Sub-APS

From Apstheory

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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