Diagonal functorIn , a branch of mathematics, the diagonal functor is given by , which maps as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the : a product is a universal arrow from to . The arrow comprises the projection maps. More generally, given a , one may construct the , the objects of which are called . For each object in , there is a constant diagram that maps every object in to and every morphism in to .
Precast concretePrecast concrete is a construction product produced by casting concrete in a reusable mold or "form" which is then cured in a controlled environment, transported to the construction site and maneuvered into place; examples include precast beams, and wall panels for tilt up construction. In contrast, cast-in-place concrete is poured into site-specific forms and cured on site. Recently lightweight expanded polystyrene foam is being used as the cores of precast wall panels, saving weight and increasing thermal insulation.
Category of small categoriesIn mathematics, specifically in , the category of small categories, denoted by Cat, is the whose objects are all and whose morphisms are functors between categories. Cat may actually be regarded as a with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a , and therefore not an object of itself.
Free objectIn mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations).
CoproductIn , the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic to the , which means the definition is the same as the product but with all arrows reversed.
Monoidal categoryIn mathematics, a monoidal category (or tensor category) is a equipped with a bifunctor that is associative up to a natural isomorphism, and an I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant s commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
Natural lawNatural law (ius naturale, lex naturalis) is a system of law based on a close observation of human nature, and based on values intrinsic to human nature that can be deduced and applied independently of positive law (the express enacted laws of a state or society). According to the theory of law called jusnaturalism, all people have inherent rights, conferred not by act of legislation but by "God, nature, or reason." Natural law theory can also refer to "theories of ethics, theories of politics, theories of civil law, and theories of religious morality.
Types of concreteConcrete is produced in a variety of compositions, finishes and performance characteristics to meet a wide range of needs. Modern concrete mix designs can be complex. The choice of a concrete mix depends on the need of the project both in terms of strength and appearance and in relation to local legislation and building codes. The design begins by determining the requirements of the concrete. These requirements take into consideration the weather conditions that the concrete will be exposed to in service, and the required design strength.
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Rough setIn computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I.
Equivalence of categoriesIn , a branch of abstract mathematics, an equivalence of categories is a relation between two that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned.
