Maths - Category Theory - Product

    Product
(pullback)
  product arrow category  
generalisation   a kind of limit
set example product set

cartesian product

{a,b,c}*{x,y}=
{{a,x},{b,x},{c,x},{a,y},{b,y},{c,y}}

group   the product is given by the cartesian product with multiplication defined componentwise.
Grp (abelian)   direct sum
vector space   direct sum
poset   greatest lower bound
meet
base topological space    
POS  

greatest lower bounds (meets)

Rng    
Top   the space whose underlying set is the cartesian product and which carries the product topology
Grf    
category   objects: (a,b)
morphism: (a,b)->(a',b')

tensor products are not categorial products.

In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).

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.

 

 


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see also:

Catsters youtube videos - Terminal and initial objects

Catsters youtube videos - Products and coproducts

Catsters youtube videos - Pullbacks and pushouts

Catsters youtube videos - General limits and colimits

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.

flag flag flag flag flag flag 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

Specific to this page here:

 

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