Integral Categories and Calculus Categories
نویسندگان
چکیده
Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing an integral transformation, sA : !A→ !A⊗A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a calculus category. Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives amounts to demanding that a certain natural transformation, K : !A→ !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we provide examples of such categories. 1998 ACM Subject Classification F.3.3 Studies of Program Constructs, F.4.1 Mathematical Logic
منابع مشابه
2010 Category Theory Octoberfest
Steve Awodey (CMU): “Sketch of the homotopy interpretation of intensional type theory” Abstract: As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people. As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people...
متن کاملNel ’ s category theory based differential and integral Calculus , or Did Newton know category theory ?
In a series of publications in the early 1990s, L D Nel set up a study of non-normable topological vector spaces based on methods in category theory. One of the important results showed that the classical operations of derivative and integral in Calculus can in fact be obtained by a rather simple construction in categories. Here we present this result in a concise form. It is important to note ...
متن کاملOn Certain Integral Tensor Categories and Integral Tqfts
We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.
متن کاملOn the classification of certain fusion categories
We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories — and consequently, semisimple Hopf algebras — of dimension pq2, where p and q are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify...
متن کاملSOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS
In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for ...
متن کامل