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.


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

B -> A   B
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:


Relation to Internal Hom-set

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

Next Steps

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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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