My players and I are starting a new campaign soon, with brand new characters set in the same world and after all of our previous campaigns. In the first session, I plan on having them run into some cultists that the players (but not the characters) have encountered in a previous campaign. One of the new characters was a student of one of the old characters, and it is mentioned in their backstory that "he showed me a museum of his previous adventures." Would a History check be used to determine if this new character identifies the cultists from the museum?
More generally, what's the time limit on a History check? Can they roll a history check to see if they remember things from the previous session? From their backstory? Or is it explicitly past events they did not experience? If this is the case, then what sort of check would I use for more recent history, like a "memory" check? Straight Wisdom, Investigation, straight Intelligence, Perception?
Each of the knowledge-based skills has some overlap with the others, and it is up to the DM to choose which skill is most appropriate (or if an ability check without the benefit of a skill proficiency is most appropriate). There are no rules for making this decision other than the descriptions of the six abilities and their associated skills, which are somewhat subjective.
Think about the answers to the following questions.
Is there a chance of failure on the proposed skill check? It sounds like the character would easily and automatically recognize the cult from the museum of adventures. All that a check would really model might be forgetfulness or lack of attention getting in the way. Is the character especially forgetful or absentminded, or do they have normal memory function?
Would failure be interesting, improve the story, or increase the fun? If you make a player roll to see if they remember something directly related to their stated backstory and they fail, the player knows they're being withheld information tied to their own backstory, and this will likely be frustrating and rob them of a sense of agency over their character. If they don't remember, will you have to contrive some less natural way to convey the information that doesn't tie in so well with the party's existing connection?
I recommend that you don't hinge what seems to be an important and interesting hook for the beginning of your new campaign on a die roll.
According to the Player's Handbook, a DM might ask for an Intelligence check when a character needs their memory. See page 177, "Using Ability Scores", "Intelligence" (emphasis mine):
Intelligence measures mental acuity, accuracy of recall, and the ability to reason.
An Intelligence check comes into play when you need to draw on logic, education, memory, or deductive reasoning.
Can they roll a history check to see if they remember things from the previous session?
Your players should roll Intelligence (History) when an only you, the DM, ask them "make a History check". They can't just "roll a history check" by themselves and then, if the roll was good, ask you to give them information. See PH, page 174:
The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
Asking for a check is not an obligation. You can ask for a dice roll, or you can use your common sense and game knowledge to resolve the outcome. It is your job as a DM to find your own approach that works best in this particular situation. See DMG, page 236 "The Role of Dice":
Some DMs rely on die rolls for almost everything.
One approach is to use dice as rarely as possible.
Many DMs find that using a combination of the two approaches works best.