mercredi 16 décembre 2015

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

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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

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

A category is a mathematical structure, like a group or a vector space, abstractly defined by axioms. Groups were defined in this way in order to study symmetries (of physical objects and equations, among other things). Vector spaces are an abstraction of vector calculus. What makes category theory different from the study of other structures is that in a sense the concept of category is an abstraction of a kind of mathematics. (This cannot be made into a precise mathematical definition!) This makes category theory unusually self-referential and capable of treating many of the same questions that mathematical logic treats. In particular, it provides a language that unifies many concepts in different parts of math. In more detail, a category has objects and morphisms or arrows. (It is best to think of the morphisms as arrows: the word “morphism” makes you think they are set maps, and they are not always set maps. The formal definition of category is given in the chapter on categories.)

Aspects of category theory Because the concept of a category is so general, it is to be expected that theorems provable for all categories will not usually be very deep. Consequently, many theorems of category theory are stated and proved for particular classes of categories. Homological algebra is concerned with Abelian categories, which exhibit features suggested by the category of Abelian groups. Logic is studied using topos theory: a topos is a category with certain properties in common with the category of sets but which allows the logic of the topos to be weaker than classical logic. It is characteristic of the malleability of category theory that toposes were originally developed to study algebraic geometry. An important use purpose of categorical reasoning is to identify within a given argument that part which is trivial and separate it from the part which is deep and proper to the particular context. For example, in the study of the theory of the GCD, the fact that it is essentially unique simply follows from the uniqueness of the product in any category and is thus really trivial. On the other hand, the fact that the GCD of the integers A and B can be expressed as a linear combination of A and B with integer coefficients—GCD(a, b) = ma + nb, for some integers m and n—is a much deeper fact that is special to a much more restricted situation.

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dimanche 6 décembre 2015

Google Ending Support For 32-Bit Chrome For Linux, Debian 7 And Ubuntu 12.04

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Google announced that it will drop support for 32-bit version of Linux, Debian 7 (Wheezy) and Ubuntu 12.04 (Precise).

In a Chromium-dev group post on Tuesday, Dec. 1, Google software engineer Dirk Pranke stated that Mountain View will drop support for Google Chrome on the aforementioned operating system

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Facebook, Twitter, Google quietly step up fight against terrorist propaganda

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Facebook, Google and Twitter are stepping up efforts to combat online propaganda and recruiting by Islamic militants, but the internet companies are doing it quietly to avoid the perception that they are helping the authorities police the web.

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