Published Scope is a parent listing field for your Shopify listings that can be found on the edit listings page for a Shopify listing.
You have two options to choose from:
WEB - LM will sync orders from your online store.
GLOBAL - LM will sync orders from your POS, aka your brick and mortar store.
Typical client set-ups will choose web. However, if you have a brick and mortar store that you sell your product out of, you will want to choose global so we can keep track of that inventory.