#
Categorical domain theory: Scott topology, powercategories, coherent categories

##
Panagis Karazeris

In the present article we continue recent work in the direction of
domain theory were certain (accessible) categories are used as
generalized domains. We discuss the possibility of using certain
presheaf toposes as generalizations of the Scott topology at this
level. We show that the toposes associated with Scott complete
categories are injective with respect to dense inclusions of
toposes. We propose analogues of the upper and lower powerdomain
in terms of the Scott topology at the level of categories. We show
that the class of finitely accessible categories is closed under
this generalized upper powerdomain construction (the respective
result about the lower powerdomain construction is essentially
known). We also treat the notion of ``coherent domain'' by
introducing two possible notions of coherence for a finitely
accessible category (qua generalized domain). The one of them
imitates the stability of the compact saturated sets under
intersection and the other one imitates the so-called ``2/3 SFP''
property. We show that the two notions are equivalent. This
amounts to characterizing the small categories whose free
cocompletion under finite colimits has finite limits.

Keywords: accessible category, Scott complete category, classifying topos, powerdomain, coherent domain, perfect topos, free cocompletion.

2000 MSC: 18C35, 18B25, 03G30, 18A35.

*Theory and Applications of Categories*, Vol. 9, 2001, No. 6, pp 106-120.

http://www.tac.mta.ca/tac/volumes/9/n6/n6.dvi

http://www.tac.mta.ca/tac/volumes/9/n6/n6.ps

http://www.tac.mta.ca/tac/volumes/9/n6/n6.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n6/n6.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n6/n6.ps

TAC Home