twisted morphism category
I've been reading Lawvere's 1970 paper "Equality in hyperdoctrines and comprehension schema...." which appeared in the AMS series Symposia in Pure Mathematics, volume 17, pp 1-14. In the first line on page 12 he refers to the " 'twisted morphism category' " B^ and the forgetful functor from B^ to (B op) x (B). Can someone explain what this twisted morphism category is and/or provide a reference to the definition? Thanks for your help. Carl Futia
Topos8@aol.com writes:
I've been reading Lawvere's 1970 paper "Equality in hyperdoctrines and comprehension schema...." which appeared in the AMS series Symposia in Pure Mathematics, volume 17, pp 1-14.
In the first line on page 12 he refers to the " 'twisted morphism category' " B^ and the forgetful functor from B^ to (B op) x (B).
Objects of B^ are morphisms (f:a->b) of B. Morphism from (f:a->b) to (g:c->d) in B^ is a pair of morphisms ((p:c->a),(q:b->d)), forming together with f and g commuting square (note the direction of arrows). Alternatively, one can define 'twisted morphism category' as Grotendieck construction for Hom functor: Hom: B^op x B -> Set.
Can someone explain what this twisted morphism category is and/or provide a reference to the definition?
Thanks for your help.
Carl Futia
Nikita.
The "twisted morphism category" arising from the category B is the category of "elements of" the set-valued hom-functor B^op x B ---> Sets of B . Its objects are the morphisms f: X ---> Y of B ; a map from f to g: X' ---> Y' is a pair of B-morphisms, x: X' ---> X , y: Y ---> Y' satisfying g = yfx . I certainly used that name in the mid or late '60s, in work I'm not able to cite you from here, sorry. Cheers, -- FEJ Linton 11-Dec-2002 19:20:36 -0400,4997;000000000001-00000000
Yikes! is everything these days `twisted' some even refer to a simple regrading as a twist .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds On Tue, 10 Dec 2002, Fred E J Linton wrote:
The "twisted morphism category" arising from the category B is the category of "elements of" the set-valued hom-functor B^op x B ---> Sets of B . Its objects are the morphisms f: X ---> Y of B ; a map from f to g: X' ---> Y' is a pair of B-morphisms, x: X' ---> X , y: Y ---> Y' satisfying g = yfx . I certainly used that name in the mid or late '60s, in work I'm not able to cite you from here, sorry.
Cheers,
-- FEJ Linton
12-Dec-2002 17:36:34 -0400,1959;000000000001-00000000
participants (4)
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Fred E J Linton -
James Stasheff -
Nikita Danilov -
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