APS functor
From Apstheory
Contents |
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.
Subs and quotients
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).
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,
Property theory
IAPS functor
An APS functor is termed an IAPS functor if the APS associated with any object is an IAPS.
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
.
