Saturday, 8 March 2008

ct.category theory - Parametrized natural numbers object.

I give two examples of categories with finite products and simple NNO. In the first example the simple NNO is also a parameterized NNO, while in the second example it is not. Although it is difficult to understand your question, I believe the examples should clarify matters.



First, consider the category mathcalC whose objects are the finite powers of mathbbN, namely mathbbN0, mathbbN1, mathbbN2, ... and morphisms are set-theoretic functions f:mathbbNktomathbbNm. This category clearly has finite products, is not cartesian-closed because there are too many morhisms mathbbNtomathbbN, and it has a parameterized NNO, namely the obvious one.



Second, consider the category mathcalD whose objects are the finite powers of mathbbN, like before, and whose morphisms are as follows:



  1. Morphisms mathbbNktomathbbNm with mneq1 are all set-theoretic functions.

  2. Morphisms mathbbN0tomathbbN1 are all set-theoretic functions, i.e., for each natural number there is one.

  3. Morphisms mathbbNktomathbbN1 with kneq0 are all set-theoretic functions f:mathbbNktomathbbN for which there exists a projection pij:mathbbNktomathbbN and g:mathbbNtomathbbN such that f=gcircpij.

In other words, in mathcalD every function into mathbbN depends on only one of its parameters (exercise: prove that these are closed under composition.) The category mathcalD has finite products and a simple NNO, namely the obvious one, but no parameterized NNO. If it did, we could construct addition +:mathbbN2tomathbbN as a morphism in the category.

No comments:

Post a Comment