APS functor

From Apstheory

Current revision as of 12:54, 30 December 2006

[edit] Definition

[edit] Symbol-free definition

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

An APS functor over a category is an APS functor from the category to itself.

[edit] Definition with symbols

[edit] Examples

[edit] 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.

[edit] 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 .

[edit] 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.

[edit] Subs and quotients

[edit] Homomorphism between APS functors

Given two APS functors Failed to parse (Can't write to or create math temp directory): F_1

and Failed to parse (Can't write to or create math temp directory): F_2
on a category, a homomorphism Failed to parse (Can't write to or create math temp directory): h:F_1
& rarr; Failed to parse (Can't write to or create math temp directory): F_2
associates, to every object Failed to parse (Can't write to or create math temp directory): C
of the category, a homomorphism Failed to parse (Can't write to or create math temp directory): h_C: F_1(C)
→ Failed to parse (Can't write to or create math temp directory): F_2(C)
such that given two objects Failed to parse (Can't write to or create math temp directory): C
and Failed to parse (Can't write to or create math temp directory): C'
and a homomorphism Failed to parse (Can't write to or create math temp directory): \\sigma: C
→ Failed to parse (Can't write to or create math temp directory): C'

, a natural constructed diagram commutes (clarify).

[edit] Sub-functor

Further information: APS sub-functor

An APS functor is termed a sub-functor of another APS functor if there is an injective homomorphism of APS functors from the first to the second.

For instance, the Failed to parse (Can't write to or create math temp directory): SL

functor (or special linear functor) is a sub-functor of the Failed to parse (Can't write to or create math temp directory): GL
functor (or general linear functor).

[edit] Quotient functor

Further information: APS quotient functor

An APS functor is termed a quotient functor of another APS functor if there is a surjective homomorphism from the second APS functor to the first.

[edit] Property theory

[edit] IAPS functor

An APS functor is termed an IAPS functor if the APS associated with any object is an IAPS.

[edit] 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

.