 I mmm in mm mmm mmm mmm ::: n m ::: 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L Fixed points of lifted functor Initial-terminal He-sequences: M0 / HM0 / ::: / HnM0 / ::: s Hs Hns ::: n ::: 1 o t H1 o o H 1 o A. Balan (UPB), A. Kurz (UL) On coalgebras over algebras CMCS 2010 12 / 31 [Adamek 2003] He has also (non empty) initial algebra I built upon this sequence in Alg(M), with unique M-algebra monomorphism f : I −! L 6 I mmm in mm mmm mmm mmm ::: n m ::: 1 o / H1 o / o / H 1 o / f t Ht Hnt hQQ QQQ pn QQQ QQQ QQQ Q L Fixed points of lifted functor Initial-terminal He-sequences: M0 / HM0 / ::: / HnM0 / ::: s Hs Hns ::: n ::: 1 o t H1 o o H 1 o Assumption 3: M0=1 A.
Load more

Recommended publications
  • Ambiguity and Incomplete Information in Categorical Models of Language
    Ambiguity and Incomplete Information in Categorical Models of Language Dan Marsden University of Oxford [email protected] We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied in [27, 26, 16] using Selinger’s CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn’t seem to provide a satis- factory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic information. In this work we adopt a different approach. We analyze different forms of ambiguous and in- complete information, both with and without quantitative probabilistic data. Each scheme then cor- responds to a suitable enrichment of the category in which we model language. We view different monads as encapsulating the informational behaviour of interest, by analogy with their use in mod- elling side effects in computation. Previous results of Jacobs then allow us to systematically construct suitable bases for enrichment. We show that we can freely enrich arbitrary dagger compact closed categories in order to capture all the phenomena of interest, whilst retaining the important dagger compact closed structure. This allows us to construct a model with real convex combination of binary relations that makes non-trivial use of the scalars. Finally we relate our various different enrichments, showing that finite subconvex algebra enrichment covers all the effects under consideration.
    [Show full text]
  • Lecture 10. Functors and Monads Functional Programming
    Lecture 10. Functors and monads Functional Programming [Faculty of Science Information and Computing Sciences] 0 Goals I Understand the concept of higher-kinded abstraction I Introduce two common patterns: functors and monads I Simplify code with monads Chapter 12 from Hutton’s book, except 12.2 [Faculty of Science Information and Computing Sciences] 1 Functors [Faculty of Science Information and Computing Sciences] 2 Map over lists map f xs applies f over all the elements of the list xs map :: (a -> b) -> [a] -> [b] map _ [] = [] map f (x:xs) = f x : map f xs > map (+1)[1,2,3] [2,3,4] > map even [1,2,3] [False,True,False] [Faculty of Science Information and Computing Sciences] 3 mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b [Faculty of Science Information and Computing Sciences] 4 Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) [Faculty of Science Information and Computing Sciences] 4 Map over binary trees mapTree also applies a function over all elements, but now contained in a binary tree > t = Node (Node Leaf 1 Leaf) 2 Leaf > mapTree (+1) t Node
    [Show full text]
  • 1. Language of Operads 2. Operads As Monads
    OPERADS, APPROXIMATION, AND RECOGNITION MAXIMILIEN PEROUX´ All missing details can be found in [May72]. 1. Language of operads Let S = (S; ⊗; I) be a (closed) symmetric monoidal category. Definition 1.1. An operad O in S is a collection of object fO(j)gj≥0 in S endowed with : • a right-action of the symmetric group Σj on O(j) for each j, such that O(0) = I; • a unit map I ! O(1) in S; • composition operations that are morphisms in S : γ : O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) −! O(j1 + ··· + jk); defined for each k ≥ 0, j1; : : : ; jk ≥ 0, satisfying natural equivariance, unit and associativity relations. 0 0 A morphism of operads : O ! O is a sequence j : O(j) ! O (j) of Σj-equivariant morphisms in S compatible with the unit map and γ. Example 1.2. ⊗j Let X be an object in S. The endomorphism operad EndX is defined to be EndX (j) = HomS(X ;X), ⊗j with unit idX , and the Σj-right action is induced by permuting on X . Example 1.3. Define Assoc(j) = ` , the associative operad, where the maps γ are defined by equivariance. Let σ2Σj I Com(j) = I, the commutative operad, where γ are the canonical isomorphisms. Definition 1.4. Let O be an operad in S. An O-algebra (X; θ) in S is an object X together with a morphism of operads ⊗j θ : O ! EndX . Using adjoints, this is equivalent to a sequence of maps θj : O(j) ⊗ X ! X such that they are associative, unital and equivariant.
    [Show full text]
  • Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl
    Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl To cite this version: Wolfram Kahl. Categories of Coalgebras with Monadic Homomorphisms. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. pp.151-167, 10.1007/978-3-662-44124-4_9. hal-01408758 HAL Id: hal-01408758 https://hal.inria.fr/hal-01408758 Submitted on 5 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl McMaster University, Hamilton, Ontario, Canada, [email protected] Abstract. Abstract graph transformation approaches traditionally con- sider graph structures as algebras over signatures where all function sym- bols are unary. Attributed graphs, with attributes taken from (term) algebras over ar- bitrary signatures do not fit directly into this kind of transformation ap- proach, since algebras containing function symbols taking two or more arguments do not allow component-wise construction of pushouts. We show how shifting from the algebraic view to a coalgebraic view of graph structures opens up additional flexibility, and enables treat- ing term algebras over arbitrary signatures in essentially the same way as unstructured label sets.
    [Show full text]
  • Algebraic Structures Lecture 18 Thursday, April 4, 2019 1 Type
    Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages Algebraic structures Lecture 18 Thursday, April 4, 2019 In abstract algebra, algebraic structures are defined by a set of elements and operations on those ele- ments that satisfy certain laws. Some of these algebraic structures have interesting and useful computa- tional interpretations. In this lecture we will consider several algebraic structures (monoids, functors, and monads), and consider the computational patterns that these algebraic structures capture. We will look at Haskell, a functional programming language named after Haskell Curry, which provides support for defin- ing and using such algebraic structures. Indeed, monads are central to practical programming in Haskell. First, however, we consider type constructors, and see two new type constructors. 1 Type constructors A type constructor allows us to create new types from existing types. We have already seen several different type constructors, including product types, sum types, reference types, and parametric types. The product type constructor × takes existing types τ1 and τ2 and constructs the product type τ1 × τ2 from them. Similarly, the sum type constructor + takes existing types τ1 and τ2 and constructs the product type τ1 + τ2 from them. We will briefly introduce list types and option types as more examples of type constructors. 1.1 Lists A list type τ list is the type of lists with elements of type τ. We write [] for the empty list, and v1 :: v2 for the list that contains value v1 as the first element, and v2 is the rest of the list. We also provide a way to check whether a list is empty (isempty? e) and to get the head and the tail of a list (head e and tail e).
    [Show full text]
  • Initial Algebra Semantics in Matching Logic
    Technical Report: Initial Algebra Semantics in Matching Logic Xiaohong Chen1, Dorel Lucanu2, and Grigore Ro¸su1 1University of Illinois at Urbana-Champaign, Champaign, USA 2Alexandru Ioan Cuza University, Ia¸si,Romania xc3@illinois, [email protected], [email protected] July 24, 2020 Abstract Matching logic is a unifying foundational logic for defining formal programming language semantics, which adopts a minimalist design with few primitive constructs that are enough to express all properties within a variety of logical systems, including FOL, separation logic, (dependent) type systems, modal µ-logic, and more. In this paper, we consider initial algebra semantics and show how to capture it by matching logic specifications. Formally, given an algebraic specification E that defines a set of sorts (of data) and a set of operations whose behaviors are defined by a set of equational axioms, we define a corresponding matching logic specification, denoted INITIALALGEBRA(E), whose models are exactly the initial algebras of E. Thus, we reduce initial E-algebra semantics to the matching logic specifications INITIALALGEBRA(E), and reduce extrinsic initial E-algebra reasoning, which includes inductive reasoning, to generic, intrinsic matching logic reasoning. 1 Introduction Initial algebra semantics is a main approach to formal programming language semantics based on algebraic specifications and their initial models. It originated in the 1970s, when algebraic techniques started to be applied to specify basic data types such as lists, trees, stacks, etc. The original paper on initial algebra semantics [Goguen et al., 1977] reviewed various existing algebraic specification techniques and showed that they were all initial models, making the concept of initiality explicit for the first time in formal language semantics.
    [Show full text]
  • The Expectation Monad in Quantum Foundations
    The Expectation Monad in Quantum Foundations Bart Jacobs and Jorik Mandemaker Institute for Computing and Information Sciences (iCIS), Radboud University Nijmegen, The Netherlands. fB.Jacobs,[email protected] The expectation monad is introduced abstractly via two composable adjunctions, but concretely cap- tures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and the duality between them. Moreover, the approach leads to a new re-formulation of Gleason’s theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval. 1 Introduction Techniques that have been developed over the last decades for the semantics of programming languages and programming logics gain wider significance. In this way a new interdisciplinary area has emerged where researchers from mathematics, (theoretical) physics and (theoretical) computer science collabo- rate, notably on quantum computation and quantum foundations. The article [6] uses the phrase “Rosetta Stone” for the language and concepts of category theory that form an integral part of this common area. The present article is also part of this new field. It uses results from programming semantics, topology and (convex) analysis, category theory (esp. monads), logic and probability, and quantum foundations.
    [Show full text]
  • Beck's Theorem Characterizing Algebras
    BECK'S THEOREM CHARACTERIZING ALGEBRAS SOFI GJING JOVANOVSKA Abstract. In this paper, I will construct a proof of Beck's Theorem char- acterizing T -algebras. Suppose we have an adjoint pair of functors F and G between categories C and D. It determines a monad T on C. We can associate a T -algebra to the monad, and Beck's Theorem demonstrates when the catego- ry of T -algebras is equivalent to the category D. We will arrive at this result by rst de ning categories, and a few relevant concepts and theorems that will be useful for proving our result; these will include natural tranformations, adjoints, monads and more. Contents 1. Introduction 1 2. Categories 2 3. Adjoints 4 3.1. Isomorphisms and Natural Transformations 4 3.2. Adjoints 5 3.3. Triangle Identities 5 4. Monads and Algebras 7 4.1. Monads and Adjoints 7 4.2. Algebras for a monad 8 4.3. The Comparison with Algebras 9 4.4. Coequalizers 11 4.5. Beck's Threorem 12 Acknowledgments 15 References 15 1. Introduction Beck's Theorem characterizing algebras is one direction of Beck's Monadicity Theorem. Beck's Monadicity Theorem is most useful in studying adjoint pairs of functors. Adjunction is a type of relation between two functors that has some very important properties, such as preservation of limits or colimits, which, unfortunate- ly, we will not touch upon in this paper. However, we need to know that it is indeed a topic of interest, and therefore worth studying. Here, we state Beck's Monadicity Theorem.
    [Show full text]
  • Monads and Side Effects in Haskell
    Monads and Side Effects in Haskell Alan Davidson October 2, 2013 In this document, I will attempt to explain in simple terms what monads are, how they work, and how they are used to perform side effects in the Haskell programming language. I assume you already have some familiarity with Haskell syntax. I will start from relatively simple, familiar ideas, and work my way up to our actual goals. In particular, I will start by discussing functors, then applicative functors, then monads, and finally how to have side effects in Haskell. I will finish up by discussing the functor and monad laws, and resources to learn more details. 1 Functors A functor is basically a data structure that can be mapped over: class Functor f where fmap :: (a -> b) -> f a -> f b This is to say, fmap takes a function that takes an a and returns a b, and a data structure full of a’s, and it returns a data structure with the same format but with every piece of data replaced with the result of running it through that function. Example 1. Lists are functors. instance Functor [] where fmap _ [] = [] fmap f (x : xs) = (f x) : (fmap f xs) We could also have simply made fmap equal to map, but that would hide the implemen- tation, and this example is supposed to illustrate how it all works. Example 2. Binary trees are functors. data BinaryTree t = Empty | Node t (BinaryTree t) (BinaryTree t) instance Functor BinaryTree where fmap _ Empty = Empty fmap f (Node x left right) = Node (f x) (fmap f left) (fmap f right) 1 Note that these are not necessarily binary search trees; the elements in a tree returned from fmap are not guaranteed to be in sorted order (indeed, they’re not even guaranteed to hold sortable data).
    [Show full text]
  • 1 Category Theory
    {-# LANGUAGE RankNTypes #-} module AbstractNonsense where import Control.Monad 1 Category theory Definition 1. A category consists of • a collection of objects • and a collection of morphisms between those objects. We write f : A → B for the morphism f connecting the object A to B. • Morphisms are closed under composition, i.e. for morphisms f : A → B and g : B → C there exists the composed morphism h = f ◦ g : A → C1. Furthermore we require that • Composition is associative, i.e. f ◦ (g ◦ h) = (f ◦ g) ◦ h • and for each object A there exists an identity morphism idA such that for all morphisms f : A → B: f ◦ idB = idA ◦ f = f Many mathematical structures form catgeories and thus the theorems and con- structions of category theory apply to them. As an example consider the cate- gories • Set whose objects are sets and morphisms are functions between those sets. • Grp whose objects are groups and morphisms are group homomorphisms, i.e. structure preserving functions, between those groups. 1.1 Functors Category theory is mainly interested in relationships between different kinds of mathematical structures. Therefore the fundamental notion of a functor is introduced: 1The order of the composition is to be read from left to right in contrast to standard mathematical notation. 1 Definition 2. A functor F : C → D is a transformation between categories C and D. It is defined by its action on objects F (A) and morphisms F (f) and has to preserve the categorical structure, i.e. for any morphism f : A → B: F (f(A)) = F (f)(F (A)) which can also be stated graphically as the commutativity of the following dia- gram: F (f) F (A) F (B) f A B Alternatively we can state that functors preserve the categorical structure by the requirement to respect the composition of morphisms: F (idC) = idD F (f) ◦ F (g) = F (f ◦ g) 1.2 Natural transformations Taking the construction a step further we can ask for transformations between functors.
    [Show full text]
  • MONADS and ALGEBRAS I I
    \chap10" 2009/6/1 i i page 223 i i 10 MONADSANDALGEBRAS In the foregoing chapter, the adjoint functor theorem was seen to imply that the category of algebras for an equational theory T always has a \free T -algebra" functor, left adjoint to the forgetful functor into Sets. This adjunction describes the notion of a T -algebra in a way that is independent of the specific syntactic description given by the theory T , the operations and equations of which are rather like a particular presentation of that notion. In a certain sense that we are about to make precise, it turns out that every adjunction describes, in a \syntax invariant" way, a notion of an \algebra" for an abstract \equational theory." Toward this end, we begin with yet a third characterization of adjunctions. This one has the virtue of being entirely equational. 10.1 The triangle identities Suppose we are given an adjunction, - F : C D : U: with unit and counit, η : 1C ! UF : FU ! 1D: We can take any f : FC ! D to φ(f) = U(f) ◦ ηC : C ! UD; and for any g : C ! UD we have −1 φ (g) = D ◦ F (g): FC ! D: This we know gives the isomorphism ∼ HomD(F C; D) =φ HomC(C; UD): Now put 1UD : UD ! UD in place of g : C ! UD in the foregoing. We −1 know that φ (1UD) = D, and so 1UD = φ(D) = U(D) ◦ ηUD: i i i i \chap10" 2009/6/1 i i page 224 224 MONADS AND ALGEBRAS i i And similarly, φ(1FC ) = ηC , so −1 1FC = φ (ηC ) = FC ◦ F (ηC ): Thus we have shown that the two diagrams below commute.
    [Show full text]
  • A Note on the Under-Appreciated For-Loop
    A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Techn. Report TR-HASLab:01:2020 Oct 2020 HASLab - High-Assurance Software Laboratory Universidade do Minho Campus de Gualtar – Braga – Portugal http://haslab.di.uminho.pt TR-HASLab:01:2020 A Note on the Under-Appreciated For-Loop by Jose´ N. Oliveira Abstract This short research report records some thoughts concerning a simple algebraic theory for for-loops arising from my teaching of the Algebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion- algebraic combinator grew recently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the powerful adjoint-catamorphism theorem of the Algebra of Programming. A Note on the Under-Appreciated For-Loop Jos´eN. Oliveira Oct 2020 Abstract This short research report records some thoughts concerning a sim- ple algebraic theory for for-loops arising from my teaching of the Al- gebra of Programming to 2nd year courses at the University of Minho. Interest in this so neglected recursion-algebraic combinator grew re- cently after reading Olivier Danvy’s paper on folding over the natural numbers. The report casts Danvy’s results as special cases of the pow- erful adjoint-catamorphism theorem of the Algebra of Programming. 1 Context I have been teaching Algebra of Programming to 2nd year courses at Minho Uni- versity since academic year 1998/99, starting just a few days after AFP’98 took place in Braga, where my department is located.
    [Show full text]