Sub-APS
From Apstheory
Revision as of 08:10, 9 March 2012 by 192.162.19.21 (Talk)
This article gives a basic definition in the APS theory. It is strictly local to the wiki
Definition
Let Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
be an APS over a monoidal concrete category. Then a sub-APS of Failed to parse (Can't write to or create math temp directory): (G,\\Phi) associates to each Failed to parse (Can't write to or create math temp directory): n a subobject Failed to parse (Can't write to or create math temp directory): H_n of Failed to parse (Can't write to or create math temp directory): G_n such that the restriction of Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n} to Failed to parse (Can't write to or create math temp directory): H_m × Failed to parse (Can't write to or create math temp directory): H_n takes it inside Failed to parse (Can't write to or create math temp directory): H_{m+n}
.
Thus we can view Failed to parse (Can't write to or create math temp directory): (H,\\Phi)
as an APS in its own right (note that since the associativity condition is satisfied for the block concatenation on Failed to parse (Can't write to or create math temp directory): G
, it is also satisfied for the block concatenation on Failed to parse (Can't write to or create math temp directory): H .
Since the Failed to parse (Can't write to or create math temp directory): \\Phi
is understood for the sub-APS, we may omit it and simply say that Failed to parse (Can't write to or create math temp directory): H is a sub-APS of Failed to parse (Can't write to or create math temp directory): G
.
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