Being that Pokémon Go is an app on a smartphone, it can (and does) leverage the functionality of said device - certainly if one has allowed for that to happen. One way that it can do so is by periodically generating app notifications (through the system OS) for in-game happenings, which appear when the game is not actively running and/or when one is in an app other than Pokémon Go. It's a fascinating part of the game and it actually has come somewhat of a long way, which has now become the prompt to make a discussion of it here.
In the beginning, the notifications were somewhat rudimentary. They would mostly consist of Pokémon returning from Gyms and friends sending Gifts - even for the latter, the sender was not identified. Later on, it was finally possible to identify these senders, and there are even ones for Egg hatches, which began to show the possible diversity of these notifications. By tapping on these and the other kinds of notifications to be mentioned, the game is opened and if possible, the screen relevant to the notification is displayed.
More recently, there have been lots of notifications for the various in-game events. One of these is often generated just as a certain event starts (within a few hours) and the Trainer doesn't have the game open. In similar fashion, another one is generated at the tail end of certain events (again within a few hours, this time before the events themselves end), often with a reminder for them to catch a debuting species or form, or to complete Timed Research and/or Collection Challenges. These may become rather useful for some.
Other notifications that have appeared include those for weekly progressions, as one for the weekly Adventure Sync after it's been completed (Mondays after 9 AM) to remind the Trainer to collect the rewards from the feature. Then, the recently (re)implemented Weekly Challenges have periodic notifications as well at every quarter milestone, gently nudging the Trainer to help make progress on and complete it. These progression notifications illustrate just how helpful and varied they can be for a Trainer to keep up with things.
Like other apps on the smartphones the game is on, Pokémon Go can extend its reach to some key aspects for dealing with the apps on the devices and the devices themselves in general. The usage of notifications is a particularly clever one to keep its player Trainers in the loop of what is going on, has gone on, and could possibly go on as well, given the ones detailed above (and more that haven't been detailed due to the scope of this write-up). With the notifications, they can stand to do much more just as the game does more.
One year ago: My Next Pokémon Figure Targets
Two years ago: Cosplay: MCP Creative Festival
Three years ago: Transfer Soon, or Forever Hold Your 3DS
Four years ago: Goh's Workstation Setup
Five years ago: The Rain and Pokémon
Six years ago: Premier Ball Hanging Ornament
Seven years ago: Pokémon Go Community Day + Local EX Raid, 1/13/2019
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Nine years ago: Pokémon and Cosplay