Maths - Category Theory Universal Contructs Sum

Maths - Category Theory - Sum

		Sum (Coproduct) (pushout)

generalisation		a kind of colimit
set example		disjoint union {a,b,c}+{x,y}= {a,b,c,x,y}
group		free product the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.
Grp (abelian)		direct sum consists of the elements of the direct product which have only finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors) the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero)
vector space		direct sum
poset		least upper bound join
base topological space		wedge
POS		least upper bounds (joins)
Rng
Top		disjoint unions with their disjoint union topologies
Grf
category

Sum

When generating a sum for objects with structure then the structure associated with the link can be added to the sum object.

group sum category

Product

product category construction

Products for groups are discussed on this page.

Next Steps

EuclideanSpace

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Overview

Here we give an example of a category - the sum category.

For more general information about categories see the page here.

Type theory is related the category theory. For information about sum types see this page.

In logic the product is 'and' denoted /\ this is discussed on the propositional logic page here.

Group Theory

Functional Programming and Category Theory

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see also:

Catsters youtube videos - Terminal and initial objects

catsters youtube video - Terminal and initial objects 1 - A first look at universal properties: definition of terminal object and some examples, sketch of proof that terminal objects are unique up to unique isomorphism.
catsters youtube video - Terminal and initial objects 2 - Full proof that terminal objects are unique up to unique isomorphism, and some examples of categories without terminal objects
catsters youtube video - Terminal and initial objects 3 - Definition of initial object and some examples

Catsters youtube videos - Products and coproducts

catsters youtube video - Products and coproducts 1 - Definition of product, example of cartesian product of sets, BxA and AxB as two products, introducing uniqueness up to unique isomorphism (no details)
catsters youtube video - Products and coproducts 2 - Uniqueness up to unique isomorphism, various examples and non-example (tensor product of vector spaces)
catsters youtube video - Products and coproducts 3 - Definition of coproduct, various examples
catsters youtube video - Products and coproducts 4 - The morphisms (f,g) and fxg, the diagonal, and Ax1.

Catsters youtube videos - Pullbacks and pushouts

catsters youtube video - Pullbacks and pushouts 1 - Definition of pullback, example: pullbacks in Set.
catsters youtube video - Pullbacks and pushouts 2 - Definition of pushouts, pushouts in Set, and the pullback/pushout example of intersection/union of sets

Catsters youtube videos - General limits and colimits

catsters youtube video - General limits and colimits 1 - Idea of a limit of an arbitrary diagram in a category: cones and universal cones
catsters youtube video - General limits and colimits 2 - Explanation of simple limits as universal cones: terminal objects, products, pullbacks and equalisers.
catsters youtube video - General limits and colimits 3 - Formal definition of limits: first formalising the notion of a diagram of a given shape, and expressing a cone over it as a natural transformation
catsters youtube video - General limits and colimits 4 - Formal definition of limits as certain natural isomorphism
catsters youtube video - General limits and colimits 5 - The notion of a category having all limits of a certain shape, via a right adjoint.
catsters youtube video - General limits and colimits 6 - Colimits as universal cones, and the natural isomorphism formula

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.

Terminology and Notation

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