Aspects of topoi

Freyd, Peter

doi:10.1017/s0004972700044828

Cited by 242 publications

(97 citation statements)

References 2 publications

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“…For an arrow f : B → A in a lccc C we write ∆ f : C/A → C/B for the associated pullback functor, which can be defined since slice categories have cartesian products. The key fact about locally cartesian closed categories is the following proposition [7]. Proposition 1.…”

Section: Locally Cartesian Closed Categoriesmentioning

confidence: 99%

Wellfounded Trees and Dependent Polynomial Functors

Gambino

Hyland

2004

Lecture Notes in Computer Science

104

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Section: Locally Cartesian Closed Categoriesmentioning

confidence: 99%

Wellfounded Trees and Dependent Polynomial Functors

Gambino

Hyland

2004

Lecture Notes in Computer Science

104

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“…(For definitions see the discussion of the work of Lawvere and Tierney in [3]. ) We plan to elaborate the details of this example in a sequel to the present paper.…”

Section: Proofmentioning

confidence: 99%

Localization at Injectives in Complete Categories

Lambek¹,

Rattray²

1973

Proceedings of the American Mathematical Society

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Abstract.We consider a complete category .c/. For each object /of sJ we define a functor Q:sJ-*-.?i and obtain a necessary and sufficient condition on /for Q, after restricting its codomain, to become a reflector of .rJ onto the limit closure of /. In particular, this condition is satisfied if / is injective in sJ with regard to equalizers. Among the special cases of such reflectors are: the reflector onto torsion-free divisible objects associated to an injective / in Mod R; the Samuel compactification of a uniform space; the Stone-Cech compactification.We give a second description of Q in terms of a triple on sets. If/ is injective and the functor Q is equivalent to the identity then, under a few extra conditions on sJ, s¿ov is triplable over sets with regard to the functor taking A to .

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“…This characterisation of N shows [4] that exact functors (in particular inverse images of geometric morphisms) between toposes preserve N.…”

Section: Discussionmentioning

confidence: 80%

An Exact Interpretation of while

Taylor

1993

Theory and Formal Methods 1993

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The behaviour and interaction of finite limits (products, pullbacks and equalisers) and colimits (coproducts and coequalisers) in the category of sets is illustrated in a "hands on" way by giving the interpretation of a simple imperative language in terms of partial functions between sets of states. We show that the interpretation is a least fixed point and satisfies the usual proof rule for loop invariants.This The languageWe shall interpret a language with the following command s in a category:Clearly skip and sequencing (;) correspond to identity and composition, so each command is a morphism, but what are the objects which serve as its source and target?Each command of a program has a "compile-time" as well as its run-time effect: here just the declaration (let) and un-declaration (discard) of variables. Before the command (in the text) there is a certain collection Γ of variables which are in scope, so in particular they may be used in forming expressions. At the end the scope, ∆, consists of Γ, plus the newly declared variable(s), minus those which have been discarded. Γ and ∆ are the source and target respectively of the morphism representing the command; by construction the source of one command is the target of the previous one, as it must be for composition to be defined. 1We do not require scopes to be nested, instead adopting the rule that if a variable name is in scope then it may not be the subject of a definition. This apparently weak convention is convenient because it allows assignment to be defined in terms of the other commands:x := a is the same as let x = a ; discard x ; let x = x ; discard x where x is a new variable, i.e. not in scope. In programming discard is, apart from this, redundant (provably so in a certain sense in our formal language) but it has been added for an exact match with the categorical concept (products) which is being used to interpret the language. Abstractly we have already said enough about the objects: they are lists of variables. (Similarly we have also given the morphisms as programs, but we shall see that we must impose an equivalence relation on them.) Variables are typed: we need the Boolean type 2 (with its elements yes and no), but shall not introduce any (other) type constructors.Concretely, a type is the set of values which may be taken by a variable to which it is attributed. The state of the machine is determined by the tuple of current values of the variables in scope, and so the object [[Γ]] representing the scope Γ is the product of (the sets interpreting) the types. In particular,and we writex :To say what expressions are, we need a language of algebraic operations.The categorical interpretation of algebra is very well known. Each operator f and constant c has an intended meaning as a morphism f : Y 1 × · · · × Y m → Z or c : 1 → Z in the category, Y and Z being the argument and result types.In the scope Γ ≡ [x 1 : X 1 , . . . , x n : X n ], expressions are interpreted assimilarly interpret the declaration and un-declaration commands.The foll...

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Aspects of topoi

Cited by 242 publications

References 2 publications

Wellfounded Trees and Dependent Polynomial Functors

Wellfounded Trees and Dependent Polynomial Functors

Localization at Injectives in Complete Categories

An Exact Interpretation of while

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