P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1--76.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Pino Rosolini; see also the first proposition in [6] To the author s knowledge, the earliest (and, it seems, the most general) version is Theorem V(2.2.2) of [15] We shall extract from it the particular case we need. First let us recall the notion of unique existentiation (u. e. pullback from [3] (proposition 2.21) Given any f : X Y , there is an object Q f , determined by fx 2 X j f Gamma1 f(x) fxgg fy 2 Y j 9 x f(x) yg, with the following universal property: there is a pullback square ( the u. e. pullback of f ) Q f , X jj y Q f , Y ; such that for any other ....

P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1 -- 76, 467 -- 480.

....approaches to types such as many sorted equational logic, equational type logic [21] unified algebras [28] and heterogeneous unified algebras [30] can be faithfully embedded in it. Similarly, a wide range of partial equational specification formalisms, including essentially algebraic theories [11, 1], hierarchical equational partial logic [32] left exact sketches [2] limit theories [10] conditional) many sorted partial equational logic with existence equations [32] and partial equational logic with strong equations [5] can all be faithfully embedded in partial membership equational ....

....of an essentially algebraic theory. Their notion of essentially algebraic theory is a special case of Reichel s hierarchical equationally partial (hep) theories [32] which can be regarded as a more general formulation of essentially algebraic theories as originally envisioned by Lawvere and Freyd [11]. Both the general hep notion and the more restricted one proposed by Ad amek and Rosick y have straightforward translations into partial membership equational logic. In fact, Reichel [32] makes explicit the embedding HEP , PMSEqtl into partial many sorted equational logic with (conditional) ....

P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc., 7:1--76, 1972.

....the fact that (again through usual topos theoretic arguments) any set theoretic construct can be described with a Horn theory over a suitable signature of markers. Indeed, constructive set theory can be interpreted in a general topos while toposes can be described in a essentially algebraic way ([20]) 4.2 Graph based algebras The simple example of integration we have considered clearly demonstrates that schema integration leads, in general, to database specifications (schemas) involving a collection of equational constraints expressing equality of certain queries against the schema. ....

P. Freyd. Aspects of topoi. Bull.Austral.Math.Soc., 7:1--72, 1972.

....provable but the stronger is not. In fact, the stronger version implies the law of the excluded middle. Theorem 3. If X is finite then there is no one to one function from X 1 into X. Proof. Suppose X were a finite set with a one to one function f : X 1 X. By (the proof of) Theorem 5. 44 of [9], there is a natural numbers object N that is a subobject of X (namely the intersection of all Y X such that f(Y 1) Y ) This is clearly absurd, but for the sake of completeness we sketch an intuitionistic proof that it is absurd. Since X is finite, it is a surjective image of [n] fa 2 N j ....

P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1--76, 467--480.

....that Reichel s theory morphisms defined in [Rei87] are slightly more general than theory morphisms in HEP ( e = e = But Reichel only defines a specification frame. If signatures and sentences have to be separated, a slight restriction has to be made. Freyd s essentially algebraic theories [Fre72] are essentially the same (Freyd does not define signatures, reducts and the like) We define two restrictions of HEP ( e = e = First, let HEP 1( e = w = be the restriction to those signatures where the hierarchy of operations has only height one, that is, the domain of each partial ....

....[Bur82, Bur86] 3. Partial Existentially Conditioned Existence Equational Logic P (D e = Bur82, Jar88, Jar93] 4. Partial Logic With Strong Equations P ( s = Bur82, Hoe81, Kle52, Slo68] 5. Hierarchical Equationally Partial Theories HEP ( e = e = HEP 1( e = w = HEP 1( w = [Fre72, Rei87] 6. Limit Theories R(R = 9 R = Cos79] 7. Left Exact Sketches LESKETCH [BW85, Gra87] 8. Coherent Order Sorted Algebras With Sort Constraints COS( MG93] These form the strongest level of a hierarchy of expressiveness of institutions consisting altogether of five levels. The hierarchy is ....

P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc. 7, 1--76, 1972.

.... liberal [16, 37] logical frameworks admitting initial and free constructions which are important for specification languages with module concepts [11] Now there are various restrictions of this logical framework, namely Reichel s Heptheories [33] called essentially algebraic theories by Freyd [14], theories formed with ECE equations [4] Jarzembski s weak varieties of partial algebras [22] and theories formed with strong equations [4, 20, 23, 31] Further, there are other logical frameworks capturing partiality as well. We only mention left exact sketches [3, 19] Coste s limit theories ....

....Note that Reichel s theory morphisms defined in [33] are slightly more general than theory morphisms in HEP ( e = e = But Reichel only defines a specification frame. If signatures and sentences have to be separated, a slight restriction has to be made. Freyd s essentially algebraic theories [14] are essentially the same (Freyd does not define signatures, reducts and the like) We define two restrictions of HEP ( e = e = First, let HEP 1( e = w = be restricted to those signatures where the hierarchy of operations has only height one, that is, the domain of each partial operation is ....

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P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc. 7, 1--76, 1972.

....of new types of logic to establish such a theory category correspondence. A prime example of this is the notion of category with finite limits (variously termed a cartesian, or a lex category) for which a number of different logics have been devised: the essentially algebraic theories of Freyd [ 1972 ] the lim theories of Coste [ 1979, 4 Andrew M. Pitts Section 2 ] and the generalized algebraic theories of Cartmell (see Section 6) can all be used for this purpose, although each has its drawbacks. Categories arising from theories via term model constructions can usually be characterized ....

P. J. Freyd. Aspects of topoi. Bull. Austral. Math. Soc., 7:1--76 and 467--80, 1972.

....specification of partial algebras. A very detailed presentation of the development of a specification logic for partial algebras as sketched here can be found in [CGW95] The standard reference for partial algebras is the book [Rei87] which is based on similar ideas, and the earlier presentations [KR72, Fre72, Fre73]. The first step in the development of a specification logic for partial algebras is straight forward. Signatures and models are defined as for total algebras, except that an operation symbol op : s 1 : s n s is interpreted by a partial function A op : A s1 Theta Delta Delta Delta ....

P. Freyd. Aspects of topoi, corrections. Bull. Austr. Math. Soc., (8):467--480, 1973.

....specification of partial algebras. A very detailed presentation of the development of a specification logic for partial algebras as sketched here can be found in [CGW95] The standard reference for partial algebras is the book [Rei87] which is based on similar ideas, and the earlier presentations [KR72, Fre72, Fre73]. The first step in the development of a specification logic for partial algebras is straight forward. Signatures and models are defined as for total algebras, except that an operation symbol op : s 1 : s n s is interpreted by a partial function A op : A s1 Theta Delta Delta Delta ....

P. Freyd. Aspects of topoi. Bull. Austr. Math. Soc., (7):1--72, 1972.

....[9] which makes use of cubical groupoids as an intermediate stage. As noted there, this works since Gpd is an equationally defined category of many sorted algebras in which the domains of the operations are defined by finite limit diagrams. General theorems on such algebraic theories (see [16, 17, 27, 3]) imply that Gpd is complete and cocomplete and that it is monadic over the category Cub of cubical sets , and because presentations can be used. Although it is the essence, this is not the whole story. Using methods of Day [10, 11] I show that the monoidal biclosed structure on cubical sets ....

P. J. Freyd. Aspects of topoi. Bull. Austral. Math. Soc., 7:1--76, 1972.

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P. Freyd, Aspects of topoi, Bull. Austral. Math. Soc. 7 (1972), 1--76.

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Peter Freyd. Aspects of topoi, corrections. Bull. Austr. Math. Soc., (8):467--480, 1973.

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Peter Freyd. Aspects of topoi. Bull. Austr. Math. Soc., (7):1--72, 1972.

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