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mq.edu.au --_000_CWXP265MB4058246989FE6E3750F2E2A9F4E72CWXP265MB4058GBRP_ Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Dear all, Let C be a locally small category. A functor C^op -> Set that=92s a colimi= t of representables is called a =93small presheaf=94. 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. Explicit= ly, 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 cas= e of a representable presheaf? Best regards, Paul PS there=92s a 2-categorical version of (1) at the start of Section 6.2 of = Saville=92s thesis: https://philipsaville.co.uk/thesis-for-screen.pdf Another related result is the cartesian closure of the category of containe= rs: https://pblevy.github.io/papers/hocont.pdf You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --_000_CWXP265MB4058246989FE6E3750F2E2A9F4E72CWXP265MB4058GBRP_ Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable

Dear all,

Let C be a locally small category.  A functor C^op -> Set that=92s a colimit of r= epresentables is called a =93small presheaf=94.

Here are two observations.

  1. Let= C be cartesian.  In the cartesian category [C^op, Set], any small pre= sheaf is exponentiating.
  2. Mor= e 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=92s a 2-categorical version of (1) at the start of Sect= ion 6.2 of Saville=92s thesis: <= /span>

https://philipsa= ville.co.uk/thesis-for-screen.pdf

Another related result is the cartesian closure of the category = of containers:

https://pblevy.github.io/p= apers/hocont.pdf

 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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