A Categorys Quotient Category of Isomorphism Types Versus Its

GLASNIK

MATEMATICKI

SERIJA III.

Vol. 20 (40) - 1985 GLMAB 2

ZAGREB 1985 DRUSTVO MATEMATICARA I FIZICARA SR HRVATSKE SADRZAJ—CONTENTS

Originalni znanstveni radovi

/. Hafner, On lower bound of the proof length in the equivalen- tial calculus 269 N. Seifter, Fixed points, fixed ends and groups of graphs ... 271 V. Vukovic, (Nonassociative) near-rings 279 R. Brü and M. Löpiz- -Pellicer, Extensions of algebraic Jordan basis 289 M. E. Harris, A remark on the Schur's lemma Situation 293 R. Fritsch, A category's quotient category of isomorphism types versus its skeleton 297 K. R. Yacoub, On finite groups with four independent generators two of which being of odd prime order p and one of order 2 301 AI. Polonijo, On transfers and closure conditions for quasigroups 313 J. Dana, On translations of sets in Banach space 319 M. Pal, On an extension of a theorem of S. Kurepa .... 327 Z. Kominek and H. L Miller, Some remarks on a theorem-of Steinhaus 337 Af. Perovic, On the problem of radius of injectivity for the map- pings quasiconformal in the mean 345 J. Schinasand A. Meimaridou, Asymptotic behavior for difference equations which cannot be linearized around a periodic orbit .... 349 I. Aganovic and Z. Tutek, On the lower-dimensional approximations of the mixed problem for Laplace equation 355 D. lubrinii, On the existence of Solutions of — Au = g (x, u, V u) 363 K. Baron, A remark on linear functional equations in the indeter- minate case 373 M. Bajraktarevic, Sur certaines Solutions continues de T6quation foncti- pnnelle Ay> = B\p 377 B. Stankozic, Theorems of Tauberian type for measures .... 383 A. Mikelic, A variational problem of the Thomas-Fermi-von Wei• zsäcker type 391. W. Kotze, The fuzzy Hausdorff property 409 Ju. 7. Lisica and S. Mardesic, Coherent prohomotopy and strong shape for pairs . 419 M. Mrsevic, I. L. ReillyandM. K. Vamanamurthy, On some generalizations of regularity 435 D. S. Janhovic and P. E. Long, A characterization of H-closed Spaces 441 R. L. Blair, A note on remote points 443 R. Scitovski, Searching method and existence of Solution of special nonlinear least Squares problems 451 GLASNIK MATEMATlCKI Vol. 20 (40) (1985), 297—300.

A CATEGORY'S QUOTIENT CATEGORY OF ISOMORPHISM TYPES VERSUS ITS SKELETON

R. Fritsch, Munich, West Germany

Abstract. It is shown —• in geometric language — that a category c€ and its quotient category [#] of isomorphism types have different fundamental groups. This contrasts with the fact any skeleton of (€ is a strong deformation retract of <€ • Given a category # one sometimes looks for another category e6\ which should be quite similar to ^ but has the additional property that the isomorphism classes of objects consist of one object only. Categorists take a skeleton while »working« mathematicians tend to form the »quotient category [&] of isomorphism types« just by for• mal identification of isomorphic objectsu. The aim of this note is to point out that the quotient category [^] differs from # much more strongly than any skeleton of ^, namely that there is not even a pair of adjoint functors connecting ^ and [^] while any skeleton of <ß is equivalent to itself [7; p. 91]. To be precise, let (€ be a category and $0 its class of objects. Then there are a class @ and a surjective map [ ] : s/ -> & such that

[A] = [C]oA = C for all A,Ces/

([4, p. 72]). Consider stf as discrete subcategory of 88 also as a discrete category and [ ] as a fiinctor in the evident manner. Then form the pushout

[ H \ to obtain [^], the quotient category of isomorphism types of r€. It has the class £8 as class of objects and as non-identity arrows all lists

(/*> •••5/25/1)

Mathematics subject Classification (1980): 18A30. Key words and phrases: Category, isomorphism type, skeleton, classifying space of a category, groupoid associated with a category, object groups of groupoid, fundamental groups. 1 > I found this tendency in many oral and by letter discussions in contrast to the opinion of a referee that nobody »would really expect the quotient category to do the joh proposed«. The main reasons for it are that quotient constructions appear everywhere in the mathematical world and that the choice of a skeleton requires a suitable axiom of choice. 298 R. Fritsch of arrovvs in (ß which satisfy (0 fi is a non-identity arrow, 1 < i < k,

(ü) dorn fi ^ cod /f_ 1? 1 < k, but («0 [dorn f^ = [cod//_ i], 1 < i < k (cf [3; Theorem 4, p. 73] and [1, p. 263/264]).

Next take the groupoids Gr (6 and Gr [#] associated with [#] respectively (in the sense of [2, p. 10]; the smallness condition there again can be avoided by the set-theoretical trick used for obtaining 88). For an object C of define the fundamental group n (c&, C) of %' based at C as the object group Gr & {C} ( = group of automorphisms of C) in the groupoid Gr^. Analogously we have the fundamental group 7z([&]> [C]) of [#] based at [C]. This notion allows to formu- late the main resuit of this note:

THEOREM. Let cß be a category and let [r£] be its quotient category of isomorphism type. Then there is for all C e Ob (€ an (in general large) free group &c such that

«an [CD c)*^c where [C] denotes the isomorphism type of C and * denotes the free product

(of groups). Moreover &c depends (up to isomorphism) only on the con• nected component of C in

(One gets an impression of the size of the free group ^c by exhi- A biting a System of free generators: Let f£ be the component of C in #

f and let {Cft} be a class of objects in € containing exactly one represen-

( tative for every isomorphism type; then the class Ob ß \ {Cft} forms such a System.) This allows the final conclusion: In general the fundamental groups it (pf ], [C]) and Tz (tf> C) will be non-isomorphic> while a pair of adjoint functors between c€ and [rß] would induce an isomorphism between these groups. For small categories # this can be expressed more geometrically, using the notion of the classifying Space of [7; p. 106]. The discussion of strong homotopy in [8; p. 199] shows that BrS" is a strong deformation retract of B ^ for every skeleton C of f€ and that any adjoint functor induces a homotopy equivalence between the corresponding classifying Spaces. Since the fundamental groups of a category f€ as defined before are just the usual fundamental groups of the classifying space B^, the discussion shows that in general B# and B [W] have already difterent fundamental groups so that there is no chance for a homotopy equivalence between them. A category's quotient category. 299

The key to the theorem lies in the two following propositions.

PROPOSITION 1. — Let 88 and cß be groupoids and assume the free product [1; p. 269/270]

to be connected. Then, for any object D e Ob 3f the object group Q) {D} is of the form

where F is a free group, G is a free product of object groups of 88, exactly one for every component of 88, and H is a free product of object groups of c€, exactly one for every component of V. This can be proved by the same method as Kurosch's Theorem is proved in [3; p. 118 f.].

PROPOSITION 2. Let

88 c 9

be a pushout of categories such that s/ and 88 are discrete, p is surjective and is connected. Then there is a free group !F such that

7i {ß, qC) ~ TzttfyC)*^ for all CeObV.

Moreover 3* is trivial iff 8P is injective. To prove Proposition 2 one can assume c€ and Q) to be groupoids; then a careful analysis of the pushout in question yields the result by means of an application of Proposition 1 in the special case, where 88 is totally disconnected.

REFERENCES:

[1] R. Brozvn, Elements of Modern Topology, European Mathematics Series, McGraw-Hill, London 1968. [2] P. Gabriel and M. Zisrnan, Caiculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 35 (1967), Springer— Verlag, Berlin—Heidelberg—New York. 300 R. Fritsch

[3] P. J. Higgins, Categories and Groupoids, Mathematical Studies 32 (1971), Van Nostrand Reinhold Co., New York—Cincinnati—Toronto—Melbourne. [4] T. Jech, Set Theory, Pure and applied mathematics, Academic Press, New York— San Francisco—London 1978. [5] D. M. Latch, R. W. Thomason and W. S. Wilson, Simplicial sets from catego• ries, Math. Z. 164 (1979), 195—214. [6] 5. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (1971), Springer—Verlag, New York —Heidelberg—Berlin. [7] G. Segal, Classifying Spaces and spectral sequences, Inst. Hautes fitudes Sei. Publ. Math. 34 (1968), 105—112. (Received June 27, 1983) Mathematisches Institut (Revised July 11, 1984) Theresienstraüe 39 D-8000 München 2

ODNOS KVOCIJENTNE KATEGORIJE TIPOVA IZOMORFNOSTI PREMA SKELETU KATEGORIJE

R. Fritsch, München, Zap. Njcmaöka

Sadrzaj

Neka c£ oznaöava kategoriju a [C] njenu kvocijentnu kategoriju tipova izomorfnosti (tj. kategoriju u kojoj se formalno identificiraju izo- morfni objekti). Pokazano je da te kategorije (izrazeno geometrijskim jezikom) imaju razliöite fundamentalne grupe. To je u opreenosti sa änjenicom da je svaki skelet od ^ strogi deformacioni rektrakt od ^.