zero morphism - Article Index

Encyclopedia - 0 Number: Encyclopedia - 0 Number
0 1 2 3 4 5 6 7 8 9 >> List of numbers -- Integers 0 10 20 30 40 50 60 70 80 90 >> 0 (zero), alternatively called naught, nil...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia - Limit Category Theory
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions ...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia - Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist an...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Elementary Properties
Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition o...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia Ii - Limit Category Theory - Examples
The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consid...   » Read the article

Encyclopedia - Initial Object: Encyclopedia Ii - Initial Object - Properties
Not all categories have initial or terminal objects, as will be seen below. Directly from the definition, one can show however that if an...   » Read the article

Encyclopedia - Biproduct: Encyclopedia Ii - Biproduct - Definition
Let C be a preadditive category. In particular, morphisms in C can be added. Given objects A1,...,An in C, suppose that we have: another...   » Read the article

Encyclopedia - Injective Cogenerator: Encyclopedia Ii - Injective Cogenerator - General Theory
In topological language, we try to find covers of unfamiliar objects. Finding a generator of an abelian category allows one to express ev...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Definitions
A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - History
0 number - Etymology. The word zero comes ultimately from the Arabic ṣifr (صفر) meaning empty or vacant, a literal translation of ...   » Read the article

Encyclopedia - Additive Category: Encyclopedia Ii - Additive Category - Elementary Properties
Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subje...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Biproducts
Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in additiv...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Additive Functors
If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. T...   » Read the article

Encyclopedia - Injective Cogenerator: Encyclopedia Ii - Injective Cogenerator - The Abelian Group Case
Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism f:Sum(G) ->...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - Distinguishing Zero From O
The oval-shaped zero and circular letter O together came into use on modern character displays. The zero with a dot in the centre seems t...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - In Computer Science
0 number - Numbering from 1 or 0?. Human beings usually number things starting from one, not zero. Yet in computer science zero has bec...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - 0 As A Number
0 is the integer that precedes the positive 1, and follows -1. In most (if not all) numerical systems, 0 was identified before the idea o...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Kernels And Cokernels
Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A&#...   » Read the article

Encyclopedia - Additive Category: Encyclopedia Ii - Additive Category - Examples
The original example of an additive category is the category Ab of Abelian groups with group homomorphisms. Ab is preadditive because it ...   » Read the article

Encyclopedia - Abelian Category: Encyclopedia Ii - Abelian Category - Elementary Properties
Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero elem...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - Distinguishing Zero From O
The oval-shaped zero (appearing like a rugby ball stood on end) and circular letter O together came into use on modern character displays...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - In Computer Science
0 number - Numbering from 1 or 0?. Human beings usually number things starting from one, not zero. Yet in computer science zero has bec...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - In Mathematics
0 number - Elementary algebra. Zero (0) is the lowest non-negative integer. The natural number following zero is one and no natural num...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia Ii - Limit Category Theory - Definition
Before defining limits, it is useful to define the auxiliary notion of a cone of a functor. Cones are also perhaps more aptly called sour...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia Ii - Limit Category Theory - Complete Categories
A category C is called complete iff every functor F : J → C, where J is any small category, has a limit; i.e. "all small limits in...   » Read the article

Encyclopedia - Additive Category: Encyclopedia Ii - Additive Category - Additive Functors
Recall that a functor F: C → D between preadditive categories is additive if it is an Abelian group homomorphism on each hom-set in C. ...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - 0 As A Numeral
The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc.,...   » Read the article

Encyclopedia - 0 Number: Encyclopedia Ii - 0 Number - 0 As A Number
0 is the integer that precedes the positive 1, and all positive integers, and follows -1, and all negative integers. In most (if not all)...   » Read the article

Encyclopedia - Limit Category Theory: Encyclopedia Ii - Limit Category Theory - Continuous Functors
It is a natural question to ask, which functors are compatible with the construction of limits in the sense that they map limits to limit...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note tha...   » Read the article