Which is more intuitive for navigating to
last week given the layout below?
Clicking up or clicking down?
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As you are showing a partial segment, a week, formatted as one normally does in a month-by-month calendar, UP would be the expected direction for the previous week and DOWN the direction one would look to find the next week.
To clue in the user, I'd add the words LAST WEEK above the top arrow and NEXT WEEK below the bottom arrow.
There's an interesting paper called How Languages Construct Time by Lera Boroditsky that talks about how the mental representation of time is affected by a number of factors. This is the part of the summary:
Across the studies cited here people in different cultures or groups have been shown to differ in whether they think of time as stationary or moving, limited or open-ended, as distance or quantity, horizontal or vertical, oriented from left to right, right to left, front to back, back to front or in cardinal space (e.g. East to West).
So the answer to your question -- what direction of time is more intuitive -- is that it depends. Certainly, in the English-speaking world, it is more common to conceptualise time as moving left to right. But in countries where RTL script is more common, time moving right to left might be more intuitive. For you example, however, I would conclude that neither up or down is intuitive, since it is not appropriate for the culture of your target audience.
It's difficult to answer a question that ultimately comes down to a vote for one of two options. But my vote is up.