Sub-APS

From Apstheory

This article gives a basic definition in the APS theory. It is strictly local to the wiki

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

.

Other notions

Smallest sub-APS containing a given collection of subobjects

If we are given a subobject of Failed to parse (Can't write to or create math temp directory): G_n

for every Failed to parse (Can't write to or create math temp directory): n

, we can talk of the sub-APS generated by these subobjects. This associates to every Failed to parse (Can't write to or create math temp directory): n , the set of elements in Failed to parse (Can't write to or create math temp directory): G_n

that arise via a block concatenation of elements upto Failed to parse (Can't write to or create math temp directory): n

.

Intersection of sub-APSes

Given two sub-APSes Failed to parse (Can't write to or create math temp directory): H

and Failed to parse (Can't write to or create math temp directory): K
of an APS Failed to parse (Can't write to or create math temp directory): G

, the intersection of Failed to parse (Can't write to or create math temp directory): H

and Failed to parse (Can't write to or create math temp directory): K
associates to each Failed to parse (Can't write to or create math temp directory): n
the subset Failed to parse (Can't write to or create math temp directory): L_n = H_n
∩ Failed to parse (Can't write to or create math temp directory): K_n
of Failed to parse (Can't write to or create math temp directory): G_n

. If the intersection of two subobjects is a subobject (which, for instance, is the case in algebraic structures) then Failed to parse (Can't write to or create math temp directory): L

is also a sub-APS of Failed to parse (Can't write to or create math temp directory): G

.

Similarly, if an arbitrary intersection of subobjects is a subobject, then an arbitrary intersection of sub-APSes is a sub-APS.