CONVENIENT CATEGORIES OF SMOOTH SPACES

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 11, November 2011, Paes S (2011)05107-X Article electronically published on June 6, 2011 CONVENIENT CATEGORIES OF SMOOTH SPACES JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Abstract. A Chen space is a set X equipped with a collection o plots, i.e., maps rom convex sets to X, satisyin three simple axioms. While an individual Chen space can be much worse than a smooth maniold, the cateory o all Chen spaces is much better behaved than the cateory o smooth maniolds. For example, any subspace or quotient space o a Chen space is a Chen space, and the space o smooth maps between Chen spaces is aain a Chen space. Souriau s dieoloical spaces share these convenient properties. Here we ive a uniied treatment o both ormalisms. Followin ideas o Penon and Dubuc, we show that Chen spaces, dieoloical spaces, and even simplicial complexes are examples o concrete sheaves on a concrete site. As a result, the cateories o such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classiier. For the beneit o dierential eometers, our treatment explains most o the cateory theory we use. 1. Introduction Alebraic topoloists have become accustomed to workin in a cateory o spaces or which many standard constructions have ood ormal properties: mappin spaces, subspaces and quotient spaces, limits and colimits, and so on. In dierential eometry the situation is quite dierent, since the most popular cateory, that o inite-dimensional smooth maniolds, lacks almost all these eatures. So, researchers are beinnin to seek a convenient cateory o smooth spaces in which to do dierential eometry. In this paper we study two candidates: Chen spaces and dieoloical spaces. But beore we start, it is worth recallin the lesson o alebraic topoloy in a bit more detail. Dissatisaction arose when it became clear that the cateory o topoloical spaces suers rom a deect: there is enerally no way to ive the set C(X, Y )o continuous maps rom a space X toaspacey a topoloy such that the natural map C(X Y,Z) C(X, C(Y,Z)) (x)(y) =(x, y) is a homeomorphism. In other words, this cateory ails to be Cartesian closed. This led to the search or a better ramework, or as Brown [6] put it, a convenient cateory. Steenrod s paper A convenient cateory o topoloical spaces [41] popularized the idea o restrictin attention to spaces with a certain property to obtain a Received by the editors September 13, 2008 and, in revised orm, October 13, Mathematics Subject Classiication. Primary 58A40; Secondary 18F10, 18F c 2011 John C. Baez and Alexander E. Honun

2 5790 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Cartesian closed cateory. It was later realized that by adjustin this property a bit, we can also make quotient spaces better behaved. The resultin cateory, with compactly enerated spaces as objects, and continuous maps as morphisms, has now been widely adopted in alebraic topoloy [30]. This shows that it is perectly possible, and at times quite essential, or a discipline to chane the cateory that constitutes its main subject o inquiry. Somethin similar happened in alebraic eometry when Grothendieck invented schemes as a eneralization o alebraic varieties. Now consider dierential eometry. Like the cateory o topoloical spaces, the cateory o smooth maniolds ails to be Cartesian closed. Indeed, i X and Y are inite-dimensional smooth maniolds, the space o smooth maps C (X, Y )is hardly ever the same sort o thin. It is a kind o ininite-dimensional maniold, but makin the space o smooth maps between these into an ininite-dimensional maniold becomes more diicult. It can be done [23, 31], but there are still many spaces on which we can do dierential eometry that do not live in the resultin Cartesian closed cateory. The simplest examples are maniolds with boundary, or more enerally maniolds with corners. There are also many ormal properties one miht want, which are lackin: or example, a subspace or quotient space o a maniold is rarely a maniold, and the cateory o maniolds does not have limits and colimits. In 1977, Chen deined a simple notion that avoids all these problems [9]. A Chen space is a set X equipped with a collection o plots, i.e., maps ϕ: C X where C is any convex subset o any Euclidean space R n, obeyin three simple axioms. Despite a supericial resemblance to charts in the theory o maniolds, plots are very dierent: we should think o a plot in X as an arbitrary smooth map to X rom a convex subset o a Euclidean space o arbitrary dimension. So instead o ensurin that Chen spaces look nice locally, plots play a dierent role: they determine which maps between Chen spaces are smooth. Given a map : X Y between Chen spaces, is smooth i and only i or any plot in X, sayϕ: C X, the composite ϕ: C Y is a plot in Y. In 1980, Souriau introduced another cateory o smooth spaces: dieoloical spaces [39]. The deinition o these closely resembles that o Chen spaces: the only dierence is that the domain o a plot can be any open subset o R n, instead o any convex subset. As a result, Chen spaces and dieoloical spaces have many similar properties. So, in what ollows, we use smooth space to mean either Chen space or dieoloical space. We shall see that: Every smooth maniold is a smooth space, and a map between smooth maniolds is smooth in the new sense i and only i it is smooth in the usual sense. Every smooth space has a natural topoloy, and smooth maps between smooth spaces are automatically continuous. Any subset o a smooth space becomes a smooth space in a natural way, and the inclusion o this subspace is a smooth map. Subspaces o a smooth space are classiied by their characteristic unctions, which are smooth maps takin values in {0, 1} equipped with its indiscrete smooth structure. So, we say {0, 1} with its indiscrete smooth structure is a weak subobject classiier or the cateory o smooth spaces (see Deinition 5.11).

3 CONVENIENT CATEGORIES OF SMOOTH SPACES 5791 The quotient o a smooth space under any equivalence relation becomes a smooth space in a natural way, and the quotient map is smooth. The cateory o smooth spaces has all limits and colimits. GivensmoothspacesX and Y,thesetC (X, Y )oallsmoothmapsrom X to Y can be made into a smooth space in such a way that the natural map C (X Y,Z) C (X, C (Y,Z)) is a smooth map with a smooth inverse. So, the cateory o smooth spaces is Cartesian closed. More enerally, iven any smooth space B, the cateory o smooth spaces over B, that is, equipped with maps to B, is Cartesian closed. So, we say the cateory o smooth spaces is locally Cartesian closed (see Deinition 5.16 or details). The oal o this paper is to present a uniied approach to Chen spaces and dieoloical spaces that explains why they share these convenient properties. All this convenience comes with a price: both these cateories contain many spaces whose local structure is ar rom that o Euclidean space. This should not be surprisin. For example, the subset o a maniold deined by an equation between smooth maps, Z = {x M : (x) =(x)}, is not usually a maniold in its own riht. In act, Z can easily be as bad as the Cantor set i M = R. But it is a smooth space. It is nice havin the solution set o an equation between smooth maps be a smooth space, but the price we pay is that a smooth space can be locally as bad as the Cantor set. So, we should not expect the theory o smooth spaces to support the wealth o ine-rained results amiliar rom the theory o smooth maniolds. Instead, it serves as a lare context or eneral ideas. For a taste o just how much can be done here, see Ilesias Zemmour s book on dieoloical spaces [18]. There is no real conlict, since smooth maniolds orm a ull subcateory o the cateory o smooth spaces. We can use the larer cateory or abstract constructions, and the smaller one or theorems that rely on ood control over local structure. Since we want dierential eometers to embrace the notions we are describin, our treatment will be as sel-contained as possible. This requires a little introduction to sheaves on sites, because the key act underlyin our main results is that both Chen spaces and dieoloical spaces are examples o concrete sheaves on a concrete site. For example, Chen spaces are sheaves on a site Chen: the cateory whose objects are convex subsets o R n and whose morphisms are smooth maps, equipped with a certain Grothendieck topoloy. However, not all sheaves on this site count as Chen spaces, but only those satisyin a certain concreteness property, which uarantees that any Chen space has a well-behaved underlyin set. Formulatin this property uses the act that Chen itsel is a concrete site. Similarly, the cateory o dieoloical spaces can be seen as the cateory o concrete sheaves on a concrete site Dieoloical. The cateory o all sheaves on a site is extremely nice: it is a topos. Here, ollowin ideas o Penon [34, 35] and Dubuc [11, 13], we show that the cateory o concrete sheaves on a concrete site is also nice, but slihtly less so: it is a quasitopos. This yields many o the ood properties listed above.

4 5792 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Various other notions o smooth space are currently bein studied. Perhaps the most eleant approach is synthetic dierential eometry [21], which drops the assumption that a smooth space be a set equipped with extra structure. This ives a topos o smooth spaces, and it allows a riorous treatment o calculus usin ininitesimals. Most other approaches treat smooth spaces as sets equipped with a speciied class o maps in, maps out, or maps in and out. We recommend Stacey s work [40] or a detailed comparison o these approaches. Chen and Souriau take the maps in approach, where a plot in a smooth space X is a map into X, anda unction : X Y between smooth spaces is smooth when its composite with every plot in X is a plot in Y. Smith [38], Sikorski [22, 37] and Mostow [32] ollow the maps out approach instead, in which a smooth space X comes equipped with a collection o coplots ϕ: X C or certain spaces C, andamap : X Y between smooth spaces is smooth when its composite with every coplot on Y is a coplot on X. Frölicher takes the maps in and out approach, in which a smooth space is equipped with both plots and coplots [15, 26]. This ives two ways to determine the smoothness o a map between smooth spaces, which are required to ive the same answer. Our work covers a wide class o deinitions that take the maps in approach. The structure o the paper is as ollows. In Section 2, we deine Chen spaces and dieoloical spaces and ive some examples. We also discuss the relation between these two ormalisms, ocusin on maniolds with corners and the work o Stacey [40]. In Section 3, we list many convenient properties shared by these cateories. In Section 4, we recall the concept o a shea on a site and show that Chen spaces and dieoloical spaces are concrete sheaves on concrete sites. Simplicial complexes ive another interestin example. In Section 5, we show that any cateory o concrete sheaves on a concrete site is a quasitopos with all limits and colimits. Most o the properties described in Section 3 ollow as a direct result. 2. Smooth spaces Souriau s notion o a dieoloical space [39] is very simple: Deinition 2.1. An open set is an open subset o R n. A unction : U U between open sets is called smooth i it has continuous derivatives o all orders. Deinition 2.2. A dieoloical space is a set X equipped with, or each open set U, a set o unctions ϕ: U X, called plots in X, such that: (1) I ϕ is a plot in X and : U U is a smooth unction between open sets, then ϕ is a plot in X. (2) Suppose the open sets U j U orm an open cover o the open set U, with inclusions i j : U j U. Iϕi j is a plot in X or every j, thenϕ is a plot in X. (3) Every map rom the one point o R 0 to X is a plot in X. Deinition 2.3. Given dieoloical spaces X and Y, a unction : X Y is a smooth map i, or every plot ϕ in X, the composite ϕ is a plot in Y.

5 CONVENIENT CATEGORIES OF SMOOTH SPACES 5793 Chen actually considered several dierent deinitions. Here we use his inal, most reined approach [9], which closely resembles Souriau s: Deinition 2.4. A convex set is a convex subset o R n with nonempty interior. A unction : C C between convex sets is called smooth i it has continuous derivatives o all orders. Deinition 2.5. A Chen space is a set X equipped with, or each convex set C, a set o unctions ϕ: C X, called plots in X, satisyin these axioms: (1) I ϕ is a plot in X and : C C is a smooth unction between convex sets, then ϕ is a plot in X. (2) Suppose the convex sets C j C orm an open cover o the convex set C with its topoloy as a subspace o R n. Denote the inclusions as i j : C j C. I ϕi j is a plot in X or every j, thenϕ is a plot in X. (3) Every map rom the one point o R 0 to X is a plot in X. Deinition 2.6. Given Chen spaces X and Y, a unction : X Y is a smooth map i, or every plot ϕ in X, the composite ϕ is a plot in Y. It is instructive to see how Chen s deinition evolved. O course he did not speak o Chen spaces ; he called them dierentiable spaces. In 1973, he took a dierentiable space to be a Hausdor space X equipped with continuous plots ϕ: C X satisyin axioms 1 and 3 above, where the domains C were closed convex subsets o Euclidean space [7]. In 1975, he added a preliminary version o axiom 2 and dropped the condition that X be Hausdor [8]. Startin in 1977, Chen used a deinition equivalent to the one above [9, 10]. In particular, he dropped the topoloy on X, the continuity o ϕ, and the condition that C be closed. This marks an important realization, emphasized by Stacey [40]: we can ive a space a smooth structure without irst ivin it a topoloy. Indeed, we shall see that a smooth structure determines a topoloy! The notion o a smooth unction : C C between convex sets is a bit subtle, particularly or points on the boundary o C. One tends to imaine C as either open or closed, but the eneric situation is more messy. For example, C could be the closed unit disk D 2 minus the set Q o points on the unit circle with rational coordinates. Both Q and its complement are dense in the unit circle. Situations like this, while ar rom our main topic o interest, deserve a little thouht. So, suppose C R n and C R m are convex subsets with nonempty interior. To deine the kth derivative o a unction rom C to C, it suices to deine the irst derivative o a unction F : C V or any inite-dimensional normed vector space V, since when this derivative exists it will be a unction df rom C to the normed vector space o linear maps hom(r n,v). We can then deine the derivative o this unction, and so on. Thereore, we say that the derivative o F exists at the point x C i there is a linear map (df ) x : R n V such that F (y) F (x) df x (y x) 0 y x as y x or y C {x}. NotethatsinceC is convex with nonempty interior, df x is unique i it exists.

6 5794 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG This is the usual deinition oin back to Fréchet, and scarcely worth remarkin on, except or the obvious caveat that y must lie in C. In the case C =[0, 1], this means we are usin one-sided derivatives at the endpoints. In the case o the convex set D 2 Q, it means we are usin a eneralization o one-sided derivatives at all points on the boundary o this set, which is the unit circle minus Q. Luckily, whenever C and C are convex sets, we can characterize smooth unctions : C C in three equivalent ways: (1) The unction : C C has continuous derivatives o all orders. (2) The unction : C C has continuous derivatives o all orders in the interior o C, and these extend continuously to the boundary o C. (3) I γ : R C is a smooth curve in C, thenγ is a smooth curve in C. The equivalence o conditions 1 and 2 is not hard; the equivalence o 2 and 3 was proved by Kriel [24], and appears as Theorem 24.5 in Kriel and Michor s book [25]. Since most o our results apply both to Chen spaces and dieoloical spaces, we lay down the ollowin conventions: Deinition 2.7. We use smooth space to mean either a Chen space or a dieoloical space, and use C to mean either the cateory o Chen spaces and smooth maps, or dieoloical spaces and smooth maps. We use the term domain to mean either a convex set or an open set, dependin on the context. Henceorth, any statement about smooth spaces or the cateory C holds or both Chen spaces and dieoloical spaces Examples. Next we ive some examples. For these it is handy to call the set o plots in a smooth space its smooth structure. So, we may speak o takin a set and puttin a smooth structure on it to obtain a smooth space. (1) Any domain D becomes a smooth space, where the plots ϕ: D D are just the smooth unctions. (2) Any set X has a discrete smooth structure such that the plots ϕ: D X are just the constant unctions. (3) Any set X has an indiscrete smooth structure where every unction ϕ: D X is a plot. (4) Any smooth maniold X becomes a smooth space where ϕ: D X is a plot i and only i ϕ has continuous derivatives o all orders. Moreover, i X and Y are smooth maniolds, then : X Y is a morphism in C i and only i it is smooth in the usual sense. (5) Given any smooth space X, we can endow it with a new smooth structure, where we keep only the plots o X that actor throuh a chosen domain D 0. When D 0 = R this smooth structure is called the wire dieoloy in the theory o dieoloical spaces [18]. While this construction ives many examples o smooth spaces, these seem to be useul mainly as counterexamples to naive conjectures. (6) Any topoloical space X can be made into a smooth space where we take the plots to be all the continuous maps ϕ: C X. Since every smooth map is continuous this deines a smooth structure. Aain, these examples mainly serve to disprove naive conjectures.

7 CONVENIENT CATEGORIES OF SMOOTH SPACES 5795 I Di is the cateory o smooth inite-dimensional maniolds and smooth maps, our ourth example above ives a ull and aithul unctor Di C. So, we can think o C as a kind o extension or completion o Di with better ormal properties. Any smooth space X can be made into a topoloical space with the inest topoloy such that all plots ϕ: D X are continuous. With this topoloy, smooth maps between smooth spaces are automatically continuous. This ives a aithul unctor C Top. In particular, i we take a smooth maniold, reard it as a smooth space, and then turn it into a topoloical space this way, we recover its usual topoloy Comparison. We should also say a bit about how Chen spaces and dieoloical spaces dier, and how they are related. To bein with, let us compare their treatment o maniolds with boundary, or more enerally maniolds with corners [19, 27]. An n-dimensional maniold with corners M has charts o the orm ϕ: X k M, where X k = {(x 1,...,x n ) R n : x 1,...,x k 0} or k =0,...,n. The case k = 1 ives a hal-space, amiliar rom maniolds with boundary. Since X k R n is convex, any chart ϕ: X k M canbemadeintoaplot in Chen s sense. So, i we make M into a Chen space where the plots ϕ: C M are just maps that are smooth in the usual sense, it ollows that any map between maniolds with corners : M N is smooth as a map o Chen spaces i and only i it is smooth in the usual sense. However, the subset X k R n is typically not open. So, we cannot make a chart or a maniold with corners into a plot in the sense o dieoloical spaces. Nonetheless, we can make any maniold with corners M into a dieoloical space where the plots ϕ: U M are the maps that are smooth in the usual sense, and then, in act, a map between maniolds with corners is smooth as a map between dieoloical spaces i and only i it is smooth in the usual sense! The key to seein this is the theorem o Kriel mentioned above. Since the issues involved are local, it suices to consider maps : X k R m. Suppose : X k R m is smooth in the sense o dieoloical spaces. Then the composite γ is smooth or any smooth curve γ : R X k. By Kriel s theorem, this implies that has continuous derivatives o all orders in the interior o X k, extendin continuously to the boundary. So, is smooth in the usual sense or maniolds with corners. Conversely, any : X k R m smooth in the usual sense is clearly smooth in the sense o dieoloical spaces. Stacey has iven a more eneral comparison o Chen spaces versus dieoloical spaces [40]. To briely summarize this, let us write ChenSpace or the cateory o Chen spaces, and DieoloicalSpace or the cateory o dieoloical spaces. Stacey has shown that these cateories are not equivalent. However, he has constructed some useul unctors relatin them. These take advantae o the act that every open subset o R n becomes a Chen space with its subspace smooth structure, and conversely, every convex subset o R n becomes a dieoloical space.

8 5796 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Usin this, Stacey deines or any Chen space X a dieoloical space SoX with the same underlyin set, where ϕ: U SoX is a plot i and only i ϕ: U X is a smooth map between Chen spaces. This extends to a unctor So: ChenSpace DieoloicalSpace that is the identity on maps. He also deines or any dieoloical space Y achen space Ch Y with the same underlyin set, where ϕ: C Ch Y is a plot i and only i ϕ: C Y is a smooth map between dieoloical spaces. Aain, this extends to a unctor Ch : DieoloicalSpace ChenSpace that is the identity on maps. Stacey shows that : X Ch Y is a smooth map between Chen space :SoX Y is a smooth map between dieoloical spaces. In other words, Ch is the riht adjoint o So. The unctor So also has a let adjoint Ch : DieoloicalSpace ChenSpace which acts as the identity on maps. This time the adjointness means that :Ch Y X is a smooth map between Chen spaces : Y SoX is a smooth map between dieoloical spaces. Furthermore, Stacey shows that both these composites, DieoloicalSpace Ch ChenSpace So DieoloicalSpace, DieoloicalSpace Ch ChenSpace So DieoloicalSpace, are equal to the identity. With a little work, it ollows that both Ch and Ch embed DieoloicalSpace isomorphically as a ull subcateory o ChenSpace: a relective subcateory in the irst case, and a corelective one in the second. The embeddin Ch is a bit strane: as shown by Stacey, even the ordinary closed interval ails to lie in its imae! To see this, he takes I to be [0, 1] R made into a Chen space with its subspace smooth structure. I I were isomorphic to a Chen space in the imae o Ch,sayI = Ch X,wewouldthenhave Ch SoI =Ch SoCh X =Ch X = I. However, he shows explicitly that Ch SoI is not isomorphic to I; it is the unit interval equipped with a nonstandard smooth structure. The embeddin Ch lacks this deect, since Ch SoI = I. For an example o a Chen space not in the imae o Ch, we can resort to Ch SoI. Suppose there were a dieoloical space X with Ch X = Ch SoI. Then we would have SoCh X = SoCh SoI; hence X = SoI. But this is a contradiction, since we know that Ch applied to SoI ives I, which is not isomorphic to Ch SoI. Luckily, the embeddin Ch works well or maniolds with corners. In particular, i Di c is the cateory o maniolds with corners and smooth maps, we have a

9 CONVENIENT CATEGORIES OF SMOOTH SPACES 5797 commutative trianle Di c DieoloicalSpace Ch ChenSpace, where the diaonal arrows are the ull and aithul unctors described earlier. 3. Convenient properties o smooth spaces Now we present some useul properties shared by Chen spaces and dieoloical spaces. Followin Deinition 2.7, we call either kind o space a smooth space, and we use C to denote either the cateory o Chen spaces or the cateory o dieoloical spaces. Most o the proos are straihtorward diaram chases, but we deer all proos to Section 5. Subspaces Any subset Y X o a smooth space X becomes a smooth space i we deine ϕ: D Y to be a plot in Y i and only i its composite with the inclusion i: Y X is a plot in X. We call this the subspace smooth structure. It is easy to check that, with this smooth structure, the inclusion i: Y X is smooth. Moreover, it is a monomorphism in C. Not every monomorphism is o this orm. For example, the natural map rom R with its discrete smooth structure to R with its standard smooth structure is also a monomorphism. In Proposition 5.7, we show that a smooth map i: Y X comes rom the inclusion o a subspace precisely when i is a stron monomorphism (see Deinition 5.5). The 2-element set {0, 1} with its indiscrete smooth structure is called the weak subobject classiier or smooth spaces, and is denoted by Ω. The precise deinition o a weak subobject classiier can be ound in Deinition 5.11, but the idea is simple: or any smooth space X, subspaces o X are in one-to-one correspondence with smooth maps rom X to Ω. In particular, any subspace Y X corresponds to the characteristic unction χ Y : X Ωivenby { 1 x Y, χ Y (x) = 0 x/ Y. In Proposition 5.13 we prove the existence o a weak subobject classiier in a more eneral context. Quotient spaces I X is a smooth space and is any equivalence relation on X, the quotient space Y = X/ becomes a smooth space i we deine a plot in Y to be any unction ϕ: D Y orwhichthereexistsanopencover{d i } o D and a collection o plots in X, {ϕ i : D i X} i I,

10 5798 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG such that the ollowin diaram commutes: ϕ i D i X ι i D ϕ Y, where p: X Y is the unction induced by the equivalence relation and ι i : D i D is the inclusion. We call this the quotient space smooth structure. It is easy to check that with this smooth structure, the quotient map p: X Y is smooth and an epimorphism in C. Not every epimorphism is o this orm: or example, the natural map rom R with its standard smooth structure to R with its indiscrete smooth structure is also an epimorphism. In Proposition 5.10, we show that a smooth map p: X Y comes rom takin a quotient space precisely when p is a stron epimorphism (see Deinition 5.8). Terminal object The one element set 1 can be made into a smooth space in only one way, namely by declarin every unction rom every domain to 1 to be a plot. This smooth space is the terminal object o C. Initial object The empty set can be made into a smooth space in only one way, namely by declarin every unction rom every domain to to be a plot. (O course, such a unction exists only or the empty domain.) This smooth space is the initial object o C. Products GivensmoothspacesX and Y, the product X Y o their underlyin sets becomes a smooth space, where ϕ: D X Y is a plot i and only i its composites with the projections p X : X Y X, p Y : X Y Y are plots in X and Y, respectively. We call this the product smooth structure on X Y. It is easy to check that with this smooth structure, p X and p Y are smooth. Moreover, or any other smooth space Q with smooth maps X : Q X and Y : Q Y, there exists a unique smooth map : Q X Y such that the ollowin diaram commutes: X X p X Q X Y p Y p Y Y. So, X Y is indeed the product o X and Y in the cateory C. Coproducts GivensmoothspacesX and Y, the disjoint union X + Y o their underlyin sets becomes a smooth space where ϕ: D X + Y is a plot i and only i or each connected component U o D, ϕ U is either the composite o a plot in X with the inclusion i X : X X + Y,orthecompositeoa

11 CONVENIENT CATEGORIES OF SMOOTH SPACES 5799 plot in Y with the inclusion i Y : Y X + Y. We call this the coproduct smooth structure on X + Y. Note that or Chen spaces the domains o the plots are convex and thus have only one connected component. So, in this case, ϕ is a plot in the disjoint union i and only i it actors throuh a plot in either X or Y. It is easy to check that with this smooth structure, i X and i Y are smooth. Moreover, or any other smooth space Q with smooth maps X : X Q and Y : Y Q, there exists a unique smooth map : X + Y Q such that X X i X Q X+Y i Y commutes. So, X + Y is indeed the coproduct o X and Y in the cateory C. Equalizers Given a pair,: X Y o smooth maps between smooth spaces, the set Z = {x X : (x) =(x)} X becomes a smooth space with its subspace smooth structure, and the inclusion i: Z X is the equalizer o and : Y Y Z i X Y. In other words, or any smooth space Q with a smooth map h X : Q X makin the ollowin diaram commute: Q h X X Y, there exists a unique smooth map h: Q Z such that i Z X h h 7 X Q commutes. Coequalizers Given a pair,: X Y o smooth maps between smooth spaces, the quotient Z = Y/((x) (x)) becomes a smooth space with its quotient smooth structure, and the quotient map p: Y Z is the coequalizer o and : Y X Y p Z.

12 5800 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG The universal property here is dual to that o the equalizer: just turn all the arrows around. Pullbacks Since C has products and equalizers, it also has pullbacks, also known as ibered products. Given a diaram o smooth maps X Y Z we equip the set X Z Y = {(x, y) X Y (x) =(y)} with its smooth structure as a subspace o the product X Y. The natural unctions p X : X Z Y X, p Y : X Z Y Y are then smooth, and it is easy to check that this diaram is a pullback square: X Z Y p X X p Y Y In other words, iven any commutative square o smooth maps like this: Z. h Y Q Y h X X Z, there exists a unique smooth map h: Q X Z Y makin the ollowin diaram commute: Q h h X h Y X Z Y p X X p Y Y More enerally, we can compute any limit o smooth spaces by takin the limit o the underlyin sets and endowin the result with a suitable smooth structure. This ollows rom Proposition 5.12, where we show that C has all small limits, toether with the act that the oretul unctor rom C to Set preserves limits, since it is the riht adjoint o the unctor equippin any set with its discrete smooth structure. Z.

13 CONVENIENT CATEGORIES OF SMOOTH SPACES 5801 Pushouts Since C has coproducts and coequalizers, it also has pushouts. Given a diaram o smooth maps Z X Y, we equip the set X + Z Y =(X + Y )/((z) (z)) with its smooth structure as a quotient space o the coproduct X + Y.The natural unctions i X : X X + Z Y, i Y : Y X + Z Y are then smooth, and in act this diaram is a pushout square: Z X Y i X i Y X+Z Y. The universal property here is dual to that o the pullback and can also be easily checked. More enerally, we can compute any limit o smooth spaces by takin the limit o the underlyin sets and endowin the result with a suitable smooth structure. This ollows rom Proposition 5.23, where we show that C has all small colimits, toether with the act that the oretul unctor rom C to Set preserves colimits, since it is the let adjoint o the unctor equippin any set with its indiscrete smooth structure. Mappin spaces Given smooth spaces X and Y,theset C (X, Y )={: X Y : is smooth} becomes a smooth space, where a unction ϕ: D C (X, Y )isaploti and only i the correspondin unction ϕ: D X Y iven by ϕ(x, y) = ϕ(x)(y) is smooth. With this smooth structure one can show that the natural map C (X Y,Z) C (X, C (Y,Z)) (x)(y) =(x, y) is smooth, with a smooth inverse. So, we say that the cateory C is Cartesian closed (see Deinition 5.15). Parametrized mappin spaces Mappin spaces are a special case o parametrized mappin spaces. Fix asmoothspaceb as our parameter space, or base space. Deine a smooth space over B to be a smooth space Y equipped with a smooth map p: Y

14 5802 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG B called the projection. For each point b B, deine the iber o Y over b to be the set Y b = {y Y : p(y) =b}, made into a smooth space with its subspace smooth structure. We can think o a smooth space over B as a primitive sort o bundle, without any requirement o local triviality. Note that iven smooth spaces X and Y over B, the pullback or ibered product X B Y is aain a smooth space over B. In act this is the product in a certain cateory o smooth spaces over B. I Y and Z are smooth spaces over B, let CB (Y,Z) = C (Y b,z b ). b B We make this into a smooth space, the parametrized mappin space, as ollows. First deine a unction p: CB (Y,Z) B sendin each element o C (Y b,z b )tob B. This will be the projection or the parametrized mappin space. Then, note that iven any smooth space X and any unction : X CB (Y,Z), we et a unction rom X to B, namelyp. I this is smooth we can deine the pullback smooth space X B Y. Then we can deine a unction : X B Y Z by (x, y) = (x)(y). This allows us to deine the smooth structure on CB (Y,Z): or any domain D, a unction ϕ: D CB (Y,Z) is a plot i and only i p ϕ is smooth and the correspondin unction ϕ: D B Y Z is smooth. With this smooth structure, one can check that p: CB (Y,Z) B is smooth. So, the parametrized mappin space is aain a smooth space over B. The point o the parametrized mappin space is that iven smooth spaces X, Y, Z over B, there is a natural isomorphism o smooth spaces CB (X B Y,Z) = CB (X, CB (Y,Z)). We summarize this by sayin that C is locally Cartesian closed (see Deinition 5.16). In the case where B is a point, this reduces to the act that C is Cartesian closed. The ollowin theorem subsumes most o the above remarks: Deinition 3.1. A quasitopos is a locally Cartesian closed cateory with inite colimits and a weak subobject classiier. Theorem 3.2. The cateory o smooth spaces, C, is a quasitopos with all (small) limits and colimits.

15 CONVENIENT CATEGORIES OF SMOOTH SPACES 5803 Proo. In Theorem 5.25 we show that this holds or any cateory o eneralized spaces, that is, any cateory o concrete sheaves on a concrete site. In Proposition 4.13 we prove that ChenSpace is equivalent to a cateory o this kind, and in Proposition 4.15 we show the same or DieoloicalSpace. 4. Smooth spaces as eneralized spaces The concept o a eneralized space was developed in the context o quasitopos theory by Antoine [1], Penon [34, 35] and Dubuc [11, 13]. Generalized spaces orm a natural ramework or studyin Chen spaces, dieoloical spaces, and even simplicial complexes. For us, a cateory o eneralized spaces will be a cateory o concrete sheaves over a concrete site. For a sel-contained treatment, we start by explainin some basic notions concernin sheaves and sites. We motivate all these notions with the example o Chen spaces, and in Proposition 4.13, we prove that Chen spaces are concrete sheaves on a concrete site. We also prove similar results or dieoloical spaces and simplicial complexes. We can deine sheaves on a cateory as soon as we have a ood notion o when a amily o morphisms : D i D covers an object D. For this, our cateory should be what is called a site. Usually a site is deined to be a cateory equipped with a Grothendieck topoloy. However, as emphasized by Johnstone [20], we can et away with less: it is enouh to use a Grothendieck pretopoloy, or coverae. The dierence is not very reat, since every coverae on a cateory determines a Grothendieck topoloy with the same sheaves. Coveraes are simpler to deine, and or our limited purposes they are easier to work with. So, we shall take a site to be a cateory equipped with a coverae. Two dierent coveraes may determine the same Grothendieck topoloy, but knowledeable readers can check that everythin we do depends only on the Grothendieck topoloy. Deinition 4.1. A amily is a collection o morphisms with common codomain. Deinition 4.2. A coverae on a cateory D is a unction assinin to each object D D a collection J (D) o amilies ( i : D i D i I) called coverin amilies, with the ollowin property: Given a coverin amily ( i : D i D i I) and a morphism : C D, there exists a coverin amily (h j : C j C j J) such that each morphism h j actors throuh some i. Deinition 4.3. A site is a cateory equipped with a coverae. We call the objects o a site domains. In Lemma 4.12 we describe a coverae on the cateory Chen, whose objects are convex sets and whose morphisms are smooth unctions. For this coverae, a coverin amily is just an open cover in the usual sense. This makes Chen into a site, and Chen spaces will be concrete sheaves on this site. To understand how this works, let us quickly review sheaves and then explain the concept o concreteness. Deinition 4.4. A preshea X on a cateory D is a unctor X : D op Set. For any object D D, we call the elements o X(D) plots in X with domain D. Usually the elements o X(D) are called sections o X over D. However, iven a Chen space X there is a preshea on Chen assinin to any convex set D the set X(D) o all plots ϕ: D X. So, it will uide our intuition to quite enerally call an object D D a domain and elements o X(D) plots.

16 5804 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Axiom 1 in the deinition o a Chen space is what ives us a contravariant unctor rom Chen to Set: it says that iven any morphism : C D in Chen, weeta unction X(): X(D) X(C) sendin any plot ϕ: D X to the plot ϕ : C X. Axiom 2 says that the resultin preshea on Chen is actually a shea: Deinition 4.5. Given a coverin amily ( i : D i D i I) ind and a preshea X : D op Set, a collection o plots {ϕ i X(D i ) i I} is called compatible i whenever : C D i and h: C D j make the ollowin diaram commute: C h D j j D i D, i then X()(ϕ i )=X(h)(ϕ j ). Deinition 4.6. Given a site D, a preshea X : D op Set is a shea i it satisies the ollowin condition: Given a coverin amily ( i : D i D i I) and a compatible collection o plots {ϕ i X(D i ) i I}, then there exists a unique plot ϕ X(D) such that X( i )(ϕ) =ϕ i or each i I. On any cateory, there is a special class o presheaves called the representable ones: Deinition 4.7. ApresheaX : D op Setiscalledrepresentable i it is naturally isomorphic to hom(,d): D op Set or some D D. The site Chen is subcanonical : Deinition 4.8. A site is subcanonical i every representable preshea on this site is a shea. We shall include this property in the deinition o a concrete site. But there is a much more important property that we shall also require. A Chen space X ives a special kind o shea on the site Chen: a concrete shea, rouhly meanin that or any D Chen, elements o X(D) are certain unctions rom the underlyin set o D to some ixed set. O course, this notion relies on the act that D has an underlyin set! The ollowin deinition ensures that this is the case or any object D in a concrete site. Deinition 4.9. A concrete site D is a subcanonical site with a terminal object 1 satisyin the ollowin conditions: The unctor hom(1, ): D Setisaithul. For each coverin amily ( i : D i D i I), the amily o unctions (hom(1, i ): hom(1,d i ) hom(1,d) i I) isjointly surjective, meanin that the union o their imaes is all o hom(1,d). Quite enerally, any object D in a cateory D with a terminal object has an underlyin set hom(1,d), oten called its set o points. The requirement that hom(1, ) be aithul says that two morphisms,: C D in D are equal when they induce

17 CONVENIENT CATEGORIES OF SMOOTH SPACES 5805 the same unctions rom points o C to points o D. In other words: objects have enouh points to distinuish morphisms. In this situation we can think o objects o D as sets equipped with extra structure. The second condition above then says that the underlyin amily o unctions o a coverin amily is itsel a coverin, in the sense o bein jointly surjective. Henceorth, we let D stand or a concrete site. Now we turn to the notion o concrete shea. There is a way to extract a set rom a shea on a concrete site. Namely, a shea X : D op Set ives a set X(1). In the case o a shea comin rom a Chen space, this is the set o one-point plots ϕ: 1 X. Axiom3impliesthatit is the underlyin set o the Chen space. Furthermore, or any shea X on a concrete site, there is a way to turn a plot ϕ X(D) into a unction ϕ rom hom(1,d)to X(1). To do this, set ϕ(d) =X(d)(ϕ). A simple computation shows that or the shea comin rom a Chen space, this process turns any plot into its underlyin unction. (See Proposition 4.13 or details.) In this example, we lose no inormation when passin rom ϕ to the unction ϕ: distinct plots have distinct underlyin unctions. The notion o concrete shea makes this idea precise quite enerally: Deinition Given a concrete site D, we say that a shea X : D op Set is concrete i or every object D D, the unction sendin plots ϕ X(D) to unctions ϕ: hom(1,d) X(1) is one-to-one. We can think o concrete sheaves as eneralized spaces, since they eneralize Chen spaces and dieoloical spaces. Every concrete site ives a cateory o eneralized spaces: Deinition Given a concrete site D, a eneralized space or D space is a concrete shea X : D op Set. A map between D spaces X, Y : D op Set is a natural transormation F : X Y. We deine DSpace to be the cateory o D spaces and the maps between them. Now let us ive some examples: Lemma Let Chen be the cateory whose objects are convex sets and whose morphisms are smooth unctions. The cateory Chen has a subcanonical coverae where (i j : C j C j J) is a coverin amily i and only i the convex sets C j C orm an open coverin o the convex set C R n with its usual subspace topoloy, and i j : C j C are the inclusions. Proo. Given such a coverin amily (i j : C j C j J) and : D C in Chen, then { 1 (i j (C j ))} is an open cover o D which actors throuh the amily i j as unctions on sets. We can reine this cover by convex open balls to obtain a coverin amily o D which actors throuh the amily i j in Chen. Since the covers are open covers in the usual sense, it is clear that the site is subcanonical. We henceorth consider Chen as a site with the above coverae. Since any 1-point convex set is a terminal object, Chen is a concrete site. This allows us to deine a kind o eneralized space called a Chen space ollowin Deinition Proposition A Chen space is the same as a Chen space. More precisely, the cateory o Chen spaces and smooth maps is equivalent to the cateory ChenSpace.

18 5806 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Proo. Let C stand or the cateory o Chen spaces and smooth maps. We bein by constructin unctors rom C to ChenSpace and back. To reduce conusion, just or now we use italics or objects and morphisms in C, and boldace or those in ChenSpace. First, iven X C, we construct a concrete shea X on Chen. For each convex set C, we deine X(C) to be the set o all plots ϕ: C X, andivenasmooth unction : C C between convex sets, we deine X(): X(C) X(C )as ollows: X()ϕ = ϕ. Axiom 1 in Chen s deinition uarantees that ϕ lies in X(C ), and it is easy to check that X is a preshea. Axiom 2 ensures that this preshea is a shea. To check that X is concrete, irst note that axiom 3 ives a bijection between the underlyin set o X and the set X(1), sendin any point x X to the one-point plot whose imae is x. Then, let ϕ X(C) and compute ϕ: hom(1,c) X(1) = X: ϕ(c) =X(c)(ϕ) =ϕ(c), where at the last step we identiy the smooth unction c hom(1,c) with the one point in its imae. So, ϕ is the underlyin unction o the plot ϕ. It ollows that the map sendin ϕ to ϕ is one-to-one, so X is concrete. Next, iven a smooth map : X Y between Chen spaces, we construct a natural transormation : X Y between the correspondin sheaves. For this, we deine C : X(C) Y (C) by C (ϕ) =ϕ. To show that is natural, we need the ollowin square to commute or any smooth unction : C C: X(C) C Y(C) X() X(C ) C Y() Y(C ). This just says that (ϕ) = (ϕ). We leave it to the reader to veriy that this construction deines a unctor rom C to ChenSpace. To construct a unctor in the other direction, we must irst construct a Chen space X rom any concrete shea X on Chen. For this we take X = X(1) as the underlyin set o the Chen space, and we take as plots in X with domain C all unctions o the orm ϕ, whereϕ X(C). Axiom 1 in the deinition o a Chen space ollows rom the act that X is a preshea. Axiom 2 ollows rom the act that X is a shea. Axiom 3 ollows rom the act that X = X(1). Next, we must construct a unction : X Y rom a natural transormation : X Y between concrete sheaves. For this we set = 1 : X(1) Y(1). Aain, we leave it to the reader to check that this construction deines a unctor.

19 CONVENIENT CATEGORIES OF SMOOTH SPACES 5807 Finally, we must check that the composite o these unctors in either order is naturally isomorphic to the identity. This is straihtorward in the case where we turn a Chen space X C into a concrete shea X and back into a Chen space. When we turn a concrete shea X into a Chen space X and back into a concrete shea X,wehave X (C) ={ϕ: C X(1)}, but the latter is naturally isomorphic to X(C) via the unction X(C) X (C) ϕ ϕ thanks to the act that X is concrete. Dieoloical spaces work similarly: Lemma Let Dieoloical be the cateory whose objects are open subsets o R n and whose morphisms are smooth maps. The cateory Dieoloical has a subcanonical coverae where (i j : U j U j J) is a coverin amily i and only i the open sets U j U orm an open coverin o the open set U R n,andi j : U j U are the inclusions. Proo. The proo is a simpler version o the proo or Chen, since we are considerin open but not necessarily convex sets. We henceorth treat Dieoloical as a site with this coverae. The one-point open subset o R 0 is a terminal object or Dieoloical, so this is a concrete site. As beore, we have: Proposition A dieoloical space is the same as a Dieoloical space. More precisely, the cateory o dieoloical spaces is equivalent to the cateory DieoloicalSpace. Proo. The proo o the correspondin statement or Chen spaces applies here as well. An example o a very dierent lavor is the cateory o simplicial complexes: Deinition An (abstract) simplicial complex is a set X toether with a amily K o nonempty inite subsets o X such that: (1) Every sinleton lies in K. (2) I S K and T S, thent K. A map o simplicial complexes :(X, K) (Y,L) isaunction : X Y such that S K implies (S) L. We can eometrically realize any simplicial complex (X, K) by turnin each n- element set S K into a eometrical (n 1)-simplex. Then axiom 1 above says that any point o X corresponds to a 0-simplex, while axiom 2 says that any ace o a simplex is aain a simplex. To view the cateory o simplicial complexes as a cateory o eneralized spaces, we use the ollowin site.

20 5808 JOHN C. BAEZ AND ALEXANDER E. HOFFNUNG Lemma Let F be the cateory with nonempty inite sets as objects and unctions as morphisms. There is a subcanonical coverae on F where or each object D in F there is exactly one coverin amily, consistin o all inclusions D D. Proo. Given a coverin amily ( i : D i D i I) and a unction : C D, each unction in a coverin amily havin C as codomain composed with clearly actors throuh some i. For instance, take i to be the identity unction on D. The coverae is clearly subcanonical since each coverin includes the identity morphism. Henceorth we make F into a concrete site with the above coverae. Since every coverin amily contains the identity, this coverae is vacuous : every preshea is a shea. Presheaves on F have been studied by Grandis under the name symmetric simplicial sets, since they resemble simplicial sets whose simplices have unordered vertices [17]. It turns out that concrete sheaves on F are simplicial complexes: Proposition The cateory o F spaces is equivalent to the cateory o simplical complexes. Proo. We deine a unctor rom the cateory o F spaces to the cateory o simplicial complexes. We use n to stand or an n-element set. Since the underlyin set hom(1,n)on F is naturally isomorphic to n, we shall not bother to distinuish between the two. Given an F space, that is, a concrete shea X: F op Set, we deine a simplicial complex (X, S) with X = X(1) and K = {imϕ ϕ X(n),n F}. To check axiom 1, we note that a point x X is a plot ϕ X(1), and {x} =imϕ K. To check axiom 2, we ix an object n, aplotϕ X(n) and a subset Y imϕ K. We consider ϕ 1 (Y ) n and let m be the object in F representin the inite set o cardinality ϕ 1 (Y ). There is an inclusion m ϕ 1 (Y ) n, and the commutativity o S(n) hom(n,s(1)) S(m) hom(m,s(1)) shows that Y is an element o K and that the structure deined is, in act, a simplicial complex. Given a natural transormation : X Y between F spaces we obtain a map = 1 : X(1) Y(1). By the commutativity o X(n) hom(n,x(1)) X(n) hom(n,y(1)) we see that this deines a map o simplicial complexes and this process clearly preserves identities and composition. Since a map : X Y o F spaces is completely determined by the unction 1 : X(1) Y(1) it is clear that this unctor is aithul. We see that the unctor is ull since iven a map o simplicial complexes :(X, K) (Y,L) and a morphism between inite sets j : m n, then the

21 CONVENIENT CATEGORIES OF SMOOTH SPACES 5809 naturality square X(n) X(j) X(m) (n) Y(n) (m) Y(m) Y(j) commutes, thus deinin a natural transormation between F spaces. We can also reverse the process described, takin a simplicial complex (X, K) and deinin an F space X whose imae is isomorphic to (X, K). For each n F, we let X(n) betheseton-element sets S K. The downward closure property o simplicial complexes uarantees that this is an F space, and it is easy to see that one can construct an isomorphism rom the imae o this F space under our unctor to (X, K). Thus, we have obtained an equivalence o cateories. 5. Convenient properties o eneralized spaces In this section we establish convenient properties o any cateory o eneralized spaces. We bein with some handy notation. In Section 4 we introduced three closely linked notions o underlyin set or underlyin unction in the context o a concrete site D. It will now be convenient, and we hope not conusin, to denote all three o these by an underline: The underlyin set o a domain: D =hom(1,d) Any concrete site D has an underlyin set unctor hom(1, ): D Set. Henceorth we denote this unctor by an underline: : D Set. So, any domain D D has an underlyin set D, and any morphism : C D in D has an underlyin unction : C D. The concreteness condition on D says that this underlyin set unctor is aithul. The underlyin set o a eneralized space: X = X(1) Any eneralized space X : D op Set has an underlyin set X(1). Henceorth we denote this set as X. Similarly, any map o eneralized spaces : X Y has an underlyin unction 1 : X(1) Y (1), which we henceorth write as : X Y. It is easy to check that these combine to ive an underlyin set unctor : DSpace Set. In Proposition 5.1 we show that this underlyin set unctor is also aithul. The underlyin unction o a plot: ϕ(d) =X(d)(ϕ) For any eneralized space X : D op Set, any plot ϕ X(D) has an underlyin unction ϕ: D X deined as above. The concreteness condition in the deinition o eneralized space says that the map rom plots to their underlyin unctions is one-to-one. One can check that this map deines a natural transormation : X(D) X D. Proposition 5.1. The underlyin set unctor : DSpace Set is aithul.