MODULATED BICATEGORIES

The concept of regular category [1] has several 2-dimensional analogue s depending upon which special arrows are chosen to mimic monics. Here , the choice of the conservative arrows, leads to our notion of faithf ully 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 e ither side) with arrows of F. In other words, when are all fibrations in K flat? Novel axioms on K are provided for this, and we call a bica tegory H modulated when H(op) is such a K. Thus, we have constructed a proarrow equipment ( ): H --> M (in the sense of [28]) with M = F(coo p). Moreover, M is locally finitely cocomplete and certain collages ex ist [23]. In the converse direction, if M is any locally countably coc omplete bicategory which admits finite collages [23], then the bicateg ory M of maps in M is modulated. (Recall from [26, p 266], that a 1-ce ll in a bicategory is called a map when it has a right adjoint.)