Dear all, Let C be a locally small category. A functor C^op -> Set that’s a colimit of representables is called a “small presheaf”. Here are two observations. 1. Let C be cartesian. In the cartesian category [C^op, Set], any small presheaf is exponentiating. 2. More generally, let C be monoidal. In the multicategory [C^op, Set], any small presheaf is exponentiating. To see (1), it suffices to prove it for a representable presheaf. Explicitly, a presheaf H exponentiated by C(-,a) is H(- * a). The construction of (2) is similar. Has either result appeared in the literature? At least for the special case of a representable presheaf? Best regards, Paul PS there’s a 2-categorical version of (1) at the start of Section 6.2 of Saville’s thesis: https://philipsaville.co.uk/thesis-for-screen.pdf Another related result is the cartesian closure of the category of containers: https://pblevy.github.io/papers/hocont.pdf You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>