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Conceptual Mathematics, Second Edition by Stephen H. Schanuel, F. William Lawvere

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SESSION 30

Exponentiation

1.  Map objects, or function spaces

Map objects, or function spaces, are sometimes also called exponential objects because they satisfy laws of which the laws of exponents in arithmetic are special cases. They are used to study the way in which an output depends on a whole process, rather than just a single input. For example, the energy expended in walking from Buffalo to Rochester depends not only on the distance traveled, but on the whole ‘motion’ you perform. This motion is itself a map, say from an interval of time to ‘space.’

We saw that a product of two objects X1 and X2 of a category images can be described as a terminal ...

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