Maths - Category Theory - Exponent

Exponent of category T is YT which represents mapping from T to Y:

T -> Y

So a mapping from one category to another has properties similar to exponent, for instance in the following mapping from set to set:

Example in Set

exponent   If we are mapping from set T to set Y and the number of elements in T is t and the number of elements in Y is y then the number of mappings from T to Y is yt.

This also behaves like an exponent in that YA * YB = YA+B. That is, if we have 3 categories, one of which is the sum of the other 2, each has a mapping to a 4th category then these mappings will be a product. See diagram on left:

exponent contra   exponent covarient

For a generalistion of this see Yoneda embedding.

Evaluation

In propositional logic there is a rule called 'modus ponens':

B -> A   B
A
modus ponens

That is: if 'B implies A' and 'B' is true then A is true.

This has a similar form to this functor:

ε: AB × B -> A

In this case we will call ε 'evaluation'.

Alternativly, for any object C:

f: C × B -> A

there is a unique arrow:

f': C -> AB

The relationship between ε, f and f' is:

ε•(f' × 1B) = f

This is represented by the following diagram:

evaluation

Relation to Internal Hom-set

An internal hom-set in a cartesian closed category is an exponential object.

 


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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

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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.

 

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