Abstract
We present two related categorical constructions. Given a category C, we construct a category C[d], the category of directed systems in C. C embeds into C[d], and if C has enough colimits, then C is monadic over C[d]. Also, if E,M is a factorization structure for C, then C[d] has a related factorization structure Ed Md such that if E consists entirely of monic arrows, then so does Ed and the Ed-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does Md and the Md-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
