TY - JOUR

T1 - Flexible limits for 2-categories

AU - Bird, G. J.

AU - Kelly, G. M.

AU - Power, A. J.

AU - Street, R. H.

PY - 1989/11/5

Y1 - 1989/11/5

N2 - Many important 2-categories - such as Lex, Fib/B, elementary toposes and logical morphisms, the dual of Grothendieck toposes and geometric morphisms, locally-presentable categories and left adjoints, the dual of this last, and the Makkai-Paré 2-category of accessible categories and accessible functors - fail to be complete, lacking even equalizers. These examples do in fact admit all bilimits - those weakenings of the limit notion that represent not by an isomorphism but only by an equivalence - but much more is true: they admit important classes of honest limits, including products, cotensor products, comma objects, Eilenberg-Moore objects, descent objects, inserters, equifiers, inverters, lax limits, pseudo limits, and idempotent-splitting. We introduce the class of flexible limits, which includes all of the above and is, in the technical sense, a closed class. Note that such honest limits, when they exist, have many advantages over bilimits: they are unique to within isomorphism, and their universal properties are both stronger and more convenient to use, a whole level of coherent families of invertible 2-cells being avoided.

AB - Many important 2-categories - such as Lex, Fib/B, elementary toposes and logical morphisms, the dual of Grothendieck toposes and geometric morphisms, locally-presentable categories and left adjoints, the dual of this last, and the Makkai-Paré 2-category of accessible categories and accessible functors - fail to be complete, lacking even equalizers. These examples do in fact admit all bilimits - those weakenings of the limit notion that represent not by an isomorphism but only by an equivalence - but much more is true: they admit important classes of honest limits, including products, cotensor products, comma objects, Eilenberg-Moore objects, descent objects, inserters, equifiers, inverters, lax limits, pseudo limits, and idempotent-splitting. We introduce the class of flexible limits, which includes all of the above and is, in the technical sense, a closed class. Note that such honest limits, when they exist, have many advantages over bilimits: they are unique to within isomorphism, and their universal properties are both stronger and more convenient to use, a whole level of coherent families of invertible 2-cells being avoided.

UR - http://www.scopus.com/inward/record.url?scp=0002476856&partnerID=8YFLogxK

U2 - 10.1016/0022-4049(89)90065-0

DO - 10.1016/0022-4049(89)90065-0

M3 - Article

AN - SCOPUS:0002476856

VL - 61

SP - 1

EP - 27

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 1

ER -