In many engineering applications it is useful to reason about "negative information". For example, in planning problems, providing an optimal solution is the same as giving a feasible solution (the "positive" information) together with a proof of the fact that there cannot be feasible solutions better than the one given (the "negative" information). We model negative information by introducing the concept of "norphisms", as opposed to the positive information of morphisms. A "nategory" is a category that has "nom"-sets in addition to hom-sets, and specifies the interaction between norphisms and morphisms. In particular, we have composition rules of the form morphism + norphism $\to$ norphism. Norphisms do not compose by themselves; rather, they use morphisms as catalysts. After providing several applied examples, we connect nategories to enriched category theory. Specifically, we prove that categories enriched in de Paiva's dialectica categories GC, in the case C = Set and equipped with a modified monoidal product, define nategories which satisfy additional regularity properties. This formalizes negative information categorically in a way that makes negative and positive morphisms equal citizens.

翻译：在许多工程应用中，对“负面信息”进行推理十分有用。例如，在规划问题中，提供最优解等同于给出一个可行解（“正面”信息），同时证明不存在比该解更优的可行解（“负面”信息）。我们通过引入“反态射”概念来建模负面信息，以区别于态射所代表的正面信息。“反范畴”是一种除同态集外还拥有“反复合集”的范畴，并规定了反态射与态射之间的相互作用。特别地，我们具有形如“态射 + 反态射 $\to$ 反态射”的复合规则。反态射自身不能复合，而是以态射作为催化剂。在提供若干应用实例后，我们将反范畴与富化范畴理论联系起来。具体而言，我们证明：在de Paiva的对偶范畴GC上（取C=Set并配备改进的幺半积）富化的范畴，可定义满足额外正则性质的反范畴。这以范畴论方式形式化了负面信息，使得负面态射与正面态射处于同等地位。