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Tuesday, December 15, 2015

Isomorphisms

1. Isomorphism between objects

somorphisms  A morphism f : A \to B in a category is said to be an isomorphism if there is a morphism g : A \to B in the category with gf = 1_A, fg = 1_B. It is easy to prove that g is then uniquely determined by f. The morphism g is called the inverse of f, written g = f^{-1}. It follows that f = g^{-1}. If there is an isomorphism from A to B, we say A is isomorphic to B, and it is easy to prove that "isomorphism" is an equivalence relation on the objects of the category. Examples      A function from A to B in the category of sets is an isomorphism if and only if it is bijective.     A homomorphism of groups is an isomorphism if and only if it is bijective.     The isomorphisms of the category of topological spaces and continuous maps are the homeomorphisms. In contrast to the preceding example, a bijective continuous map from one topological space to another need not be a homeomorphism because its inverse (as a set function) may not be continuous. An example is the identity map on the set of real numbers, with the domain having the discrete topology and the codomain having the usual topology.  Monomorphisms and Epimorphisms  A morphism f in a category \mathcal{C} is a monomorphism if, for any morphisms g and h, if fg and fh are defined and fg = fh then g = h.   A morphism f in a category \mathcal{C} is an epimorphism if, for any morphisms g and h, if gf and hf are defined and gf = hf then

2. Goupoïdes

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:      Group with a partial function replacing the binary operation;     Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory.[1] Notice that a groupoid where there is only one object is a usual group.  Special cases include:      Setoids, that is: sets that come with an equivalence relation;     G-sets, sets equipped with an action of a group G.  Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt

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