Products in categories of relations

AbstractThe relational product construction is often consider as an abstract version of cartesian products. The existence of those products is strongly connected with the representability of that category. In this paper we investigate a canonical weakening of the notion of a relational product. Unlike the strong version, any (small) category of relations can be embedded into a suitable category providing all weak relational products. Furthermore, we provide several examples, and we study the categorical properties of the new construction

On selfadjoint functors satisfying polynomial relations

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors, are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products, Serre subcategories, quotients and centralizer subalgebras.Comment: 24 page

Integrating Marginal Cost into Pricing-to-market Models for U.S. Agricultural Products

This article investigates the markup pricing behaviour of U.S. exporters of agricultural products. Agricultural products studied are feed, flour, frozen potatoes, frozen orange juice, five categories of beef, five categories of pork, and two categories of chicken. The popular pricing-to-market (PTM) approach of Krugman (1987) is used to examine market power and imperfect competition for the markets under study. The PTM model can directly investigate whether there is any evidence of market power in international trade.The sensitivity of U.S. export prices to exchange rate fluctuations may indicate price discrimination and imperfect competition in the international markets. The PTM approach is popular because of its relatively simple specification and empirical testing.Demand and Price Analysis, International Relations/Trade,

Stacky Lie groups

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added; coherence relations for groups in 2-categories given (section 4

TransNFCM: Translation-Based Neural Fashion Compatibility Modeling

Identifying mix-and-match relationships between fashion items is an urgent task in a fashion e-commerce recommender system. It will significantly enhance user experience and satisfaction. However, due to the challenges of inferring the rich yet complicated set of compatibility patterns in a large e-commerce corpus of fashion items, this task is still underexplored. Inspired by the recent advances in multi-relational knowledge representation learning and deep neural networks, this paper proposes a novel Translation-based Neural Fashion Compatibility Modeling (TransNFCM) framework, which jointly optimizes fashion item embeddings and category-specific complementary relations in a unified space via an end-to-end learning manner. TransNFCM places items in a unified embedding space where a category-specific relation (category-comp-category) is modeled as a vector translation operating on the embeddings of compatible items from the corresponding categories. By this way, we not only capture the specific notion of compatibility conditioned on a specific pair of complementary categories, but also preserve the global notion of compatibility. We also design a deep fashion item encoder which exploits the complementary characteristic of visual and textual features to represent the fashion products. To the best of our knowledge, this is the first work that uses category-specific complementary relations to model the category-aware compatibility between items in a translation-based embedding space. Extensive experiments demonstrate the effectiveness of TransNFCM over the state-of-the-arts on two real-world datasets.Comment: Accepted in AAAI 2019 conferenc