F.W. Lawvere (1968), Some algebraic problems in the context of functorial semantics of algebraic theories, in Proceedings of the Midwest Category Seminar II, Lecture Notes in Mathematics 61, 41--61, Springer-Verlag.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....H: T T 0 has a left adjoint F H : PAlg T PAlg T 0 . A notion of tensor product for partial algebraic theories is used in [22] to obtain, among other things, a very elegant definition of the theory of monoidal categories (see Example A. 2 in the Appendix) The tensor product (see for instance [16, 28]) is a well known construction for ordinary algebraic (Lawvere) theories. Its importance can be understood by observing that the algebraic structures of a theory T can be defined not only on sets, the standard case denoted by PAlg T (Set) where Set is the category having small sets as objects ....

F.W. Lawvere (1968), Some algebraic problems in the context of functorial semantics of algebraic theories, in Proceedings of the Midwest Category Seminar II, Lecture Notes in Mathematics 61, 41--61, Springer-Verlag.

....which we use the notation A : h Fnan Fnan v u g. In addition, cells can be composed both horizontally ( and vertically ( Delta ) as follows: given A : h Fnan Fnan v u g, B : f Fnan Fnan u w k, and C : g Fnan Fnan z s h 0 , then 1 Tensor product (see for instance [20]) is a well known construction for ordinary algebraic (Lawvere) theories. It can be extended to theories with partial operations (e.g. PMEqtl [26] with essentially the same properties. a Fnan Fnan fflffl v b Delta = a Fnan Fnan fflffl v Deltau b Fnan Fnan fflffl u c c ....

F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In Proc. Midwest Category Seminar II, number 61 in Springer Lecture Notes in Mathematics, pages 41--61, 1968.

....term tile systems (tTS) 2.2 Term Tile Systems In what follows we consider one sorted signature only. The many sorted case can be handled very easily in a similar way, but requires a more complex notation that is not necessary for our case study and therefore avoided. An algebraic theory [27, 28, 24] is just a cartesian category having underlined natural numbers as objects. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma , and is denoted by Th[ Sigma] the arrows from m to n are in a one to one correspondence with n tuples of ....

F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In Proc. 2th Midwest Category Seminar. Springer Lecture Notes in Mathematics 61, 41--61 (1968).

....of this kind, where the operation of arrow and cell composition is defined only if certain conditions are satisfied. A self contained description of a simplified version of , is included. In particular, for this version of the notion of of theories is developed. Tensor product (see for instance [25, 37]) is a well known construction for ordinary algebraic (Lawvere) theories. Its importance can be understood by observing that the algebraic structures of a theory can be defined not only on sets, the standard case ( but also on any category with suitable products or limits, to yield a category ( ....

....a subalgebra inclusion , and each partial ary operation in will be interpreted as a subalgebra inclusion ( together with a homomorphism ( Furthermore, in order for these interpretations to yield a algebra structure in , the axioms must be satisfied. In a way analogous to algebraic theories [25, 17], to lim theories [40] and to sketches [23] there is then a theory in partial membership equational logic such that ( Notice that we could have chosen a bigger poset of subalgebra inclusions, yielding a looser definition of ( A natural choice for would have been the set of weak ....

F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In , pages 41--61. Springer Lecture Notes in Mathematics No. 61, 1968.

....categories Kff from Km to K n that describes the effect of substituting (a type expression whose meaning is) ffi for each type variable i in m. The rest of the story is that the sets n of type variables and the assignments ff can be organized into a base category jKj that is a Lawvere theory [14, 15] of type expressions, and that K itself then becomes a functor from jKj op to the large category CCC of Cartesian closed categories (i.e. a model of the theory jKj in CCC) Thus we have: Definition A pre PL category K = hjKj; Ki consists of a base category jKj such that ffl Ob jKj is the set ....

Lawvere, F. W. Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories. in: Reports of the Midwest Category Seminar II, edited by S. Mac Lane. Lecture Notes in Mathematics, vol. 61, Springer-Verlag, Berlin, 1968, pp. 41--61.

....and the functoriality axiom (of tensor product Omega ) expresses a basic fact about the true concurrency of the model. A second example, showing that the use of categories offer a general and convenient characterization also of configurations, is given by Lawvere theories. An algebraic theory [44, 45, 40] is just a cartesian category having natural numbers as objects. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma, and is denoted by Th[ Sigma] also L Sigma ) the arrows from m to n are in a one to one correspondence with n tuples ....

.... and that Omega Gammat 15696 5 inclusions A B constitute a poset category of canonical inclusions, it is possible to define the category PAlg T (PAlg T 0 ) for any two theories T = Omega ; Gamma) and T 0 = Omega 0 ; Gamma 0 ) Moreover, in a way analogous to algebraic theories [45, 29], to lim theories [31] and to sketches [42] in [58] the construction of a tensor theory T Omega T 0 in partial membership equational logic is given such that PAlg T Omega T 0 PAlg T (PAlg T 0 ) PAlg T 0 (PAlg T ) Notice that we could have chosen a bigger poset of subalgebra ....

F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In Proc. 2th Midwest Category Seminar. Springer Lecture Notes in Mathematics 61. 1968. pp. 41--61

....composition. To give a formal definition of auxiliary structure we can assume that they form two monoidal categories having the same class of objects. In particular, a general and convenient categorical characterization of configurations and effects can be given in terms of algebraic theories [25,26,24]. The free algebraic theory associated to a (one sorted) signature Sigma is called the Lawvere theory for Sigma, and is denoted by L Sigma : the objects are underlined natural numbers, the arrows from m to n are in a oneto one correspondence with n tuples of terms of the free Sigma algebra ....

F.W. Lawvere. Some algebraic problems in the context of functorial semantics of algebraic theories. In Proc. 2th Midwest Category Seminar. Lecture Notes in Mathematics 61, 41--61 (1968).

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