Editing APS 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 <math>S</math>, the power APS over <math>S</math> has as its <math>n^{th}</math> member the set <math>S^n</math>, with the block concatenation map being the concatenation of tuples.

===Over the category of Abelian groups===

Given an Abelian group <math>G</math> the [[constant functor]] sends <math>G</math> to an APS whose <math>n^{th}</math> member is <math>G</math> for all <math>n</math>, and such that the block concatenation map is simply addition in <math>G</math>.

===Over the category of rings===

Given a ring <math>R</math> the [[matrix IAPS|matrix functor]] sends <math>R</math> to the matrix IAPS <math>Mat(R)</math>, which associates to each <math>n</math> the matrix ring <math>Mat_n(R)</math> and for which the block concatenation maps are the block concatenations of matrices.

==Subs and quotients==

===Homomorphism between APS functors===

Given two APS functors <math>F_1</math> and <math>F_2</math> on a category, a homomorphism <math>h:F_1</math> & rarr; <math>F_2</math> associates, to every object <math>C</math> of the category, a homomorphism <math>h_C: F_1(C)</math> &rarr; <math>F_2(C)</math> such that given two objects <math>C</math> and <math>C'</math> and a homomorphism <math>\\sigma: C</math> &rarr; <math>C'</math>, a natural constructed diagram commutes (clarify).

===Sub-functor===

{{further|[[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 <math>SL</math> functor (or special linear functor) is a sub-functor of the <math>GL</math> functor (or general linear functor).

===Quotient functor===

{{further|[[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.

==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 <math>S</math> is <math>S^1</math> which is the same as <math>S</math>.
* The first member of the constant APS over an Abelian group <math>G</math> is again <math>G</math>.
* The first member of the matrix IAPS over a ring <math>R</math> is <math>Mat_1(R)</math> which is the same as <math>R</math>.

[[Category: Functors]]
[[Category: Terminology adapted from a standard context]]