TY - JOUR

T1 - Modulated bicategories

AU - Carboni, Aurelio

AU - Johnson, Scott

AU - Street, Ross

AU - Verity, Dominic

PY - 1994/7/8

Y1 - 1994/7/8

N2 - The concept of regular category [1] has several 2-dimensional analogues depending upon which special arrows are chosen to mimic monics. Here, the choice of the conservative arrows, leads to our notion of faithfully conservative bicategory K in which two-sided discrete fibrations become the arrows of a bicategory F = DFib(K). While the homcategories F(B,A) have finite limits, it is important to have conditions under which these finite "local" limits are preserved by composition (on either side) with arrows of F. In other words, when are all fibrations in F flat? Novel axioms on F are provided for this, and we call a bicategory F modulated when Fop is such a F. Thus, we have constructed a proarrow equipment ( )*: → M (in the sense of [28]) with M = Fcoop. Moreover, M is locally finitely cocomplete and certain collages exist [23]. In the converse direction, if M is any locally countably cocomplete bicategory which admits finite collages [23], then the bicategory M* of maps in M is modulated. (Recall from [26, p 266], that a 1-cell in a bicategory is called a map when it has a right adjoint.).

AB - The concept of regular category [1] has several 2-dimensional analogues depending upon which special arrows are chosen to mimic monics. Here, the choice of the conservative arrows, leads to our notion of faithfully conservative bicategory K in which two-sided discrete fibrations become the arrows of a bicategory F = DFib(K). While the homcategories F(B,A) have finite limits, it is important to have conditions under which these finite "local" limits are preserved by composition (on either side) with arrows of F. In other words, when are all fibrations in F flat? Novel axioms on F are provided for this, and we call a bicategory F modulated when Fop is such a F. Thus, we have constructed a proarrow equipment ( )*: → M (in the sense of [28]) with M = Fcoop. Moreover, M is locally finitely cocomplete and certain collages exist [23]. In the converse direction, if M is any locally countably cocomplete bicategory which admits finite collages [23], then the bicategory M* of maps in M is modulated. (Recall from [26, p 266], that a 1-cell in a bicategory is called a map when it has a right adjoint.).

UR - http://www.scopus.com/inward/record.url?scp=0001386066&partnerID=8YFLogxK

U2 - 10.1016/0022-4049(94)90009-4

DO - 10.1016/0022-4049(94)90009-4

M3 - Article

AN - SCOPUS:0001386066

VL - 94

SP - 229

EP - 282

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 3

ER -