Editing APS
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* For each natural number <math>n</math>, an associated object of the category, denoted <math>G_n</math>. | * For each natural number <math>n</math>, an associated object of the category, denoted <math>G_n</math>. | ||
- | * For each ordered pair <math>(m,n)</math> of natural numbers, a homomorphism <math>\\Phi_{m,n}:G_m | + | * For each ordered pair <math>(m,n)</math> of natural numbers, a homomorphism <math>\\Phi_{m,n}:G_m X G_n</math> → <math>G_{m+n}</math>. |
Satisfying the following compatibility conditions: | Satisfying the following compatibility conditions: | ||
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{{further|[[sub-APS]]}} | {{further|[[sub-APS]]}} | ||
- | Given an APS <math>(G,\\Phi)</math>, a sub-APS <math>H</math> associates, to each <math>n</math>, a subobject <math>H_n</math> of <math>G_n</math>, such that the image of <math>H_m | + | Given an APS <math>(G,\\Phi)</math>, a sub-APS <math>H</math> associates, to each <math>n</math>, a subobject <math>H_n</math> of <math>G_n</math>, such that the image of <math>H_m X H_n</math> under <math>\\Phi_{m,n}</math> lies inside <math>H_{m+n}</math>. |
===Quotient APS notion=== | ===Quotient APS notion=== | ||
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===Padding-injectivity=== | ===Padding-injectivity=== | ||
- | An APS is termed padding-injective, or is termed a [[PIAPS]], if for any fixed <math>g</math> in <math>G_m</math>, the map sending <math>h</math> in <math>G_n</math> to <math | + | An APS is termed padding-injective, or is termed a [[PIAPS]], if for any fixed <math>g</math> in <math>G_m</math>, the map sending <math>h</math> in <math>G_n</math> to <math\\Phi_{m,n}(g,h)</math> is injective. |
==See also== | ==See also== |