1. Covariant Functor

2. Contravariant Functor

Libellés : Category and Functor |

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mardi 15 décembre 2015

1. Isomorphism between objects

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

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Libellés : Category and Functor |

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Category defintions & Examples

09:41 |

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Extract from : CRing Project, Chapter 0 source : https://math.berkeley.edu/~amathew/chcategories.pdf

1. Category definition

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

2. Examples and notations

$1.6 Preorder 2 Examples of categories 2.1 Baby examples 2.2 The category of sets 2.3 Mathematical structures as categories 2.3.1 Preorders 2.3.2 Groups 2.3.3 Matrices 2.4 Categories of sets with structure 3 Properties of objects and morphisms erminology and fine points If f: A \to B in a category, A is called the domain or source of f, and B is called the codomain or target of f. \text{Hom}(A,B) is called a hom class or a hom set if it is indeed a set. In general a hom set may be empty, but for any object A, \text{Hom}(A,A) is not empty because it contains the identity morphism. The hom class \text{Hom}(A,B) may be denoted by \text{Hom}_\mathcal{C}(A,B) or \mathcal{C}(A,B) if it is necessary to specify which category is referred to. Morphisms may also be called maps. This does not mean that every morphism in any category is a set function (see #Baby examples and #Preorders). Arrow is a less misleading name. The composite gf may be written g\circ f. It might be more natural to write the composite of f:A\to B and g:B\to C as f g instead of g f but the usage given here is by far the most common. This stems from the fact that if the arrows are set functions and x\in A, then (gf)(x)=g(f(x)). Thus g f is best read as "do g after f".$