## M4M 2007 Continuous Functions on Final Coalgebras

### BibTeX

@MISC{Ghani_m4m2007,

author = {Neil Ghani and Peter Hancock and Dirk Pattinson},

title = {M4M 2007 Continuous Functions on Final Coalgebras},

year = {}

}

### OpenURL

### Abstract

It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, A-branching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for power-series functors on the category of sets and functions. While our main technical contribution is the characterisation of all continuous functions, defined on a final coalgebra and taking values in a discrete space by means of inductive types, a methodological point is that these inductive types are most conveniently formulated in a framework of dependent type theory.

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(Show Context)
Citation Context ...signment (an)n∈ω ↦→ does not define a contin1 otherwise uous function. To define the function requires decidability of a Π0 1 statement, which corresponds to Bishop’s limited principle of omniscience =-=[4]-=-. This is of course constructively invalid. Computationally speaking, we are interested in how much information about the argument a a function f might need to access in order to compute the value f(a... |

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(Show Context)
Citation Context ...n 1. The connecting morphism of the sequence 1 ← F1 ← F 2 1 . . . are denoted by pnm : F m 1 → F n 1. Where need arises, we will consider B to be equipped with the discrete topology. It is well known =-=[2]-=- that the representation of the final coalgebra as projective limit gives rise to an ultrametric, that we now introduce. Definition 3.1 (Baire Topology) Let’s define an ultrametric on νF by stipulatin... |

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14 |
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(Show Context)
Citation Context ...ive terms of the streams. Choice sequences are paradigm examples of ‘infinite objects’, known to us only through finite information observed of them, as it were externally. For a thorough account see =-=[5,8,9]-=-. According to Brouwer, all functions on choice sequences are continuous. Continuity of a function means that if you want it to provide you some finite amount of information about the value of the fun... |

13 |
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(Show Context)
Citation Context ...ive terms of the streams. Choice sequences are paradigm examples of ‘infinite objects’, known to us only through finite information observed of them, as it were externally. For a thorough account see =-=[5,8,9]-=-. According to Brouwer, all functions on choice sequences are continuous. Continuity of a function means that if you want it to provide you some finite amount of information about the value of the fun... |

13 |
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(Show Context)
Citation Context ...ive terms of the streams. Choice sequences are paradigm examples of ‘infinite objects’, known to us only through finite information observed of them, as it were externally. For a thorough account see =-=[5,8,9]-=-. According to Brouwer, all functions on choice sequences are continuous. Continuity of a function means that if you want it to provide you some finite amount of information about the value of the fun... |

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(Show Context)
Citation Context ...orts of some kind in this way, and so bring some powerful techniques to bear on these tricky situations. Related work On a superficial level, our results appear to be dual to the insights reported in =-=[1]-=-, where it is shown that function types with certain domain types can be represented as final coalgebras, whereas we show that continuous functions are represented by initial algebras, at least when t... |

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(Show Context)
Citation Context ...ras, at least when the codomain is discrete. Both ideas have at their core the adjunction (A →) ⊣ (A×) between exponentiation and multiplication. There is some similarity between the work reported in =-=[3]-=- and our results. This work concerns methods for extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and non-constructive choice principles. One me... |

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