APS functor

From Apstheory

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Revision as of 03:21, 30 December 2006

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Examples 2.1 Over the category of sets 2.2 Over the category of Abelian groups 2.3 Over the category of rings 3 Property theory 3.1 Ground-matched functor

Definition

Symbol-free definition

An APS functor over a category is a functor from the category to the corresponding APS category. In other words, an APS functor over a category takes as input an object of the category and outputs an APS over the category.

Definition with symbols

Examples

Over the category of sets

The power APS is an example of an APS function. Given any set Failed to parse (Can't write to or create math temp directory): S , the power APS over Failed to parse (Can't write to or create math temp directory): S

has as its Failed to parse (Can't write to or create math temp directory): n^{th}
member the set Failed to parse (Can't write to or create math temp directory): S^n

, with the block concatenation map being the concatenation of tuples.

Over the category of Abelian groups

Given an Abelian group Failed to parse (Can't write to or create math temp directory): G

the constant functor sends Failed to parse (Can't write to or create math temp directory): G
to an APS whose Failed to parse (Can't write to or create math temp directory): n^{th}
member is Failed to parse (Can't write to or create math temp directory): G
for all Failed to parse (Can't write to or create math temp directory): n

, and such that the block concatenation map is simply addition in Failed to parse (Can't write to or create math temp directory): G .

Over the category of rings

Given a ring Failed to parse (Can't write to or create math temp directory): R

the matrix functor sends Failed to parse (Can't write to or create math temp directory): R
to the matrix IAPS Failed to parse (Can't write to or create math temp directory): Mat(R)

, which associates to each Failed to parse (Can't write to or create math temp directory): n

the matrix ring Failed to parse (Can't write to or create math temp directory): Mat_n(R)
and for which the block concatenation maps are the block concatenations of matrices.

Property theory

Ground-matched functor

An APS functor is termed ground-matched if the ground member of the APS functor applied to an object is the same as the object itself. All the above examples of APS functors are ground-matched:

• The first member of the power APS of a set Failed to parse (Can't write to or create math temp directory): S

is Failed to parse (Can't write to or create math temp directory): S^1
which is the same as Failed to parse (Can't write to or create math temp directory): S

.

• The first member of the constant APS over an Abelian group Failed to parse (Can't write to or create math temp directory): G

is again Failed to parse (Can't write to or create math temp directory): G

.

• The first member of the matrix IAPS over a ring Failed to parse (Can't write to or create math temp directory): R

is Failed to parse (Can't write to or create math temp directory): Mat_1(R)
which is the same as Failed to parse (Can't write to or create math temp directory): R

.