Theory and Applications of Categories, Vol. 1, No. 9, 1995, pp. 174–178.

ON THE SIZE OF CATEGORIES

PETER FREYD AND ROSS STREET

Transmitted by Michael Barr

ABSTRACT. The purpose is to give a simple proof that a category is equivalent to a small category if and only if both it and its presheaf category are locally small.

In one of his lectures (University of New South Wales, 1971) on Yoneda structures [SW], the second author conjectured that a category A is essentially small if and only if both A and the presheaf category PA are locally small. The first author was in the audience and at the end of the lecture suggested a proof of the conjecture using some of his own results. This was reported on page 352 of [SW] and used to motivate a definition of “small” in [St]; yet the proof was not published. The proof given in the present paper evolved via correspondence between the authors in 1976-77 while the second author was on sabbatical leave at Wesleyan University (Middletown, Connecticut) but has remained unpublished despite our expectation at various times that it would appear as an exercise in some textbook.

In 1979, a longer, but related, proof appeared in [F1]. We advised the author of this history and sent him our proof. This was reported in [F2], but our proof was still not published.

Now that there is actually an application [RW], we decided publication was in order. We have expressed the construction in a form we believe begs generalization to, for example, parametrized categories [SS]. Note throughout that “small” can mean “finite”.

For an object A of a category A, we let

Idem(A) = {e : A ⟶ A : ee = e}

denote the set of idempotents on A. The category A has small idempotency [SS] when Idem(A) is a small set for all objects A. It is clear that every locally small category (that is, category with small homsets) has small idempotency. We shall see conversely that, if binary products exist, small idempotency implies local smallness.

We write S for the category of small sets and functions between them. We write S^{A} for the category of functors F : A ⟶ S and natural transformations between them. [We work with S^{A} rather than the presheaf category to avoid contravariant functors.]

A retraction pair (m, r) at an object A in a category A consists of arrows m : X ⟶ A, r : A ⟶ X with rm = 1_{X}. Two retraction pairs (m, r), (n, s) at A are equivalent when there is an invertible arrow h : X ⟶ Y such that m = nh and hr = s. We write Ret(A) for the set of equivalence classes [m, r] of retraction pairs at A. There is a well-defined injective function

Received by the editors 28 October 1995.

Published on 30 December 1995

1991 Mathematics Subject Classification : 18A25.

Key words and phrases: small, locally small, small homsets, idempotent, presheaf category.

\begin{matrix}c \\ ◯ \end{matrix} Peter Freyd and Ross Street 1995. Permission to copy for private use granted.

Ret(A)[figure] Idem(A)

taking the equivalence class [m, r] to the idempotent mr : A ⟶ A. So Ret(A) is small if Idem(A) is. For our purposes here we could in fact assume A to admit splittings of all idempotents; so the above injective function would be bijective and we could avoid introducing Ret.

As foreshadowed, we can use this to show that small idempotency is equivalent to local smallness when A has binary products. For, we have an injective function

A(A, B)[figure] Ret(A × B)

taking f : A ⟶ B to the equivalence class [m, p] where pm = 1_{A}, qm = f and p : A×B ⟶ A, q : A × B ⟶ B are the projections.

A split monic is an arrow m : X ⟶ A with a left inverse. Two split monics m : X ⟶ A, n : Y ⟶ A are equivalent when there is an invertible arrow h : X ⟶ Y with nh = m. An equivalence class [m : X ⟶ A] of split monics into A is called a split subobject of A. We write Ssub(A) for the set of split subobjects of A. There is a well-defined surjective function

Ret(A) [figure] Ssub(A)

taking the equivalence class [m, r] to the split subobject [m : X ⟶ A]. So Ssub(A) is small if Ret(A) is.

It is clear from the above that, if A has small idempotency then Ssub(A) is a small set for all objects A of A. In this case we define a functor

T : A ⟶ S

on objects by

T A = Ssub(A) + {0},

and, for f : A ⟶ B in A, the function T f : T A ⟶ T B is given by

(T f)[m : X ⟶ A] = \{ \begin{matrix}[fm] & when fm has a left inverse \\ 0 & otherwise \end{matrix}

To see that T is a functor we use the fact that if gfm has a left inverse then so does fm.

We shall now introduce a function θ from the objects of A to the endomorphisms of T. We show that θ may be viewed as a function, not from the objects but, from the isomorphism classes of objects, and when so viewed, is an injection. The definition of the function

θ : objA ⟶ S^{A}(T, T)

is as follows. For each K ∈ objA, the natural transformation θK : T ⟶ T has component

(θK)_{A} : T A ⟶ T A

at A given by

(θK)_{A}[m : X ⟶ A] = \{ \begin{matrix}[m] & when X ≅ K \\ 0 & otherwise \end{matrix}

Clearly K ≅ L implies θK = θL; but the converse is true too. For, if θK = θL then

(θK)_{L}[1_{L} : L ⟶ L] = (θL)_{L}[1_{L} : L ⟶ L] = [1_{L} : L ⟶ L] ≠ 0;

so, from the definition of (θK)_{L}, we have L ≅ K. It is clear that (θK)(θK) = θK, so we have an injective function

objA /≅ ⟶ Idem(T, T)

induced by θ. We have proved:

Theorem. If A and S^{A} have small idempotency then the set of isomorphism classes of objects of A is small.

Corollary. A category A is equivalent to a small category if and only if A and S^{A} are locally small.

Question. Suppose A is a locally small site such that the category ShA of sheaves on A is locally small. Does it follow that ShA is a Grothendieck topos? We do not know. Notice that, if the Grothendieck topology on A is such that every presheaf is a sheaf, then A is equivalent to a small category by the above Corollary; so ShA is a Grothendieck topos in that case.

Remarks. Lest our functor T : A ⟶ S seem mysterious, we provide two constructions of T which may help the reader. The first construction demands more of the ambient set theory, but makes T more transparent. The second is choice free and invokes no large sets.

(1) Suppose A is locally small. Each representable functor H_{A} = A(A, −) : A ⟶ S has a unique maximal subfunctor M_{A} : A ⟶ S; an element f ∈ H_{A}B = A(A, B) is in M_{A}B if and only if it is not split monic. Let K_{A} denote the result of smashing M_{A} to a point; that is, M_{A} is defined by the pushout

[figure: 1; K_{A}; M_{A}; H_{A}]

in S^{A} where 1 is the terminal object (which, of course, is the functor constantly valued at the one-point set also denoted by 1). The functor K_{A} can be regarded as landing in the category S_{∗} of pointed small sets wherein 1 is both terminal and initial (= coterminal). Note that K_{A} takes to 1 precisely those objects for which A is not a retract. Let Λ denote a set of representatives of the isomorphism classes of objects of the category A. Put

S = ∑_{A∈Λ} K_{A}

where the summation denotes the coproduct of pointed sets. In fact, S takes values in the category S_{∗} of small pointed sets since, for all B ∈ A, only a small set of terms in the coproduct are non-trivial (because K_{A}B is non-trivial if and only if B is a retract of A). It is clear that S has at least as many endomorphisms as the coproduct has terms; so we have reproved the Corollary above.

Moreover, the functor T defined earlier is a quotient of S. For each object A, note that the group Aut(A) of automorphisms of A acts on the right of K_{A} (since it acts by composition on H_{A} and M_{A}). Let J_{A} denote the “orbit space” (that is, J_{A}B = K_{A}B/Aut(A) for all B). Then there is a natural isomorphism

T ≅ ∑_{A∈Λ} J_{A}

of pointed-set-valued functors.

(2) Now we turn to the second construction of T assuming A has small idempotency. There is a preorder on each set Idem(A) motivated by the natural order on the “images” of the idempotents; the preorder is defined by: e ≤ e′ when e = e′e. Let e ≈ e′ be the equivalence relation generated; so e ≈ e′ means e = e′e and ee′ = e′. Let UA be the set of equivalence classes together with the empty set (so, as a set, UA is the free partial order with first element on the preorder Idem(A)). For each f : A ⟶ B, define the function Uf : UA ⟶ UB by, for all elements E ∈ UA,

(Uf)(E) = {feu : u ∈ A(B, A), uf ∈ E, e ∈ E}

It can be verified that U : A ⟶ S is a functor. For each object X of A, let ϕ(X) be the endomorphism of U whose component ϕ(X)_{A} at A takes E ∈ UA to the element

ϕ(X)_{A}(E) = {xy : yx = 1_{X}, xy ∈ E}

of UB. This ϕ induces an injective function

objA /≅ ⟶ End(U)

Our functor T is isomorphic to a subfunctor of U, namely, the smallest subfunctor containing the images of all the natural transformations ϕ(X), X ∈ A. In other words, T is isomorphic to the subfunctor of U obtained by discarding the idempotents which do not split.

References

[F1] François Foltz, Légitimité des catégories de préfaisceaux, Diagrammes 1 (1979) F1-F5.

[F2] François Foltz, A propos de: “Légitimité des catégories de préfaisceaux”, Diagrammes 2 (1979) 1.

[ML] Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Math. 5 (Springer-Verlag, Berlin 1971).

[RW] Robert Rosebrugh and Richard Wood, An adjoint characterization of the category of sets, Proceedings AMS 122 (1994) 409-413.

[SS] Dietmar Schumacher and Ross Street, Some parametrized categorical concepts, Communications in Algebra 16 (1988) 2313-2347.

[St] Ross Street, Elementary cosmoi I, Lecture Notes in Math. 420 (1974) 134-180.

[SW] Ross Street and R.F.C. Walters, Yoneda structures on 2-categories, J. Algebra 50 (1978) 350-379.

Department of Mathematics University of Pennsylvania Philadelphia, Pennsylvania 19104-6395 U.S.A. School of Mathematics, Physics, Computing & Electronics Macquarie University New South Wales 2109 Australia

Email: pjf@saul.cis.upenn.edu and street@mpcs.mq.edu.au

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

mous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1995/v1n9.{dvi,ps}

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Université de Paris 13: breen@dmi.ens.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: mthfwl@ubvms.cc.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Paré, Dalhousie University: pare@cs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca R. W. Thomason, Université de Paris 7: thomason@mathp7.jussieu.fr Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca