Super comma category

Super comma category

[Serge P. Kovalyov]

Dear Category Theory gurus,

Is there any literature in which MacLane's "super comma category"
(Cat./.C) of all small diagrams in a category C is studied in details?
Actually I work in its covariant form, where a morphism from a diagram
D:X->C to G:Y->C is a pair (e,F) consisting of a functor F:X->Y and a
natural transformation e:D->GF. For example, is it known that the
embedding of an ordinary comma category Cat/C into Cat./.C preserves
colimits? Also there exists a monad (Cat./.-, d, m) on CAT, where d_C
takes each C-object X to a discrete diagram {X}, and m represents
"drawing" a diagram of diagrams as a diagram. Is its Eilenberg-Moore
category isomorphic to some "familiar" construction?

Thanks,
Serge.




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Re: Super comma category

[Andree Ehresmann]

Dear Serge,

An answer to your question is the paper "Decompositions et
lax-completions" par Rene Guitart et Luc Van den Bril (Cahiers Top. et
Geom. Diff. XVIII-4, 1977, pp. 333-407) where the category is called
'categorie des diagrammes". This paper is freely accessible on the
NUMDAM site
http://www.numdam.org/numdam-bin/feuilleter?j=CTGDC

With my best wishes
Andree




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