The following work has recently been re-printed as Macquarie University Mathematics Report no. 93-123: Enriched Categories, Internal Categories and Change of Base. By Dominic Verity (Doctoral Thesis, prepared under the supervision of Dr. Martin Hyland, submitted to Cambridge University in April 1992 and sucessfully examined in July 1992) Copies may be obtained by sending a request to me at the following address: domv@macadam.mpce.mq.edu.au A brief description follows: Part I: Change of Base. ----------------------- As the utility of generalised category theories has become more apparent, a coherent description of what happens when we change our base category has become more and more important. For instance suppose we have two monoidal categories V and W and a homomorphism H:V--->W which admits a right adjoint, then we might ask for a full description of the structures that H induces between the bicategories of V- and W- enriched categories and functors (or profunctors). A closely related problem asks for a description of the analogous structures induced on bicategories of internal categories, fibrations or stacks by a geometric morphism of toposes f:E--->F. Some partial results in this direction are well known, for instance in all of the situations described above we will get biadjoints between the bicategories of generalised categories and functors. In general the action of base change on bicategories of profunctors (aka bimodules) is more difficult to describe. Various "local adjointness" notions, of differing utility and complexity, have been introduced to cope with this problem, but none of these have been fully satisfactory. A significant difficulty with this approach is that a given morphism of bicategories may admit many different local adjoints. Here we give a complete solution to this problem by considering base change structures between bicategories of functors and profunctors at one in the same time. To do this we consider proarrow equipments (M,K,*) (in the sense of Wood [4]), which we show are the objects of a family of closely related "bicategory enriched" categories, called EHom, EMor, EcoMor and EMap. These are bicategory enrichment in the sense that they are enriched in the strict traditional way over the closed category of (small) bicategories and normalised homomorphisms, wherein B^C is the bicategory of normalised homomorphisms, strong transformations and modifications form C to B. Just as in a 2-category we may describe adjunctions equationally in terms of unit and counit 2-cells, we may interpret analogously the notion of (normalised) 'biadjunction' in such bicategory enriched categories. It turns out that base change gives rise to biadjoints of this type in the enriched categories of proarrow equipments introduced above. This fact is proved in detail for all of the (closely related) examples above, culminating in a "comparison lemma" for equipments of stacks. An important application of this framework is a precise general result about the effect of base change on the weighted colimits (or limits) that a generalised category possesses. This follows directly from the description of weighted limits in terms of kan extensions/liftings of profunctors and representability. It is worth pointing out that this approach to base change is very close in spirit to that of Carboni, Kelly and Wood [2] (wherein all bicategories are locally ordered) which it generalises. Part II: Double Limits. ----------------------- At the International Category Theory meeting at Bangor North Wales in 1989, Bob Pare introduced an alternative approach to describing limits in 2-categories which used Double Categories rather than 2-weights (cf [3]). He went on to describe a particularly nice class of well behaved 2-limits, which he dubbed "Persistent" and defined in terms of stability properties with respect to equivalences. These he characterised in terms of structural properties of the double categories that parameterise them. However, while he was able to construct a double category which parameterised the same limits as any given 2-weight, he admitted that constructing a 2-weight from a double category was more problematic. We might think of this as a change of base problem, from Cat-enriched category theory to the theory of categories internal to Cat (which is exactly what double categories are). To make this precise we must first recognise that coequalisers in Cat are not well behaved, being unstable under pullback, so we cannot construct a bicategory of profunctors internal to it. However we can embed Cat in the far better behaved category of simplicial sets and do our work there. All this we make precise and then apply the work of part I to relate closed classes of 2-weights to corresponding classes of double categories satisfying certain closure properties. Once all this has been done it becomes a triviality to provide a "basis" of fundamental double limits which collectively generate the closed class of persistent limits. In fact products and splitting of idempotents along with constructions called inserters and equifiers are enough, which demonstrates that Pare's persistent limits coincide with the Flexible ones defined (in the context of 2-weights) by Street and Kelly et al [1]. References: [1] Bird, Kelly, Power and Street, Flexible limits for 2-categories, JPAA 61, November 1989. [2] Carboni, Kelly and Wood, A 2-categorical approach to change of base and geometric morphisms I, Cahiers de Top. et Geom. Diff. Categoriques 32(1991), 47-95. [3] Pare, Double Limits, July 1989, Unpublished notes from Bangor summer meeting. [4] Wood, Abstract proarrows I, Cahiers de Top. et Geom. Diff. 23-3(1982), 279-290. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++