Should attributes be getting inherited (openStudio)?

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Inherited resources should show up in the Facility tab. You can identify an inherited object by two characteristics; the text will be green and there won't be an "x" to delete the assignment. In our default templates we typically apply construction sets to the building, but schedules sets typically get assigned by space type. But you can really do it any way you want. You can even have building default constructions but then have a custom exterior wall by building story. For reference this page has a diagram at the bottom describing inheritance.

You can drill down to surfaces and manually assign surfaces, similar to how you can assign schedules to a specific load instance. For internal loads, if they are inhered from a space type, then you can't edit them from within a space. You can add additional instances, but you can't change the instances that are inherited from the space type without editing or cloning the space type. You can always add extra loads to a space on top of what come with the space type.

Note: On GridView (new in 1.5.0 for space type and thermal zone tabs) inherited items show up, but they are not green so it isn't obvious at first if they are inherited or hard assigned.

They are being inherited, if you assigned them at a "higher" level of the building. It won't show the inheritance at the lower levels - I am not sure if this is the design intent, or if it just hasn't been addressed yet.

That is to say, if you put construction and schedule sets in at the building level, or to a single storey of the building, they will apply to all surfaces and spaces, unless overridden at a lower level (say, applying a different construction directly to a surface, or a schedule to a space).

You can verify this by applying a construction/schedule set to the whole building, then going into Sketchup and checking the constructions for each surface.