Subject: categories: Re: twisted arrow categories

From: [email-masked] (Ross Street)
Date: Wed, 21 Oct 1998 10:03:33 +1000

>In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow
>category of C such that objects are arrows f : a -> b of C and arrows
>are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The
>construction is part of a proof of the reduction of ends to limits.
>
>I would be grateful for pointers to other occurrences of this construction
>in the literature.

The twisted arrow category of  A  is the category of elements of the hom
functor  A(-,-) : A^op x A --> Set.  In internal and variable category
theory there is a sense in which the twisted arrow fibration comes first
and then can be used to define hom functors and local smallness.  For
example, see page 292 of my paper

Cosmoi of internal categories, Transactions American Math. Soc. 258 (1980)
271-318; MR82a:18007

Regards,
Ross
  www.mpce.mq.edu.au/~street/