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Concept# Morphism

Summary

In mathematics, particularly in , a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in analysis and topology, continuous functions, and so on.
In , morphism is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism.
The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from , where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.
A C consists of two classes, one of and the other of . There are two objects that are associated to every morphism, the and the . A morphism f from X to Y is a morphism with source X and target Y; it is commonly written as or the latter form being better suited for commutative diagrams.
For many common categories, objects are sets (often with some additional structure) and morphisms are functions from an object to another object. Therefore, the source and the target of a morphism are often called and respectively.
Morphisms are equipped with a partial binary operation, called . The composition of two morphisms f and g is defined precisely when the target of f is the source of g, and is denoted g ∘ f (or sometimes simply gf). The source of g ∘ f is the source of f, and the target of g ∘ f is the target of g.

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Related publications (6)

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Epimorphism

In , an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms , Epimorphisms are categorical analogues of onto or surjective functions (and in the the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion is a ring epimorphism. The of an epimorphism is a monomorphism (i.e. an epimorphism in a C is a monomorphism in the Cop).

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation . In the more general setting of , a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow f : X → Y such that for all objects Z and all morphisms g1, g2: Z → X, Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below.

Restriction (mathematics)

In mathematics, the restriction of a function is a new function, denoted or obtained by choosing a smaller domain for the original function The function is then said to extend Let be a function from a set to a set If a set is a subset of then the restriction of to is the function given by for Informally, the restriction of to is the same function as but is only defined on .

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We exhibit sufficient conditions for a monoidal monad T on a monoidal category C to induce a monoidal structure on the Eilenberg-Moore category C^T that represents bimorphisms. The category of actions in C^T is then shown to be monadic over the base category C.

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

2019Hannes Bleuler, Ricardo Pérez Suárez

This work focuses on some promising applications of polyvinylidene fluoride (PVDF) based bimorph actuators in microengineering. The actuator of several centimeters in length and 100 mu m in thickness takes advantage of the structural and electromechanical capabilities of the PVDF. These characteristics make the proposed PVDF based bimorph ideal for a laser scanner/switcher where high speed laser beam manipulation is feasible. This domain requires light and efficient actuators capable of actuating the mirror surface at kHz frequency range and having a weight of less than 200 mg and volume of less than 150 mm(3). Analytical and finite element (FE) modeling were used to design the bimorph actuator with an attached mirror on the tip. Results of the modeling were also used to establish dimension criteria for the proposed bimorphs with an attached mirror. Some clean and gray room processing was needed to fabricate the bimorph actuators. Static and dynamic characterization was carried out to validate the response of the proposed actuator. Other bimorph based configurations were analyzed and compared to the proposed one. Results indicate that the PVDF bimorph actuator responds conveniently in a range up to 3 kHz for beam scanning tasks. The same principle may be implemented to 2 DoF actuators. Results obtained from modeling tools indicate a similar performance in terms of displacement. In addition, due to the intrinsic properties of the PVDF, the actuation system is compact which is advantageous for other domains such as biomedical and aerospace. (C) 2012 Elsevier B.V. All rights reserved.