On satellites in arbitrary categories

ON SA TELLITES IN ARBITRAR Y CA TEGORIES G. JANELIDZE (Commun icated by Academician G. C ho goshvili on 20 December 1 975) 1 W e generalize the definition of satellites with r esp ect to presheaves (and co- presheav es) with trace in the sense of [1]; a preshea f with trace is replaced b y a graph with a pair of diagrams defined o n it. W e show that the right satellite functor is left adjoint to the left sa tellite functor, and that a functor having a right (left) adjoint preserves rig ht (left) satellites . In particular cases the cons truction of satellites is given. W e shall denote b y S ( X , Y ), where X and Y are categ ories, the categ o ry of which the ob jects are graphs S together with tw o diagrams F = F ( S ) : S − → X and G = G ( S ) : S − → Y ; the mor phisms in S ( X , Y ) are defined in the natural wa y . W e hav e an isomorphism S ( X , Y ) ◦ ≈ S ( Y ◦ , X ◦ ) ( 1 ) given by ( X S F l r G , 2 Y ) 7− → ( Y ◦ S ◦ G l r F , 2 X ◦ ). Definition 2. In the diagra m A X T : D         Y V Z d ? ? ? ? ? ? ? ? S F Z d ? ? ? ? ? ? ? ? G : D         of graphs and their mor phisms, le t the graphs X , Y and A be categ o ries, a nd let the diagra ms T and V b e functors . W e will ca ll a morphism of diagr ams δ : T F − → V G an S -c o nne cting morphism (more pr ecisely , an ( S , F , G ) -c onne cting morphism ) δ : T − → V . In this case we s hall call the triple ( T , δ, V ) an S -c onne ct e d p air (of fun ctors with values in A ). If an S -connected pair ( T , δ, V ) is rig h t univer- sal (left universal), then we say that V is a right satel lite for T ( T is a left satel lite for V ) r elative to S and w e write V = S 1 T ( T = S 1 V ). The satellites of functors w ith respect to copr e s heav es (with trace) in the sense of [1] are sp ecia l cases of satellites in the sense of Definition 2. This also applies to s a tellites of co nt rav ariant functors relative to presheav es. Moreov er, unlike [1], Definition 2 allows to consider satellites of (cov ariant) functors relative t o presheaves and satellites o f contrav a riant functors relative to copresheaves. 1 Originally published in Russian as G. Z. Janelidze, On satel lites in arbit r ary c ategor ies , Bull. Georgian A cad. Sci. 82 (1976), no. 3, 529–532. T ranslated b y Jone In txaurraga Larr a˜ naga and Tim V an der Li nden with the help of Alexander F rolkin and Julia Go edec ke. 1 2 G. JANELIDZE Example 3 . In Situation 2, let X = Y , let S b e a subgraph of the ca teg ory of diagrams X Σ , where Σ is the graph of the form Σ = ( Σ ′′ , 2 Σ , 2 Σ ′ ), and let F ( S ) and G ( S ) be defined by F ( S )( − ) = ( − )(Σ ′ ) a nd G ( S )( − ) = ( − )(Σ ′′ ). If now S is a full subcateg o ry of X Σ and | S | satisfies the conditions of [1, Def. 1.3], then Definition 2 coincides with [1, Def. 1 .5, 1.6] a nd, in par ticular, is a generaliza tion of the clas sical definition. Example 4. Let K : Y − → A be a functor and Σ a graph with a distinguished ob ject Σ . W e shall deno te by Σ ′ the full subgraph of Σ such that | Σ ′ | = | Σ | − { Σ } . W e will furthermore choose: a subg raph S of the category Y Σ , whose o b jects will be called r esolutions ; a sub categor y X of the category A Σ ′ , whos e ob jects will be called c omplexes ; and a functor T : X − → A , which will be called the homolo gy (or homotopy ) functor. W e shall req uire the functor K to induce a diag r am S − → X , which w e sha ll denote by F ( S ). Next we define the dia gram G ( S ) : S − → Y by G ( S )( − ) = ( − )(Σ). The functor S 1 T : Y − → A (provided it exis ts) will be called the deriv ative of t he functor K , r elative to the triple ( S , X , T ). Under an appropria te c hoice of this triple, o ne o btain the der iv atives in the sense of [2] (from which the idea of co nsidering derived functors as homolog ies of satellite functors is als o taken) and [3, 4, 5, 6 ]; here it makes no difference whether left or rig h t deriv atives are used. In Examples 3 and 4, only the cov ariant case was considered; the contrav a riant case ca n be consider ed analo gously . W e return again to Situation 2 a nd will supp ose for simplicit y that A admits satellites re la tive to ( X , Y ), i.e. S 1 T and S 1 V exist for all S ∈ S ( X , Y ), T : X − → A and V : Y − → A . Then the assignments ( S , T ) 7− → S 1 T and ( S , V ) 7− → S 1 V determine bifunctors S ( X , Y ) × ( X , A ) − → ( Y , A ) , ( 5 ) S ( X , Y ) ◦ × ( Y , A ) − → ( X , A ) , ( 6 ) (where ( X , A ) is the category of functors fro m X to A and ( Y , A ) has the analo g ous meaning) and there ar e natural isomor phisms S 1 T ≈ ( S ◦ ) 1 T , S 1 V ≈ ( S ◦ ) 1 V ( 7 ) (if we make | ( X , A ) | = | ( X ◦ , A ◦ ) | and | ( Y , A ) | = | ( Y ◦ , A ◦ ) | ). In this wa y , we obtain the bifunctors (5 ) and (6) from each other using (1). Let us add tw o more prop erties of satellites: Theorem 8. F or each S ∈ S ( X , Y ), the functor S 1 : ( X , A ) − → ( Y , A ) is left adjoint to the functor S 1 : ( Y , A ) − → ( X , A ). Theorem 9. Let K : A − → B be a functor. If the pair ( T , δ, V ) is right uni- versal (left universal) and the functor K has a right (left) a djoin t, then the pair ( K T , K δ, K V ) is also right (left) univ er sal. W e turn to the construction o f satellites. L e t A = Sets, X and Y small categor ies, and T : X − → A , S ∈ S ( X , Y ) fixed. W e s ha ll define a functor Y · S · T : Y − → A as follows. W e put Y · S · T ( Y ) = a X ∈| X | ,Y ′ ∈| Y | Y ( Y ′ , Y ) × ( G − 1 ( Y ′ ) ∩ F − 1 ( X )) × T ( X ) ∼ , ( 10 ) ON SA TELLITES IN ARBITRAR Y CA TEGORIES 3 where ∼ is the smallest equiv alence relation under which cls( y G ( s ) , S 1 , t ) = cls( y , S 2 , T F ( s )( t )) for each S -morphism s : S 1 − → S 2 . If f ∈ Y , then we define Y · S · T ( f ) by cls( y , S, t ) 7− → cls( f y , S, t ). Theorem 11. L et A = Sets and let X , Y b e smal l c ate gories, S ∈ S ( X , Y ) and T : X − → A , V : Y − → A functors. Then (a) t he assignment δ ( S ) : t 7− → cls(1 G ( S ) , S, t ) defines an S -c onne cting morphism δ : T − → Y · S · T and the S -c onne cte d p air ( T , δ, Y · S · T ) is right universal; (b) the assignment ϑ ( S ) : ϕ 7− → ϕ ( G ( S ))(cls(1 G ( S ) , S, 1 F ( S ) ) defines an S -c onne cting morphism ϑ : ( Y , A )( Y · S · ( X ( − , ?)) , V (?)) − → V ( − ) and the S - c onne cte d p air (( Y , A )( Y · S · ( X ( − , ?)) , V (?)) , ϑ, V ) is left uni- versal. This also makes it poss ible to construct satellites, and in the case when A is a v ariety of universal alge br as (for instance, the ca tegory of abelia n g roups), the passage from Sets to A works in the sa me wa y as it do es for limits: for right satellites one should take the A -free algebra over Y · S · T ( Y ) (for every Y ∈ | Y | ), then take the (minimal) quotient that makes every δ ( S ) a homomorphism; for left satellites one only needs to equip ( Y , A )( Y · S · ( X ( X , ?)) , V (?)) (for every X ∈ | X | ) with an appropria te A -algebra structure. A. Razmadze Mathematica l Institute Georgian Academ y o f Sciences Tbilisi, Georgia (Received 16 January 1976 ) References [1] H. Inassaridze, Quelques p oints d’alg ` e b r e homol o gique et d’alg` ebr e hom otopique e t leurs ap- plic ations , Proceedings of Tbilisi A. Razmadze Mathematical Institute 48 (1975), 5–137, in Russian. [2] H. Inassaridze, Cohomolo gy wit h values in semigr oups , Ph.D. thesis abstract, Tbilisi , 1970, in Russian. [3] A. Dold and D. Pupp e, H omolo gie nic ht-additiver F unktor en. Anwendungen , Ann. Inst. F ourier (Grenoble) 11 (1961), 201–312. [4] S. Eilenberg and J. Mo ore, F oundations of r elative homolo gic al algebr a , M emoirs AMS, vol. 55, American Mathematical So ciety , 1965. [5] M. Ti erney and W. V ogel, Simplicial r esolutions and derive d funct ors , Math. Z. 111 (1969), no. 1, 1–14. [6] R. G. Swan, Some r elations betwe en higher K-funct ors , J. A lgebra 21 (1972), 113–136. 4 G. JANELIDZE Author ’s remarks, September 2008 1. Many th anks from the author to Tim V an der Linden, Julia Goed ec ke, and everyo ne who help ed them in th is translation! 2. The pap er w as written in Russian, b ecause publishing a p aper in English w as not allo wed in Soviet mathematical journals. Ev ery Soviet author had “tw o initials” b efore his surname, the first of whic h w as indeed initial while the second w as his father’s initial (“patronymic”), according to the Russian tradition, although it was inappropriate for Georgians (just as it w ould b e inappropriate for W estern-Europeans). 3. The author did not h a ve his PhD yet, and so subm itt in g a pap er he n eeded an “ap- prov al” from a professor—in this case Hvedri I n assaridze, who was not yet an A cademician (Soviet “Academician” means “Member of Academy”, which is a title far ab ov e Profes- sor, which itself is far ab ov e t h e PhD level)—and therefore h is approv al sh ou ld hav e b een follo wed by a “fin al ap p ro val” — in this case by Academician George Chogoshvili. 4. There is a misprin t in the formula (isomorphism) ( 1 ): the first “ ◦ ” should b e remov ed (this obvious misprint w as originall y noticed by then young Beso Pac huash vili). In fact there is another du alit y th ere, but it is n ot really explored in the p aper. 5. The original Russian version uses the term “diagram sc h eme” instead of “graph”. 6. Most importantly , bac k in 1975 the author did not know ab out K an exten sions, but “redisco vered” them and their basic prop erties. In particular Theorem 11 follo ws from v arious kno wn observ ations. A few y ears later the author sa w “All concepts are Kan extensions” in Saun ders Mac Lan e’s “Categories for th e w ork in g Mathematician” (written in 1971); how ever, in 1990, Mac Lane suggested to him to pub lish his results on satellites fully . This has not h app ened so far, but some of those results are in G. Janelidze, Satel lites and Galois extensions of c om mutative rings , PhD Thesis, Tbil- isi, 1978 G. Janelidze, Satel l ites with r esp e ct to Galoi s extens ions , Proc. Math. In st. Georgian Acad. Sci. LXII (1979), 38–48 ( in Russian) G. Janelidze, Computation of Kan ext ensions by me ans of inje ctive obje cts and the functors Ext in nonadditive c ate gories , Pro c. Math. I n st. Georgian A cad. Sci. LXX (1982), 42–51 (in Ru ssian) G. Janelidze, Internal c ate gories and Galois the ory of c ommutative rings , DSc Thesis, Tbilisi, 1991 (defended at St.-Petersburg State Universit y in 1992) H. Inassaridze, Nonab elian homolo gic al algebr a and its applic ations , Mathematics and its Applications 421 , Kluw er Academic Pu blishers, Dordrech t, 1997. 7. The first sen tence of the pap er is not form u lated very w ell: one can indeed get Inassaridze satel lites as a sp ecial case, but what the author is doing is a “b etter” ap- proac h rather t han a generalization. In fact the reason of “b etter” is that it is a situation where Grothendieck fibrations are b etter than indexed categories (which Inassaridze called “preshea ves of categories”), and moreov er, fi brations can b e replaced with arbitrary func- tors. And furth ermore, things are “inspired by Y oneda rather than Grothend ieck”, but explaining all th ese is a long story ... Note also that (nearly the same) d escription of Inassaridze satellites in t erms of Kan extensions w as obtained by Polish mathematician Stanisla w Balce rzyk in S. Balcerzyk, On I nassaridze satel lites r elatively to tr ac es of pr eshe aves of c ate gories , Bull. Acad. Pol. S ci., S´ er. S ci. Math. Astron. Phys. 25 (1977 ), 857–861.