Sekai is the mother of Ami Arakawa, the wife of Nozomu Arakawa, and the sister-in-law of User. It is implied that, in life, she was a school teacher with a passionate love for poetry. Sometime in the past, Sekai and her husband were the victims of a tragic car accident, killing the two of them and orphaning their daughter.
Sekai's appearances are often brief and very sudden, leaving little time to fully grasp her personality. This is made even more difficult by the ambiguous status of her supposed "revival", making the delineation between which traits are true aspects of her personality nearly impossible.
What can be determined, though, is that she was an avid fan of William Shakespeare, often quoting or making allusions to his work. She's also seems to be very playful, often making quips or sarcastic jokes towards User or about her status of being dead.
Sekai's relationship with User is very complicated. While User seems to harbor a lot of complex and confusing feelings towards her, Sekai herself often approaches him with playfulness and flirtatious quips. She enjoys pointing out flaws in User's personality, but doesn't seem to view them as inherently negative, seemingly agreeing with the sentiment expressed by Nodoka.
Due to her being an important figure from User's past, and User suffering from amnesia, a lot of details about her are obfuscated through surreal imagery or otherwise completely lost.