On The Existence Of Category Bicompletions

On The Existence Of Categ ory Bicompletio ns Brian J. D A Y Ma y 4, 2009 . Ab str act: A c ompleteness c onje ctur e is advanc e d c onc erning the fr e e smal l-c olim it c om- pletion P ( A ) of a (p ossibly l ar ge) c ate gory A . The c onje ctur e is b ase d on the existenc e of a smal l gener ating-c o gener ating s et of obje cts in A . W e sketch how the validity of the r esult would le ad to the existenc e of an Isb el l -L amb ek bic ompletion C ( A ) of such an A , wi thout a “change-of-univer se” pr o c e dur e b eing ne c essary to describ e or discuss the bic ompletion All categ o ries, functors , and natural transforma tio ns, etc., shall b e r elative to a basic c o mplete and co co mplete symmetric monoidal closed category V with all intersections of s ubo b jects. A tent ative conjecture, ba s ed par tly on the re- sults o f [3], is that if A is a (la rge) categ ory containing a small g enerating a nd cogenera ting set of ob jects, then P ( A ) (which is the free small-colimit co m- pletion o f A with resp ect to V ) is not o nly co complete (a s is well known), but also co mplete with a ll intersections of sub o b jects. If this conjecture is tr ue , then one can esta blish the existence of a r esulting “Isb ell-Lambek” bicompletion o f such an A , along the line s of [1] § 4 , using the Y oneda embedding Y : A ⊂ P ( A ). This pro po sed bicompletion, denoted here by C ( A ), has the same “size” a s A and is, roughly sp eaking , the (replete) closure in P ( A ), under b oth itera ted limits and intersections of sub ob jects, of the clas s (i.e. larg e set) of all r epresentable functors from A op to V . More pr e cisely , one can construct C ( A ) directly us ing the Is bell- conjugacy adjunction P ( A ) Lan Y ( Z ) / / P ( A op ) op R o o whose existence (see [3] § 9) follows fro m the conjectured completenes s of b oth P ( A ) and P ( A op ), and where Z : A → P ( A op ) op is the dual of the Y oneda embedding A op ⊂ P ( A op ). Thu s we pro ceed by factor ing the left adjoint Lan Y ( Z ) as a reflection followed by a cons erv ative left adjoint P ( A )   Lan Y ( Z ) / / P ( A op ) op R o o v v l l l l l l l l l l l l l l C ( A ) ∪ cons 6 6 l l l l l l l l l l l l l l . Such a fa c torization exists by [1] Theo rem 2.1 a nd is essentially unique by [1] Prop ositio n 5.1 . Moreov er, the induced full embedding A ⊂ C ( A ) 1 then pres erves any sma ll limit or s mall colimit that a lready exists in A . One imp orta nt co nsequence is that v ario us results from [2] on mo noidal biclosed completion o f categor ies can b e according ly rev ampe d using such a bi- completion C ( A ); see also [3] § 7, which descr ibes some exa mples wher e P ( A ) is monoidal or monoidal biclosed. Note that here e s pec ia lly one could conve- nient ly av oid the awkward “ change-of- V -universe” pro cedure employ ed in [2]. References. [1] B. J. Day , “On Adjoint-F unctor F actor ization”, Lecture Notes in Ma thema tics, 420 (Springer-V erlag 1974 ), P g. 1-19. [2] B. J. Day , “On C lo sed Catego ries O f F uncto rs I I”, Lecture Notes in Ma thema tics, 420 (Springer-V erlag 1974 ), P g. 20-54. [3] B. J. Day a nd S. La ck, “Limits Of Sma ll F unctor s ” J. Pure Appl. Alg., 210 (200 7), Pg. 651 -663. Mathematics D ept., F acult y of S cience, Macquarie Universit y , N SW 2109, Au stralia. Any replies are w elcome through T om Bo oker (thomas.bo oker@students.mq.edu.au), who kin dly typ ed the manuscri pt. 2