An informal introduction to topos theory

An informal in tro duction to top os theory T om Leinster ∗ Con ten ts 1 The definition of top os 3 2 T op oses and s et theory 7 3 T op oses and ge o metry 13 4 T op oses and uni v ersal algebra 22 This short tex t is for rea ders who are confident in basic categor y theory but know little or nothing ab out top oses. It is based on some impromptu talks given to a small gro up of ca tegory theor ists. I am no exp ert on top os theory . These notes are for p eople even less exp e r t than me. In keeping with the spirit of the talks, what follows is light on b o th deta il and references. F o r the rea der wis hing for more , almost everything here is pres ent ed in resp ectable form in Ma c Lane and Moer dijk’s very pleasant introductio n to top os theory ( 1994 ). Nothing here is new, not even the ex po sitory viewp oint (very lo o sely inspired by Johnstone ( 2003 )). As a rough indicatio n of the level of knowledge assumed, I will take it that you are totally comfor table with the Y o neda Lemma and the co ncept of cartesia n closed categor y , but I will not assume that you know the definitio n of s ubo b ject classifier or of top os. Section 1 expla ins the definition of top os . The r emaining three sections discuss some of the connections b etw een top os theory a nd other sub jects. There are ma ny more such connections than I will ment ion; I hop e it is abundantly clear that these notes are , b y design, a quick sk etch o f a large sub ject. Section 2 is on connections b etw een top os theor y and set theory . There are t wo themes here. One is that, using the language o f topo s es, we can wr ite ∗ Sc ho ol of Mathematics and Statistics, Univ ersity of Glasgow, Glasgow G12 8QW, UK; T om.Leinster@glasgo w.ac.uk. Supported by an EPSR C Adv anced Researc h F ellowship. 1 down an axiomatization of sets that stic ks closely to how sets are actually used in mathematics. This pr ovides an app ealing alter na tive to ZFC. The other, related, theme is that a t op os is a gener alize d c ate gory of sets. Section 3 is on co nnections with geo metr y (in a broa d sense); there the thought is that a top os is a gener alize d sp ac e. Section 4 is on connections with universal algebra: a t op os is a gener alize d the ory. What this mea ns is that there is one topo s embo dying the concept o f ‘ring’, an- other embo dying the c oncept of ‘field’, and s o on. This is the stor y of cla ssifying top oses. Sections 2 – 4 can be r ead in an y order, ex cept that ideally § 3 (g eometry) s hould come befor e § 4 (univer- sal alg ebra). Y ou c an r ead § 4 without having read § 3 , but the price to pay is that the notion of ‘geometric morphism’—defined in § 3 and used in § 4 —might se e m rather m ysterio us. 1 2 3 4    ❏ ❏ ❏ ❏ ❏ Algebraic geometers beware: the w ord ‘topos ’ is used by mathematicians in t wo slightly different senses, according to circumstance and cultur e . Ther e a re elementary top o ses and Grothendieck topos es. Ca tegory theor ists tend to use ‘top os’ to mean ‘elementary topo s’ by default, although Grothendieck top oses are a lso imp o r tant in category theory . But when an alge br aic g eometer says ‘top os’, they almost certa inly mean ‘Grothendieck t op os ’ (what else?). Grothendieck to po ses are ca tegories of sheav es. Elementary top os es are slightly more gener al, and the definition is simpler. They are wha t I will em- phasize her e. Grothendieck top oses are the sub ject of Section 3 , and app ear fleetingly elsewhere; but if you only wan t to learn a b o ut categories of sheav es, this is probably not the text for you. Ac knowledgement s I tha nk Andre i Akhvlediani, Eug enia Cheng, Richard Garner, Nick Gurski, Ig nacio Lop ez F r anco and Emily Riehl for their pa rtic- ipation and encoura g ement. Aspe c ts of Section 4 draw on a v aguely s imilar presentation of v aguely s imilar material b y Richard Garner . I thank the or - ganizers of Catego ry Theory 20 10 for making the talks p ossible, even though they did no t mean to: F r a ncesca Caglia ri, Eugenio Mog g i, Marco Grandis, San- dra Ma nt ov ani, Pino Rosolini, a nd Bob W alters. I thank Jon Phillips, Urs Schreiber, Mike Shulman, Alex Simpson, Danny Stevenson a nd T o dd Wilson for sugges tions and co rrections. I am esp ecially gr a teful to T o dd T rimble fo r carefully r eading an earlier version and sug g esting ma n y improvemen ts. The commutativ e diag rams w ere made using Paul T aylor’s macros. 2 1 The d efinition of top os The hardest part o f the definition of top os is the concept of sub ob ject classifier, so I will b egin ther e. F or motiv a tio n, I will sp ea k o f ‘the categor y of sets’ (and functions). What exactly this means will be discussed in Section 2 , but for no w we pr o ceed informally . In the ca tegory of sets, inv erse images a r e a sp e c ial case of pullba cks. That is, given a map f : X → Y of sets and a subs et B ⊆ Y , we hav e a pullback square f − 1 B > B X ∨ ∩ f > Y . ∨ ∩ In particular , this holds when B is a 1-element subset { y } of Y : f − 1 { y } > { y } X ∨ ∩ f > Y . ∨ ∩ There is no virtue in distinguishing b etw e en o ne-element s e ts, so we might as well write 1 instead of { y } ; then the inclusio n { y } ֒ → Y b ecomes the map 1 → Y picking out y ∈ Y , and w e hav e a pullback s quare f − 1 { y } ! > 1 X ∨ ∩ f > Y . y ∨ Next consider characteristic functions o f subsets. Fix a tw o- element set 2 = { t , f } (‘true’ and ‘false’). Then for an y s et X , the subsets of X ar e in bijectiv e corre s po ndence with the functions X → 2. In one direction, given a subset A ⊆ X , the co rresp onding function χ A : X → 2 is defined b y χ A ( x ) = ( t if x ∈ A f if x 6∈ A ( x ∈ X ). In the other, given a function χ : X → 2, the corres p o nding subset of X is χ − 1 { t } . T o say that this latter pro cess χ 7→ χ − 1 { t } is a bijection is to say that for all A ⊆ X , there is a unique function χ : X → 2 s uch that A = χ − 1 { t } . In other words: fo r a ll A ⊆ X , there is a unique function χ : X → 2 such that A ! > 1 X ∨ ∩ χ > 2 t ∨ 3 is a pullback s q uare. This proper t y o f sets can no w b e stated in purely categorical terms . W e use ֌ to indicate a mono (= monomor phism = monic). Definition 1 .1 Let E be a ca tegory with a terminal ob ject, 1. A sub ob ject classifier in E is a n ob ject Ω together with a ma p t : 1 → Ω such that for every mono A m ֌ X in E , there exists a unique ma p χ : X → Ω such that A ! > 1 X m ∨ ∨ χ > Ω t ∨ is a pullback s q uare. So, we ha ve just obser ved tha t Set has a subo b ject classifier, na mely , the t wo-element set. I n the gener al setting, we may write χ as χ A (or prop erly , χ m ) and call it the c haracteristic function of A (or m ). T o unders tand this further, we need t wo lemmas. Lemma 1.2 In any c ate gory, the pul lb ack of a mono is a mono. That is, if · > · · m ′ ∨ > · m ∨ is a pul lb ack squar e and m is a mono, then s o is m ′ . Lemma 1.3 In any c ate gory with a terminal obje ct 1 , every map out of 1 is a mono. So, pulling t : 1 → Ω ba ck a long any map X → Ω gives a mono int o X . It will also help to know the res ult of the following little e x ercise. It says, roughly , that in the definition of sub ob ject classifier, the f act that 1 is terminal comes for free. F act 1.4 L et E b e a c ate gory and let T t ֌ Ω b e a mono in E . Supp ose that for every mono A m ֌ X in E , ther e is a un ique map χ : X → Ω su ch that t her e is a pu l lb ack squar e A > T X m ∨ ∨ χ > Ω . t ∨ ∨ Then T is terminal in E . 4 This leads to a second descr iption of sub o b ject classifiers . Let M ono ( E ) be the categ ory whos e ob jects ar e monos in E and whose maps are pullback squares. Then a subob ject classifier is exactly a ter mina l ob ject of Mono ( E ). Here is a third way of lo ok ing at s ubo b ject classifier s. Given a c ategory E and a n ob ject X , a sub o b ject of X is officially a n isomorphism class of monos A m ֌ X (where isomor phism is taken in the slice categ ory E /X ). F or example, when E = Set , tw o mono s A m ֌ X , A ′ m ′ ֌ X are is omorphic if and o nly if they have the same imag e ; s o sub ob jects of X corres p o nd one-to-o ne with subsets of X . I say ‘officially’ because half the time peo ple use ‘subo b ject of X ’ to mean simply ‘mono in to X ’, or slip betw een the t wo meanings without w arning. It is a har mless abuse of language, whic h I will adopt. F or X ∈ E , let Sub( X ) b e the cla ss of sub ob jects (in the official s e nse) o f X . Assume that E is well-p owered, that is, each Sub( X ) is a set rather than a prop er class . Assume also that E has pullbacks. B y Lemma 1.2 , every map X f → Y in E induces a map Sub( Y ) f ∗ → Sub( X ) of sets, by pullback. This defines a functor Sub : E op → Se t . Third description: a sub ob ject classifier is a repres ent ation of this functor Sub. This makes intuitiv e sense, since for Sub to b e repres ent able means tha t there is an ob ject Ω ∈ E satis fying Sub( X ) ∼ = E ( X, Ω) naturally in X ∈ E . In the motiv ating cas e of the category of sets, this directly captures the thought that subsets o f a set X corr esp ond naturally to maps X → { t , f } . Now we s how that this is equiv alent to the origina l definition. By the Y o neda Lemma, a re presentation of Sub : E op → Set a mounts to an ob ject Ω ∈ E together with an element t ∈ Sub(Ω) that is ‘g eneric’ in the following sense: for every ob ject X ∈ E a nd element m ∈ Sub( X ), there is a unique map χ : X → Ω s uch that χ ∗ ( t ) = m . In other words, a representation of Sub is a mono T t ֌ Ω in E satisfying the condition in F a ct 1.4 . In other words, it is a sub ob ject class ifie r . Definition 1 .5 A top os (or e lementar y topos ) is a ca rtesian closed categ ory with finite limits and a sub ob ject classifier. Examples 1 . 6 i. The primordia l top os is Set . It has sp ecial pro p er ties not shared by most other topo ses. This is the sub ject of Section 2 . 5 ii. F or any set I , the catego ry Se t I of I - indexed families o f sets is a top os. Its sub o b ject cla ssifier is the constant family (2) i ∈ I , where 2 is a tw o-e le men t set. iii. F or any gr oup G , the ca tegory Set G of left G -s ets is a top os. Its subo b ject classifier is the set 2 with trivial G -action. iv. Encompassing all the previous examples, if A is any sma ll catego ry then the category b A = Set A op of presheav es o n A is a top os. W e can discov er what its sub ob ject classifier m ust b e by a thought ex pe r iment: if Ω is a sub o b ject classifier then by the Y oneda Lemma , Ω( a ) ∼ = b A ( A ( − , a ) , Ω) ∼ = Sub( A ( − , a )) for all a ∈ A . So Ω( a ) must b e the set of subfunctors of A ( − , a ); a nd one c an check that defining Ω( a ) in this w ay do es indeed g ive a subob ject classifier. A subfunctor of A ( − , a ) is ca lled a siev e on a ; it is a collection of maps into a satisfying a certain condition. v. F or any to p o logical spa ce S , the ca tegory Sh ( S ) of sheaves on S is a top os. This is the sub ject o f Section 3 . Mo dulo a small lie that I will come bac k to there, the spac e S can be rec overed from the topo s Sh ( S ). Hence the class o f spaces em b eds into the cla ss of topo ses, and this is why top oses can be viewed as generalized spaces. Sheav es will be defined and explained in Section 3 . T o give a brief sketc h: denote by Op en ( S ) the p o set of op en subsets of S ; then a pres heaf on the space S is a pr esheaf on the c ategory Op en ( S ), a nd a sheaf on S is a presheaf with a further pr o p erty . I will consistent ly use ‘sheaf ’ to mean what some w ould call ‘shea f of sets’. A sheaf of gr oups, rings, etc. is the same as an internal group, ring etc. in Sh ( S ). vi. The categor y FinSet of finite sets is a topo s. Similarly , Se t ca n be r e- placed by FinSet in all of the previous examples , giv ing top o ses of finite G -sets, finite sheaves, etc. Y ou might ask ‘why is the definition of top o s what it is? Why that p articular collection of axio ms? What’s the mo tiv ation?’ I will not a ttempt to a nswer, ex- cept by explaining several wa ys in which the definition has been found useful. It is als o worth no ting that the top os axioms have many non- obvious conseq uenc e s, giving topo ses a far richer s tr ucture tha n most catego ries. F o r example, every map in a topo s factorizes, ess entially uniquely , as an epi follow ed b y a mono. More spectacular ly , the axioms imply that ev ery topos has fin ite c o limits. This can b e proved by the following very elegant s trategy , due to Par ´ e ( 19 74 ). F o r every top o s E , we hav e the contrav a r iant p ow er set functor P = Ω ( − ) : E op → E . It can be shown that P is monadic. But monadic functors create limits, and E has finite limits. Hence E op has finite limits; that is, E has finite colimits. 6 2 T o p oses and set theory Here I will descr ib e what makes ‘the’ categor y o f sets sp ecia l among all top os es, and e x plain w hy I just put ‘the’ in q uotation marks. This is the stuff of rev- olution: it can completely c hange your view o f set theor y . It also pr ovides a n inv a lua ble insight into topos theory as a whole. W e b egin b y listing so me sp ecia l properties of the top os Set , using only the most commonplace assumptions ab o ut how sets and functions b ehav e. 1. The terminal o b ject 1 is a separator (generator). That is, given maps X f > g > Y in Set , if f ◦ x = g ◦ x for all x : 1 → X then f = g . It is worth dwelling on what this says. Maps 1 → X corres p o nd to el- ement s of X , and we make no notational distinction betw een the tw o. Moreov er, given a n ele ment x ∈ X a nd a map f : X → Y , we can com- po se the maps 1 x → X f → Y to obtain a map f ◦ x : 1 → Y , and this is the map co rresp onding to the element f ( x ) ∈ Y . (W e might harmlessly write b oth f ◦ x and f ( x ) a s f x .) Th us, elements are a sp ecial case of functions, and ev a luation is a sp ecial cas e of comp os ition. The prop erty above says that if f ( x ) = g ( x ) for all x ∈ X then f = g . In other words, a function is determined b y its effect o n elemen ts. 2. W rite 0 for the initial ob ject o f Set (the empty set). The n 0 6 ∼ = 1. Equiv- alently , Set is not equiv ale nt to the terminal categor y 1 . A top os satisfying prop er ties 1 and 2 is called well-p ointed . 3. This prop erty says, informally , that there is a set c onsisting of the na tur al nu mbers. What are the ‘the natur a l num b ers’, though? One wa y to g et at an a nswer is to use the pr inciple that s equences can b e defined re c ur sively . That is, given a set X , an element x ∈ X , and a map r : X → X , there is a unique sequence ( x n ) ∞ n =0 in X such that x 0 = x, x n +1 = r ( x n ) ( n ∈ N ) . (1) A sequence ( x n ) ∞ n =0 in X is just a map f : N → X , and if we w r ite s : N → N for the function n 7→ n + 1 (‘successor’), then ( 1 ) says exactly that the diagram N s > N 1 0 > X f ∨ r > x > X f ∨ (2) commutes. 7 Definition 2 .1 Let E be a category with a terminal ob ject, 1. A natural n um b e rs ob ject in E is a triple ( N , 0 , s ), with N ∈ E , 0 : 1 → N , and s : N → N , that is initial as such: fo r an y triple ( X , x, r ) o f the same t yp e, ther e is a unique map f : N → X such that ( 2 ) commut es (with N in place of N ). Prop erty 3 is, then, that Set has a natura l n umbers ob ject. 4. Epis split. That is, for any epimorphism (s ur jection) e : X → Y in Set , there exists a ma p m : Y → X such that e ◦ m = 1 Y . The splitting m chooses for each y ∈ Y an elemen t of the nonempty set e − 1 { y } . The existence of such splittings is precis ely the Axiom o f Choice. Genera lly , a category is said to sa tisfy the Axio m of Choice (or to ‘hav e Choic e ’) if epis split. In summary , sets and functions form a wel l-p ointe d t op os with n atur al num b ers obje ct and Choic e. The categor y of sets has many other elementary pr op erties (such as the fact tha t the sub ob ject classifier has exa ctly t wo elements), but they are all co ns equences of the prop erties just mentioned. But what is this thing c a lled ‘the categ o ry of sets’ ? What do we have to assume ab out sets in order to prove that these prop erties hold? Many mathematicians do not like to b e b othered with such questions, b e- cause they know that the standar d answ er will be so mething lik e ‘sets are any- thing satisfying the axioms o f ZFC’—and they feel that ZFC is irr elev ant to what they do, and prefer not to hea r ab out it. The standard answer is valid , in the sense that for every mo del of ZFC, there is a resulting category of sets satisfying the properties ab ov e. B ut it may seem irr elevant , b ecause at no p oint in es tablishing the prop erties did it feel necessary to call on a n axio m system: all the pro p e r ties ar e suggested dir ectly by the na ive ima gery of a set as a bag of dots. There is, how ever, another type of answer—and this was Lawvere’s radical idea. It is this: we take the pr op erties ab ove as our axioms on sets. In other words, we do aw ay with ZFC entirely , a nd as k ins tead tha t se ts a nd functions form a well-pointed top os with natural num b ers ob ject and Choice. ‘The’ catego r y of sets is an y ca tegory satisfying th ese axioms. In fact we should say a ca tegory of sets, since there may b e many different such categ ories, as we shall see. This is Lawv ere’s Elementary The o ry of the Categor y of Sets (ETCS), stated in mo dern language . (See Lawv ere ( 1964 ), or Lawv ere and Rosebrugh ( 2003 ) for a g o o d e x po sitory account.) It is near ly fift y years old, but still has not gained the currency it deserves, for reasons on whic h o ne ca n sp ecula te. 8 Digressio n Y ou mig ht be thinking that this is circular: that this axiomati- zation of sets depends on the notio n of ca tegory , and the notion of c a tegory depe nds on some notion of co lle ction o r set. But in fact, ETCS do es not de- pend on the general notion o f categor y . It can b e stated without using the word ‘category ’ o nce . T o see this, w e need to back up a bit. The ZFC axiomatization of sets lo ok s , informally , lik e this: • there ar e some things called ‘sets’ • there is a binary re la tion ‘ ∈ ’ on sets • some axioms hold. People seeing this (or the formal version) o ften ask certa in q uestions. What do es ‘some things’ mean? Do you mea n that there is a set o f sets? (No.) What exactly is meant by ‘binar y r elation’ ? (It means tha t for eac h set X and set Y , the s tatement ‘ X ∈ Y ’ is deemed to b e either true or false.) What do you mean, ‘deemed’ ? Etc. This is not a logic course, and I will not attempt to a nswer the questions ex cept to sa y tha t there is an assumed common unders tanding of these terms. T o hide b ehind ja rgon, ZFC is a first-order theory . The ETCS axioma tiza tion of sets loo ks like th is: • there ar e some things called ‘sets’ • for each set X and set Y , there ar e some things called ‘functions from X to Y ’ • for each set X , set Y and set Z , there is a bina ry oper ation as signing to each pair of functions f : X → Y , g : Y → Z a function g ◦ f : X → Z • some axioms hold. Y ou can a sk the same kind of log ical questio ns as for ZFC—what exactly is meant by ‘binar y op era tion’ ? etc.—which again I will not attempt to a nswer. The difficulties are no w ors e than for ZF C, and again, in the jargon, ETCS is a first-order theory . Stated in this wa y , the E TCS axioms be gin by saying that comp osition is asso ciative and has identities (so that sets, functions and compo sition of functions define a categ o ry); then they say that binary pro ducts and equalizers of sets exist, a nd there is a termina l set (so that the category of sets has finite limits); and so on, until we have said that sets and functions form a well-po inted top os with natural num b ers ob ject a nd Choice. Y ou ca n do it in ab out ten axioms. Here ends the digre s sion. 9 ZFC axiomatizes sets and mem b ership, whereas ETCS axiomatizes sets and functions. Anything that ca n be expressed in one la ng uage can b e expres sed in the other: in the usual implementation o f ZFC, a function X → Y is defined as a suita ble s ubs e t of X × Y , and in ETCS, an element of X is defined as a function from the terminal set to X . But an adv antage of the catego rical approach is that it av oids the chains of elements of elements of elements that are s o impor ta nt in traditional set theor y , yet s e em so distant from most of mathematics. ZFC is slight ly s tronger than E TCS. ‘Stronger ’ means that everything that can b e deduced a b o ut sets from the ETCS axioms ca n als o b e deduced in ZF C, but not vice versa. ‘Slig ht ly’ is meant in a so cio logical sens e. I believe it has bee n sa id that the mathematics in an o r dinary undergr aduate sylla bus (exclud- ing, naturally , any cour se in ZFC) mak es no more ass umptions ab out sets than are made by ETCS. If that is so, it must also be the case that for many math- ematicians, nothing in their entire research car eer requires more than ETCS. The tec hnica l relationship b etw een ZF C and ETCS is well understo o d. It is known exactly which fragment of ZFC is equiv ale nt to ETCS (namely , ‘b ounded’ or ‘restric ted’ Zer melo with Choice; see Mac Lane and Mo erdijk ( 19 9 4 )). It is also known what needs to b e a dded to ETCS in o rder to obtain a s ystem of equal strength to ZFC. This e xtra ingredient is a n ax io m scheme (a countably infinite family o f axioms) that set theo rists in the traditional mould would call Replacement, and categor y theorists would call a for m of co completeness . It says, informally , that given a ny set I a nd family ( X i ) i ∈ I of sets sp e cified b y a first-order formula, the copro duct P i ∈ I X i exists. The e xistence of this copr o d- uct is expressed by saying that there exist a set X and a map p : X → I (to b e thought of as the pr o jection P i ∈ I X i → I ) such that for each i ∈ I , the in verse image p − 1 { i } is isomor phic to X i . See Section 8 of McLarty ( 2004 ) for details. T op o s theor y therefor e pr ovides a different viewp oint on set theor y . Let us take a br ie f lo ok fr om this new vie w p o int at a famous theo rem of s et theory : that the Contin uum Hyp othesis is indep endent of the usual set-theor etic axioms, as prov ed b y G¨ odel and Cohen. T emp or arily , let us say that a ‘categor y o f sets’ is a w ell-p ointed topos with natural num b er s ob ject and Choice, satisfying the ax iom scheme of Replace - men t. A c a tegory of sets is sa id to satisfy the Con tin uum Hyp o thesis if f or all ob jects X , there exist monos N ֌ X ֌ 2 N = ⇒ X ∼ = N or X ∼ = 2 N . (As usual, N denotes the natural n umber s ob ject; 2 is the subo b ject classifier.) Stated catego rically , the theorem is this: g iven any catego ry of sets, y ou can build one that sa tisfies th e Contin uum Hypothesis and one that do es not. This is only a rephra sing of the standard statement, but if y ou are more a t home with the term ‘categor y’ than with ‘mo del o f a first-o rder theory’, you might find it less mysterious. So far we hav e seen the b enefits of viewing the/a catego r y o f s e ts as a sp ecia l 10 top os. But the other wa y r ound, there are gr eat b enefits to viewing a top os as a generalized categor y of sets. F or example, we migh t view Set N as the categor y of sets v arying thro ug h (discrete) time. The set of h uman being s alive today is an ob ject of Set N : a s the meaning of ‘to day’ c ha ng es, the set c hanges. A sheaf can similarly b e understo o d as a set v arying through spac e . People (esp ecially Lawvere) sometimes refer to the ca teg ory of sets as the (or a) to po s of c onstant sets, to con tras t it with top oses of v ariable sets. There are also top oses whos e ob jects can informa lly b e thoug ht of as ‘co hesive’ sets, which mea ns the following. In an ordina ry set, the points hav e no r e lation or a tta chment to each o ther: they do no t ‘co here’. But a co hesive set carries something like a top olo gy or smo oth str uc tur e, so that the points are in some sense stuck together. F or exa mple, there are top oses o f smo o th spaces, which are the setting for s ynth etic differential geometry . F rom this p oint of view, the category o f or dinary sets is e x treme a mong all topo s es: its ob jects ar e sets with no v ariatio n or cohesion at all. Viewing the ob jects of a top os a s genera lized sets is muc h more than a useful men tal tec hnique. In fa c t, it is v a lid to use set-like langua ge and re a soning in any topo s, provided that we stick to certain rules. This languag e is called the ‘int erna l lang uage’ of the topo s. Many of the central ideas of top os theory ar e simple, but that simplicity can easily be obscur ed b y the ric hness of structure a v a ila ble in a topos. Such is the case for the int erna l language. I will therefore describ e the idea in a mu ch more basic setting. First let E b e any ca tegory wha tso ever, and let A b e an ob ject of E . A g e n- eralized element of X is simply a map in E with co domain X . A g eneralized element x : S → X may be said to b e o f shap e S , or to b e an S -eleme nt of X . In the sp ecia l case that S is terminal, S -elements are called g lobal el ements . (See Example 3.2 ( iii ) for a hint on the reason for the name.) In the catego ry of sets, the global elements a re the o r dinary elemen ts, but in other categories , the global e le ment s might b e very uninteresting: consider the ca tegory of gr oups, for instance. Given a map f : X → Y in E , any generalized element x o f X gives r is e to a g eneralized element f x of Y . This is the comp osite f ◦ x , but can also be thought o f a s ‘ f ( x )’: see the rema rks on prop erty 1 at the b eginning of this section. F or maps X f > g > Y , we ha ve f = g ⇐ ⇒ f x = g x for all generalized elements x of X. (Pro of of ⇐ : take x = 1 X .) This is e mpha tically not true if we r e place ‘gener- alized’ by ‘g lobal’: a gain, consider groups. This language of generalized elemen ts is the in ternal l anguage of the cat- egory . It fits well with ordinary categorical terminolog y and notation. F or example, let E be a categ o ry w ith finite pro ducts. In the int erna l la nguage, the definition of pro duct r eads, infor mally: a n S -element of X × Y cons is ts of an S -e le ment of X tog ether with an S -element of Y . Apa r t from the ‘ S -’ 11 prefixes, this is identical to the ordina r y description o f the cartesian pr o duct of sets X and Y . And in standard categ o rical no tation, th e map S → X × Y with comp onents x : S → X and y : S → Y is denoted by ( x, y ), thus extending the set-theoretic notation for a (glo bal) elemen t of a car tes ian pro duct. T o s ee why the internal languag e is us eful, co nsider, for instance, internal groups in a finite pro duct ca tegory E . A group in E is an ob ject X together with maps m : X × X → X , i : X → X , e : 1 → X satisfying some axioms. Those axioms are usually express ed as co mm utative diagrams, whic h ha ve be e n obtained b y transla ting the class ical axio ms into di- agramma tic form. But there is no need to translate them: the classic al axioms can simply b e rep ea ted verbatim and interpreted as s tatements a b o ut gener- alize d elements. This is equiv alent. F or example, it is eas y to show that the commutativ e diag ram for asso ciativity is e quiv alent to the statement that m ( m ( x, y ) , z ) = m ( x, m ( y , z )) (3) for all generalize d elements x, y , z of X of the sa me shap e. (They ha ve to be the same shap e in o rder for expressions such as ( x, y ) to make sens e .) And just as for ordinary elements in Se t , there is no harm in wr iting xy instead of m ( x, y ), and similarly x − 1 instead of i ( x ). More v alua bly still, pr o ofs written down in the clas sical s et-theoretic sce- nario will actually be v alid in an arbitrary finite pro duct category E , as long as whatever was said ab out ele men ts in Set is a lso true for gener alized elements in E . F o r example, whenever X is a gr oup in Set and x, y , a ∈ X , w e have xa = y a = ⇒ x = y . (4) Pro of: xa = y a = ⇒ ( xa ) a − 1 = ( y a ) a − 1 = ⇒ x ( aa − 1 ) = y ( aa − 1 ) = ⇒ xe = y e = ⇒ x = y . W e can immediately co nclude that the implication ( 4 ) ho lds whenever X is a gr oup in a n arbitra ry finite pro duct ca tegory E and x, y , a are g eneralized elements o f X of the sa me shap e. Indeed, each step in the pro of is an application of an axiom such as ( 3 ) v alid in the g eneral setting. The in ter nal language is a massively lab our-saving device. T o prov e that an equation v alid in ordinary g roups is also v alid for internal groups, you merely need to cast an eye over the proof and convince yourself that it holds for gener- alized elements to o. In contrast, tr y proving the internal version of the equation y − 1 x − 1 = ( xy ) − 1 (5) by diagra mmatic metho ds . First it has to be state d diag rammatically . It says 12 that the diagr am X × X sym > X × X i × i > X × X X m ∨ i > X m ∨ commutes. Then it has to b e pr ove d , by filling the inside of this dia gram with instances of the diag rams enco ding the gr oup axioms. (It seems to need at least ten or so inner diagr ams.) But once you hav e an elemen twise pro o f, a ll this effort is unnecessar y . And the exa mple ( 5 ) chosen was v ery s imple: for more complex statements, th e b enefits of the internal language become clear er still. The internal lang uage of top oses is s imilar to that of finite pr o duct cate- gories, but m uch richer. As well as b eing able to form pair s ( x, y ) of g eneralized elements, w e can ta ke g eneralized elements o f exp onentials Y X (to be thought of as families of maps X → Y ), form subob jects suc h as { x ∈ X | f x = g x } (the equalizer of X f > g > Y ), and so on. Almost anything tha t can b e expr essed or prov ed in the category of sets can b e repro duced in an arbitr ary topo s. The only sticking p o int s are the la w o f the excluded middle and the a xiom of choice. An y pr o of that a voids those—any c onstructive pro of, in a sense that c a n b e made precise—gener alizes to an arbitrary topos . Phrases with more o r less the sa me meaning as ‘internal la nguage’ ar e ‘Mitc hell–B´ enabou lang uage’ and ‘internal logic ’. See , for instance, Mac Lane and Mo erdijk ( 1994 ) or Jo hnstone ( 2003 ). Ther e you can a lso find more s pe c ta cular applica- tions of top os theory to set theo r y , including topics suc h as forcing. 3 T o p oses and geometry This sec tion covers concepts such as sheaf, geometric morphism (map of top o s es), Grothendieck top os, and lo cale. But the most impor tant thing I wan t to e xplain is how a nd why g eometry has inspired so muc h of topo s theory . Shea ves Let X be a top ologica l space. (F ollowing tra ditio n, I will switch fro m my previ- ous con ven tion of using X to denote an ob ject of a to po s.) W rite Op en ( X ) for its pos et of o p en subsets. A presheaf on X is a functor F : Op en ( X ) op → Se t . It a ssigns to each o p en subset U a s et F ( U ), whose e lement s are called sectio ns o v er U (for reas ons to b e expla ined). It also assigns to each op en V ⊆ U a func- tion F ( U ) → F ( V ), called restriction from U to V and denoted by s 7→ s | V . I will write Psh ( X ) for the categ o ry of presheav es on X . 13 Examples 3 . 1 i. Let F ( U ) = { contin uo us functions U → R } ; restriction is restriction. ii. The same, but with ‘b ounded’ in pla c e of ‘contin uo us’. Examples ( i ) and ( ii ) are qualita tively different: contin uity is a lo cal prop- erty , but boundedness is not. This difference can be captured by asking the following question. Let ( U i ) i ∈ I be a family of op en subsets of X , and take, for each i ∈ I , a section s i ∈ F ( U i ). Might there be some s ∈ F ( S i ∈ I U i ) suc h that s | U i = s i for all i ? F or this to stand a chance of b eing tr ue, functor iality demands tha t the sections s i m ust satisfy a ‘matching condition’: s i | U i ∩ U j = s j | U i ∩ U j for all i and j . A sheaf is a presheaf such that for every family ( U i ) i ∈ I of op en sets and every matchin g family ( s i ) i ∈ I , there is a unique s ∈ F ( S i ∈ I U i ) suc h that s | U i = s i for all i ∈ I . Examples 3 . 2 i. The first exa mple ab ov e, with contin uo us functions, is a sheaf. The pro of can b e split into t wo pa rts. Given ( U i ) and ( s i ), there is certainly a unique function s : S U i → R (contin uous or not) suc h that s | U i = s i for all i . The question no w is whether s is contin uous; and bec ause contin uity is a lo cal prop erty , it is. ii. The seco nd example a bove, with b ounded functions, is not a sheaf (for a general space X ). This is be cause b oundedness is not a loc a l prop erty . iii. The sheaf of c o ntin uous re al-v alued functions is ra ther floppy , in the sense that there ar e usually many w ays to extend a contin uous function from a smaller set to a larger o ne. Often peo ple consider sheav es made up of holomorphic or ratio nal functions, which are muc h mor e rigid: there are t ypically few or no wa ys to ex tend. It is quite nor ma l for there to b e no global sections (sections over X ) at all. iv. T ake any contin uous map Y p → X of top olo gical spaces (which c a n b e thought o f as a kind o f bundle over X ). Then ther e aris es a shea f F on X , in which F ( U ) is the set of co nt inuous maps s : U → Y such that the triangle on the left c ommut es: Y U ⊂ > s > X p ∨ U U X Y ֒ → ↓ p ր s Such an s is precisely a r ight in verse, or ‘sectio n’, of the map p − 1 U → U induced by p . 14 There is also a n abstract catego rical explanation of where the concept o f sheaf comes from. Fix a space X . W e hav e a functor I : O p en ( X ) → T opSp / X where T opSp is the categor y of topolo gical spaces, T opSp /X is the slice cate- gory , and I ( U ) = ( U ֒ → X ). This functor I embo die s the simple thought that an o p e n subset of a topo logical space can b e treated as a space in its o wn right. W e now apply to I tw o v ery general categor ic a l constructions, from which the sheaf concept will app ear automa tically . First, purely b ecause the domain of I is small a nd the co domain ha s s ma ll colimits, there is an induced a djunction Psh ( X ) = Set Open ( X ) op −⊗ I > ⊥ < Hom( I , − ) T opSp /X . The right a djoint is giv en b y (Hom( I , Y ))( U ) = T opSp /X ( I ( U ) , Y ) where Y =   Y X p ∨   ∈ T opSp /X and U ∈ O p en ( X ). This is, in fact, the pro cess describ ed in Exa mple 3.2 ( iv ): the s heaf F defined there is Hom( I , Y ). The left adjoint can be describ ed as a co e nd or colimit: for F ∈ Psh ( X ), F ⊗ I = Z U F ( U ) × I ( U ) =   lim → U,s U  → X  where the co limit is ov er all U ∈ Op en ( X ) and s ∈ F ( U ), and the map from the colimit to X is the c a nonical one. Second, every adjunction res tr icts cano nically to an equiv a lence b etw een full sub c ategories : one consists o f th e ob jects at which the unit of the adjunction is an isomorphism, and the other of the ob jects at whic h the counit is an isomor- phism. W rite the equiv alence obtained from the adjunction ab ove a s Sh ( X ) > ≃ < Et ( X ) . It can b e shown tha t this Sh ( X ) is the sa me categor y of sheaves as b efore. In this way , the notion of shea f aris e s canonically fro m the very simple functor I : O p en ( X ) → T o pSp /X . The notion of ´ etale bundle also ar ises canonica lly : ´ etale bundles over X are (by definition, if you like) the ob jects of Et ( X ). Among other things, this equiv alence s hows that every sheaf is of the f or m described in Example 3.2 ( iv ). See Mac Lane and Moer dijk ( 1994 ) for details. One wa y o r another, we have the categor y Sh ( X ) of sheaves on X . It is a top os. Its s ubo b ject classifier Ω is given b y Ω( U ) = { op en subsets of U } . The cr ucial fact abo ut Sh ( X ) is that—mo dulo a small lie that I will repair later— 15 X c an b e r e c over e d fr om Sh ( X ) . So the class o f to po logical spaces embeds into the class of top os es. W e can think of topo ses as genera lized spaces. A common tec hnique in top os theory is to take a concept fro m top ology o r geometry and extend it to to p o ses. F or exa mple, s uppo se you hear someone talking ab out ‘connected topo ses’. Y o u may ha ve no idea what one is, but you can bet that the definition has b een obtained by determining what prop erty o f the to po s Sh ( X ) c orresp o nds to connectedness of the space X , then taking that as the definition of connectedness for all topo ses. The next few subsec tio ns are all examples of this gener alization proces s. Geometric morphisms So far I hav e said nothing ab out maps b etw een top oses. There is an obvious candidate for what a map of top os es should b e: a functor preser ving finite limits, exp onentials, and sub ob ject cla ssifiers. Such a functor is c a lled a logical morphism . They hav e a pa rt to play , but there is another notion o f map of top oses that has b een found muc h more useful. It can b e derived by gener alizing from top ology . Every map f : X → Y in T opSp induces an adjunction Sh ( X ) < f ∗ ⊥ f ∗ > Sh ( Y ) . (6) This is not obvious. The right a djoint f ∗ is easy to constr uct— ( f ∗ F )( V ) = F ( f − 1 V ) ( F ∈ Sh ( X ), V ∈ Op en ( Y ))—but the left adjoint f ∗ is harder . It can b e made easy by inv o king the equiv alence b etw een sheaves and ´ etale bundles; but I will not go int o that, or g ive an y other des cription of f ∗ . It is a fact that f ∗ preserves finite limits. It is also a fact (mo dulo the usual sma ll lie) that there is a natura l corre sp ondence b etw ee n co ntin uous maps X → Y and adjunctions ( 6 ) in which the left adjoint pr eserves finit e limits. So now we k now what contin uous maps lo o k like in top os- theoretic terms. W e duly generalize: Definition 3 .3 Let E and F b e top os es. A geom etric morphism f : E → F is an adjunction E < f ∗ ⊥ f ∗ > F in which the left adjoint f ∗ preserves finite limits. (People often say ‘left exact left adjoint’.) T he right adjoint f ∗ is ca lled the d i rect im age part o f f , and f ∗ is the in v erse image part. 16 I will wr ite T op os for the category o f top oses a nd g eometric morphisms. (Really it’s a 2 -categor y , in an o bvious wa y .) By co nstruction, we hav e a functor Sh : T opSp → T op os which is (2-categor ic ally) full a nd faithful, mo dulo the usua l small lie. Examples 3 . 4 i. Every functor f : C → D induces a string of adjoint func- tors b C f ! > ⊥ < f ∗ ⊥ f ∗ > b D betw een preshea f categ ories. Here f ∗ = − ◦ f , a nd f ! and f ∗ are left a nd right Kan extension alo ng f , resp ectively . Since f ∗ has a left adjoint, it preserves limits. Hence ( f ∗ , f ∗ ) is a geometric morphism b C → b D . ii. It turns o ut that, for a ny to po logical s pa ce X , the inclusio n Sh ( X ) ֒ → Psh ( X ) has a finite-limit-preserv ing left adjoint. It is ca lled sheafifica- tion or the ass o ciated sheaf functor. So the inclusion of sheav es into presheav es is a geometric morphism. Since Sh ( X ) is a ful l subca tegory , the inclusion is full and faithful; and for totally genera l reas o ns, this is equiv alent to the c ounit of the adjunction being a n isomo rphism. In other words, sheafifying a sheaf do es not change it. P oin ts Let us generalize another concept of top o logy . The points of a top ologic al space X corresp ond to the maps 1 → X (wher e 1 is the one-p oint space), which corr esp ond to the geometric mor phisms Sh (1) → Sh ( X ). But Sh (1) = Psh (1) = Set , s o we make the following definitio n. Definition 3 .5 A p oint o f a top os E is a geometric morphism Set → E . Em b eddings and Grothendiec k top oses F or a ny subspace Y of a spa ce X , the inc lus ion Y ֒ → X is an embeddi ng , that is, a ho meomorphism to its ima ge. It can b e shown that a map f : Y → X o f spaces is an embedding if and only if the direc t image part f ∗ of the corres p o nding geometr ic mor phism f : Sh ( Y ) → S h ( X ) is full and faithful. So, as usual, we g eneralize: Definition 3 .6 A g eometric mo rphism f : F → E is a n emb edding (or in- clusion ) if the direct image functor f ∗ is full and faithful. 17 W e then say that F is a subtop os of E . At least, this is the right thing to say up to eq uiv alence. Perhaps we should reserve that word for when F is actually a (full) sub categor y of E a nd f ∗ is the inclusion F ֒ → E , rather than allowing f ∗ to b e any o ld full a nd faithful functor . But a full and faithful functor induces an equiv a le nce to its image, so it makes no real difference. Probably the ea siest top oses are the presheaf top oses : those equiv alent to b C = Set C op for so me small categor y C . So maybe subtop os es of pres heaf top oses are relatively easy too. They hav e a spe cial name: Definition 3 .7 A to p o s is Grothendieck if it is (equiv alent to) a subtop os of some presheaf top os. F or instance , we saw in Example 3.4 ( ii ) tha t Sh ( X ) is a subtop o s of Psh ( X ) = \ Op en ( X ), for any top olo gical spa ce X . Hence Sh ( X ) is a Grothendieck to po s. Being Grothendieck is generally thought o f a s a mild condition o n a top os. A Grothendieck top o s has all small limits, whic h immediately disqualifies topo ses such as FinSet , FinSet C op , etc. But o ther than topos es ar ising fro m finite sets (or s ets s ub ject to s o me other cardina lit y bound), most o f the top ose s that p eople hav e worked with are Gro thendieck. A notable exce ptio n is the effective top os, the maps in which can be thoug ht of as computable functions. Other non-Grothendieck toposes o ccur in the top os-theo retic approach to non- standard analysis. There is a theorem of Gira ud giv ing a lis t of conditions on a category equiv- alent to it being a Grothendiec k topos. It includes non-elementary axioms such as ‘there is a small gener ating set’. (‘No n- elementary’ mea ns that it refers to a pre-existing notion o f set.) The Grothendieck topos es are sometimes reg arded as the nice topo ses, but p erhaps the definition of Gr othendieck topos is not as nice as the definition of elementary topos. Definition 3.7 is not the definition of Grothendieck topo s that you will find in most b o oks . I will now give a brief indica tio n of what the standard definition is and why it is equiv alen t to the one ab ove. Fix a small categ ory C . There is a one-to- one c o rresp ondenc e betw een the subtop oses of b C and the Grothendi ec k top ol ogies on C . A Grothendieck top ology is a kind of explicit, combinatorial structure; it sp ecifies which dia- grams . . . c i . . . c > c j > . . . in C a re to b e thought of as ‘covering families’ and whic h are not. (There a re axioms.) The motiv ating example is that g iven a to po logical space X , there 18 is a canonical Grothendieck top olog y o n O p en ( X ): a fa mily ( U i ֒ → U ) i ∈ I of subsets of U ∈ O p en ( X ) is cov ering if and only if U = S i ∈ I U i . The bijection { Grothendieck t op olo gies on C } ∼ = { subtop oses of b C } is written J ↔ Sh ( C , J ) . A pair ( C , J ), consisting o f a small c a tegory C equipp ed with a Grothendieck top ology J , is called a site , and Sh ( C , J ) is the categ ory of sheav es on tha t site. F or ex a mple, let X b e a topo logical space, ta ke C = Op en ( X ), and take J to b e the Gro thendieck topolog y mentioned ab ov e; then Sh ( C , J ) = Sh ( X ). Most bo oks pro ceed as follows: define Gro thendieck top ology , define site, define the category of sheav es on a site, then define a Grothendieck top o s to b e a category equiv alent to the category of sheav es on some site. I do no t know a s hort wa y to explain why the s ubto po ses of b C corr esp ond to the Grothendieck topo logies on C . The following t wo paragraphs may mak e it seem easier, or harder . First, ther e is an explicit classificatio n of t he subtoposes o f any top os E . In- deed, it can b e shown that the subto po ses of E cor resp ond to the ma ps j : Ω → Ω satisfying certa in equatio ns. (Such a j is called a La wvere–Tierney topol o gy on E , although this is so distant fr om the origina l usage of the w ord ‘top o logy’ that some p eople o b ject; Peter Johnstone , for instance, uses lo cal op erator instead.) By definitio n of sub ob ject cla ssifier, it is equiv a lent to s ay that a subtop os of E a mounts t o a subob ject of Ω satisfying certain axioms. Second, take E = b C . W e know (Example 1.6 ( iv )) that Ω ∈ b C is given by Ω( c ) = { sieves on c } . Hence a subtop os of b C corresp o nds to a collection of sieves in C , satisfying certain axioms . Calling these the ‘cov e r ing sieves’ gives the notion of Grothendieck topolog y . Lo cales Here I will explain the ‘small lie’ mentioned several times ab ove, and mak e amends. I will also explain why top os theorists ar e fond of jokes a b o ut p ointless top ology . The definition of sheaf on a topo logical space X doe s not mention the po int s of X . It mentions only the op en s ets and inclusio ns b etw een them, and use s the fact that it is p o ssible to take arbitrary unions and finite intersections of op en sets. Having obser ved this, you can see why the space X cannot always b e recov ered from the top os Sh ( X ). F o r instance, if X is indiscrete (has no op en sets except ∅ and X ) and no ne mpty , then Sh ( X ) is the sa me no matter how many p o in ts X ha s. The idea now is to split the pro ce s s X 7→ Sh ( X ) into t wo s teps. Fir s t, we forget the points of X , leaving just the se t of op en sets, ordered by inclusion. Then, we for m the ca tegory of ‘sheaves’ on that ordered set (defined a s for top ologica l spaces, almost verbatim). 19 Definition 3 .8 A frame is a partially ordered set such that every subset has a join (= least upp er b o und = sup), every finite subset has a meet (= grea test low e r b o und = inf ), and finite meets distribute over joins. A map of frames is a map preserv ing order, joins and finite meets. A top olog ic a l space X has a frame Op en ( X ) of op en subsets, and a con- tin uous map f : X → Y induces a map f − 1 : Op en ( Y ) → Op en ( X ) of frames. This gives a fu nctor Op en : T opSp → F rame op . W e now p erfor m a linguistic mano euvre. F rame op is the des ired catego r y of ‘po intless spa ces’. But we cannot wholehear tedly say that a fr a me is a p ointless space, because the maps of frames are the wrong wa y ro und. So w e in tr o duce a new w ord— l o cale —and define t he ca tegory Lo c of lo cales by Lo c = F rame op . W e ca n wholeheartedly say that a lo c ale is a po in tless space . There is a functor Sh : Lo c → T op os , defined ju st as for topologica l spa ces except that unions beco me joins and in tersec tio ns b eco me meets. The functor Sh : T opSp → T op os factorizes as T opSp Open → Lo c Sh → T o p os . This is the tw o -step pro cess mentioned a b ov e. Whenever I have said ‘mo dulo a small lie’, you ca n interpret that as ‘us e lo cales instead of topolo gical spaces’. F o r example, Sh : Lo c → T op os r e- ally is full and faithful, in a suitably up-to -isomor phis m sense: lo cale maps X → Y corr esp ond one- to -one with isomorphism classe s of g eometric morphisms Sh ( X ) → Sh ( Y ). This means that Lo c is equiv alent to a full sub ca tegory of T op os . (Actually it is an e q uiv alence of 2 - categor ies, but I will gloss ov er that po int.) Every lo c ale gives rise to a to po s—but the conv erse is a lso true. Given a top os E , the sub ob jects of 1 fo rm a p o set Sub E (1). Ass uming that E has enoug h colimits, Sub E (1) is a frame. This pro cess defines a functor T op os → Lo c E 7→ Sub E (1) . I am no w quietly c ha nging T op os to mea n the topose s with small colimits; this includes all Grothendieck topos es. Y ou might think that 1 could ha ve no interesting sub ob jects, since that is the case in the most obvious topos , Set . But ther e are top os e s that ar e nearly as ob vious in whic h Sub E (1) is not tr ivial. F or instance, tak e E = Se t I for an y set I : then Sub E (1) is the p ow er set of I . Now a wonderful thing is tr ue. The functor just defined is left adjoint to the inclusion Sh : Lo c ֒ → T op os . This means that Lo c is (equiv alent to) a r efle ctive sub catego ry of T op os . Hence the counit is an isomor phism: X ∼ = Sub Sh ( X ) (1) 20 for any locale X . This is how you recov er a locale fro m its to p o s of sheav es. So Lo c sits inside T op os a s a sub categor y of the b est kind: full and re - flective, like abelian groups in groups. It is reaso nable to say that a lo cale is a sp ecial s o rt o f top o s. More formally , a top os is lo cali c if it is of the form Sh ( X ) for some lo cale X . Lo calic top oses are easy to work with; if you were having trouble proving something for ar bitr ary to po ses, you might start b y trying to prov e it in this specia l case. Since e very lo c ale is of the form Sub E (1) fo r so me top os E , loca le theory can be re garded as the fragment o f topos theor y concerning sub ob jects of 1. A sub o b ject of 1 is a map 1 → Ω, which can reas o nably called a truth v alue. In that sense, lo cale theory is the s tudy of truth v alues. The notion of lo c ale can also be seen a s a decategor ifica tion of the notion of Gr othendieck top o s. A p ose t P is a category enriched in the tw o-element totally o rdered set 2 . There is a Y o neda embedding P → 2 P op , which has a finite-meet-preserv ing left adjoint if a nd only if P is a frame. Analogously , it is almost true that for a ca tegory E , the Y o neda embedding E → Set E op has a finite-limit-preserving left adjoint if and only if E is a Grothendieck top os. (This result is due to Street ( 1981 ). ‘Almo s t’ refers to a se t-theoretic size condition.) A map of fra mes is a function preser ving joins and finite meets, a nd the inv erse image par t of a geo metric morphism is a functor preserving colimits and finite limits. Thus, locales pla y roughly the same role among 2-enriched categories as Grothendieck toposes play among Set -enriched catego ries. How muc h has b een lost by passing fr om top olog ical spa ces to loca les? In most pe ople’s view, no t muc h. F or e x ample, we observed that all nonempty indiscrete spaces give ris e to the sa me lo cale; but man y mathematicians regar d indiscrete spaces with ≥ 2 p oints as ‘pathologica l’ and would b e p ositively happy to see them go. In fact, some things ar e g ained. F or example, a s ubgroup o f a topolo gi- cal gr oup need not b e closed, and non-closed subgr oups ar e often regar ded a s pathologica l (since the corres p o nding quotient s are non-Haus dorff ). But it is a theor e m that every subg roup of a lo c alic g r oup is clos ed. See for instance Section C5.3 of Johnstone ( 2003 ). The functor Op en : T o pSp → Lo c has a r ight adjoint, which I will not describ e. As men tioned on page 15 , ev ery adjunction restricts canonica lly to an equiv alence b etw ee n full sub categ ories. In this case, this gives an equiv alence betw een: • a full sub categ ory of T opSp , whose ob jects are calle d the s o b er spaces • a full sub categ ory of Lo c , whose ob jects are called the sp atial lo ca les. Another wa y of in terpreting the phr a se ‘modulo a small lie’ is ‘true for sob er spaces’. Sobriet y amounts to a rather mild separatio n condition. F or example, every Hausdorff space is so b e r. So in passing fr om a Hausdo rff space to a lo ca le, or to a top os, nothing whatso ever is lost. There is a kind of attitudinal para dox here. Ma n y algebr a ic top o logists think only ab out Hausdor ff spaces, and r egard non-Ha us dorff spaces as pa thological. 21 But these are often the same p eople who feel s trongly that top olog ical spaces are not r eally a bo ut op en se ts ; they think in ter ms of po ints and paths and homotopies. So it is pe rhaps par adoxical that the Hausdorff condition guaran- tees that a spa ce can b e unders to o d in terms of its o p en sets a lone: the top os of sheav es depends on nothing else, and co nt ains all the information ab out the original space. 4 T o p oses and univ ersal algebra The point of this section is to explain what p eo ple mean when they talk abo ut the cla ssifying top os of a theory . Ano ther wa y to lo ok at it is this: I will explain how to po ses can b e viewed as cousins of oper a ds and Lawv ere theories. In clas sical universal alg ebra, a n algebr aic theory (or strictly , a presen tation of an algebra ic theo ry) co ns ists of a bunch of o pe ration s ymbols of sp ecified arities, together with a bunc h of e q uations. T o take the s tandard example, the (usual presentation of the) theory of groups consists of • an op era tion sym b ol 1 of arity 0 • an op era tion sym b ol ( ) − 1 of arity 1 • an op era tion sym b ol · of ar ity 2 together with the usual equatio ns. Y o u can sp eak of ‘mo dels’ of an alge braic theory in any categor y E with finite pro ducts. In our example, they a re the int erna l gr oups in E . But there ar e other w ays o f lo oking at such theories. Consider the free finite pr o duct categ ory T e quippe d w ith a n internal gr o up. (There a re ge ne r al rea sons why such a thing must exist.) I ts univ ersa l prop erty is that for any finite pro duct category E , the finite- pr o duct-preser ving functors T → E c o rresp ond to the internal groups in E . Concretely , T lo oks something like this. It must cont ain an ob ject X , the underlying ob ject of the internal gr oup. Since T has finite pro ducts, it must also contain an ob ject X n for ea ch n ∈ N . There is no reaso n for it to hav e any other o b jects, and since it is free, it do es not. A map X n → X m is (b y definition of pro duct) an m -tuple of maps X n → X ; and the maps X n → X are (by fr eeness) whatever maps G n → G must exist for any internal g roup G in any finite pr o duct categ ory . T ha t is, they are the n -ary op erations in the theory of groups: the words in n letters. This category T is called the La wv ere theory of g roups . The same goes for rings, lattices, e tc. In a ll these cases, T is a finite pro duct category with the further prop er ty that the ob jects are in bijection with the natural n umbers, the pro duct of o b jects corr esp onding to addition of n umbers. This further property holds b ecause the theo ries describ ed so far have been single-sorted: a mo del is a single ob ject equipp ed with s ome structure. But ther e a re a lso ma ny-sorted theories, such a s the tw o-sor ted theory o f pairs ( R, M ) in which R is a ring a nd M an R -mo dule. So we can widen the 22 notion of algebr aic theory to include all (small) finite pro duct categories . Some peo ple say that an algebra ic theo r y is just a finite pro duct categor y . O thers say that algebraic theories c orr esp ond to finite pro duct categories . Others still, more traditionally , say that algebraic theories cor resp ond to only c ertain finite pro duct categorie s. T er minology aside, we ca n pla y the same game for other clas ses o f limit. F o r example, it makes no sens e to talk ab out internal ca tegories in an arbitra ry finite pro duct categ ory , b ecause the definition of internal catego ry needs pullbacks. (Comp o sition in an internal catego ry C is a map C 1 × C 0 C 1 → C 1 .) But w e can talk a b out internal ca tegories in a finite limit category; a nd as b efore, there is a fr e e finite limit categor y T equipp ed with an internal ca tegory . This mea ns that for an y finite limit categ ory E , the finite-limit-preserving functors T → E corres p o nd to the in terna l categ ories in E . A small finite limit ca tegory is ca lled (or corre s po nds to ) an essentially algebraic theory . In a categ ory with finite pr o ducts you can talk ab out internal groups but not, in general, in ternal categories. In a categor y with finit e limits y o u ca n talk ab out b oth. By extending the list of prop erties that the categor y is ass umed to sa tisfy , you can accommo da te more and more so phisticated k inds of theory . (The theory o f internal categor ies is more ‘sophisticated’ than that o f groups in the s ense that co mpo sition is only defined for some pair s of ma ps, wherea s classical universal algebr a ca n only handle op eratio ns defined o n al l pair s.) The prop erties need not b e of the form ‘limits o f such-and-such a type ex ist’. F or example, it is sometimes useful to assume epi-mono factor iz a tion, as we shall see. There is a trade-o ff here. As you allow more sophisticated language, you widen the class o f theories that can b e expressed, but you nar row the class o f categorie s in which it makes sense to take mo dels. (Y o u a lso make more work for yourself.) In the sa me way , if you trade in your motorbike for a double-deck er bus, you incr ease the num b er of passe ngers you can carr y , but you restr ict where you can car ry them: no low br idg es or tigh t alleywa ys. (Y ou also increa s e y our fuel costs.) It is se nsible, then, to use the smallest cla ss of theor ies containing the ones you a r e interested in. F or exa mple, you c ould trea t gro ups as an essentially algebraic theor y , but that would mea n you could o nly take mo dels in categor ies with al l finite limits, when in fact just pro ducts w ould do . Before I get onto to po ses, I wan t to p oint out a slightly different direction that you ca n take things in. Rather tha n just alter ing the pr op erties that the categorie s are assumed to have, you can a lso a lter the structur e with which they are equipped. T ake mo noidal categories , for instance . W e ca n sp eak of internal monoids in any monoidal categ o ry . Hence, the theory of monoids c a n be regar ded as the free monoidal category con taining an in ternal monoid. (This is in fac t the category of finite ordina ls.) Similarly , it makes sense to sp eak of algebra s for an op er a d P in an y monoidal category , and we can asso ciate to P the free monoidal category T c o ntaining a P -algebr a. Thus, for any monoidal category E , mono idal functors T → E corres po nd to P -algebra s in E . W e mig ht define a monoi dal theory to b e a small monoidal categor y . This 23 gets us into the territor y o f PROPs, where there are non trivia l theorems such as the classification of 2-dimensional topo logical quantum field theor ies: the symmetric monoidal theory of (o r, ‘P R OP for’) co mmu tative F r ob enius alge br as is the categ ory of smo oth 1 -manifolds and diffeomo r phism classes of cob o rdisms. All of this is to give an impression of how far -reaching these ideas are. It is a sketc h of the context in whic h clas sifying topos es can be understo o d. Y ou will hav e guessed tha t the same kind of thing can b e sa id for top oses as for categories with finite products, finite limits, etc. Since topos es ha ve very rich structure ( much more than just finite limits), they corres po nd to a very wide class of theories indeed. An example of the kind of theory that can be interpreted in a top o s is the theory of fields. (This is rather a feeble example, but I w a nt to k eep it simple.) A field is, of cours e, a comm utative ring R satisfying the axioms 0 6 = 1 (7) and ∀ x ∈ R , x = 0 or ∃ y : xy = 1 . (8) By a mechanical pr o cess, this definition can b e turned into a definition of ‘in- ternal field in a top os’. As comp ensa tion for the imprecision of the res t of this section, I will give the definition in detail; but if y ou wan t to sk ip it, the p oint to retain is that it is a mechanical process . Let E b e a topos. W e certainly know how to define ‘commutativ e r ing in E ’: that makes sense in any categ ory with finit e products. Let R be a commutativ e ring in E . The non trivia lit y axiom, 0 6 = 1, is expres sed by saying that the equalizer of 1 0 > 1 > R is the initial o b ject 0. F or the other a xiom, let us first define the sub ob ject U ֌ R consisting of the units (inv ertible elemen ts). The ‘set’ P = { ( x, y ) | xy = 1 } is the pullback P > 1 R × R ∨ ∨ · > R. 1 ∨ ∨ Now we w ant to define the ‘set’ U of units a s the image of the comp osite ma p f =  P ֌ R × R pr 1 → R  . W e can talk ab out ima ges in a top os, since every ma p in a top os factorizes essentially uniquely as an epi fo llow e d by a mono. So, define U ֌ R by the factorization f = ( P ։ U ֌ R ) . 24 The seco nd field axio m states that every element of R lies in either the sub ob ject 1 0 ֌ R or the sub ob ject U ֌ R . In other words, it states that the map 1 + U → R is epi. Here w e hav e used the fact that every topos has co pro ducts, written +. If you hav e read Section 2 , you will recognize that the informal talk of ‘sets’ (r eally , ob jects of E ) and the us e of set-theoretic notation { . . . | . . . } are something to do with the in ter nal la nguage of a topos. This gives a hint o f ho w the pro cess can b e mechanized. (There are actually sever al pos sible theories of fields, depending on exac tly how you wr ite down the a xioms. They all have the s ame mo dels in Set —namely , fields—but they do not hav e the sa me mo dels in other top os es. F or example, a genuinely different theor y is o bta ined b y c hanging axiom ( 8 ) to ‘ ∀ x ∈ R , ( 6 ∃ y : xy = 1) = ⇒ x = 0 ’. But this do es not affect the main p oint: given a list of formally-expressed axioms such a s ( 7 ) and ( 8 ), there is an automatic pro ce s s conv erting it in to a definition that makes sense in an arbitra ry topos.) Y ou now hav e the choice betw een a shor t story and a long story . The short story is that what we did for finite pro duct and finite limit cat- egories ca n a lso b e done for top oses. The theories c o rresp onding to topos es are ca lled the geometric theo ries, and the top os c orresp o nding to a particular geometric theory is called its class ifying topo s. The long story is lo nger b ecause there are t wo different notio ns o f map o f top oses—and y ou need to decide what a map of topo s es is in order to state the universal pr op erty o f the top os resulting from a theory . The mo r e obvious but less used no tion of map of top oses is a functor pre- serving all the structure in sight: finite limits, expo nentials, and the subo b ject classifier. These a r e called logical morphis m s . Now in a top os, you can in- terpret a really v ast range of theories: any ‘higher-order theory’, in fact. (First order means that you can only q uantif y over elements of a set; in a seco nd order theory y ou can also quantif y ov er subsets of a set; and so on.) Mo d- els of any such theory g et along well with logica l morphisms, b eca use lo g ical morphisms preser ve everything. So you can tell a simila r story for topose s, logical morphisms and higher order theories a s for finite pro duct categor ies, finite-pro duct-preser ving functors and algebraic theories. The mor e po pular notion of map of topo ses is that of geometric mor phism. (Here it helps to have read Section 3 , whe r e the definition is motiv ated.) A ge- ometric morphi sm b etw een topos e s is a functor w ith a finite-limit-pres e rving left adjo int. The co rresp onding theorie s are the geome tric theories . I will not giv e the definition, but it is not too bad an approximation to say that they are the s a me as the fir s t-order theories: every g eometric theory is first-order , and almost every fir st-order theory that o ne encount ers is geo metric. Given a geo metric theo r y , a classi fying top os for the theor y is a co complete top os T with the prop erty that for any co complete top os E , mo de ls of the theory in E c orresp o nd naturally to geometric morphisms E → T . Every geometric theory has a classifying top o s. 25 There a re tw o surprises here. O ne is the a ppe arance o f the word ‘co com- plete’, which I will not ex pla in and will not b other inserting b elow. It is gen- erally thoug ht of a s a mild condition (sa tisfied b y any Grothendieck topo s, for instance). The bigger surprise is the reversal of dire ction. The previous ca ses lead us to exp ect mo dels in E to cor resp ond to maps T → E . Howev e r , since a geometric morphism is a pair of adjoint functors , the choice of direction is a matter of conv e nt ion. As the na me sug gests, the choice that so ciety made was motiv ated b y geometr y . Perhaps if the motiv ation had been universal alg ebra, it would have b een the other w ay ro und. (This is an asp ect o f the thought that g e ometry is dual to algebra .) A map of top oses would then have b een a finite-limit-preserving functor with a rig ht adjoint, whic h is mo re or less the same thing as a functor preser ving finite limits and small colimits. If a top o s is thought of as a generalize d spa ce (as in Section 3 ) then the classifying top os of a theor y ca n be thought of as its space of models . Indeed, a p oint of the classifying top os T is (b y Definition 3.5 ) a geometric mor phism Set → T , which is exactly a mo del of the theory in Set . Some familiar top o- logical spaces can be construed as classifying to po ses. F o r example, there is a ‘theory of Dedekind cuts’ whose classifying top os is Sh ( R ), that is, R reg arded as a top os. Given how muc h s tructure a top os contains, it is s urprising how many cla ssi- fying topo ses can be describ ed simply . I will now desc r ib e the classifying top os of any algebraic theory , b y the venerable exp o s itory device of doing it just for groups. W e will need the notion of finite present ability . A group (in Set ) is finitely presen table if it admits a presentation by a finite set of generators sub ject to a finite set of relations. The category of finitely presentable groups and a ll homomorphisms b etw een them will be written Grp fp . Aside Finite presentabilit y is a mor e categorical concept tha n it might seem. W riting T : Set → Set fo r the free group monad, a re lation (equation) in a set X of g e ne r ators is an element of T X × T X . So , a family ( r i ) i ∈ I of r e lations is a map I → T X × T X , or equiv alently a diagram I ⇒ T X in Set , or equiv ale ntly a diagram F I ⇒ F X in Grp , wher e F : Set → Gr p is the fr ee g roup functor. The group presented by these generato rs and relations is the co equalizer of this diagra m in Grp . Hence a group is finitely presen table pr ecisely when it is the co equalizer of some diagram F I ⇒ F X in which I and X ar e finite sets. This for mulation of finite presentabilit y in Grp uses the free gr oup functor F . But in fact, ther e is a ge ner al definition o f finite pre s entabilit y of a n ob ject of any ca tegory . I will not go into this. 26 As promised, the clas sifying topos for groups is easy to describ e: Theorem 4.1 The classifying top os for gr oups is Set Grp fp . In other w or ds , for an y top os E , a gro up in E is the same thing as a geo metr ic morphism E → Set Grp fp . The same go es for o ther algebr aic theories . This yields so mething interesting even fo r very triv ia l theor ie s. T ake the theo ry of ob jects, whose mo dels in a category E are simply ob jects of E . A finitely presentable set is just a finite set. Hence for any top o s E , ob jects o f E corre sp ond to geometric morphisms E → Set FinSet . The topo s Set FinSet is therefore called the o b ject classi fier . W e ha ve b een asking, for a given theory , ‘what topo s classifies it?’ But we can turn the question round and ask, for a given top os T , ‘wha t do e s T classify?’ In other w ords, what are the geometric morphisms from an arbitra ry top os E in to T ? It is a fa ct that every top os T is the classifying top o s of some geometric theory—althoug h given how wide a class of theories that is, per ha ps this do es not say v ery m uch. There ar e clean answers to this r eversed question for many top oses T . In particular, this is so when T is the topo s Sh ( C , J ) of sheav e s o n a site (Sec- tion 3 ). Her e I will just tell you the answer for a sma ller class of top oses. Theorem 4.2 Le t C b e a c ate gory with fin ite limits. Then the pr eshe af top os b C classifies fin ite-limit-pr eserving functors out of C . In other words, for a ny top os E , a g e ometric mo rphism E → b C is the same thing as a finite-limit-preser ving functor C → E . (If you know a b o ut flat functors, you can dro p the ass umption that C has finite limits: for any small categ o ry C , the presheaf to po s b C classifies flat functors out of C . T his is one v ers io n of Diaconescu’s Theorem.) So there is a back-and-forth translation b etw een geometric theo r ies and the topo ses that cla s sify them. In ma ny cases, this transla tion is sur pr isingly straightforward. References Johnstone, P . T., 2003. Sketches of an Elephant: A T op os Theory Compendium . Oxford Logic Guides. Oxford Un iversit y Press. La wvere, F. W., 1964. An elemen tary theory of the category of sets. Pr o c e e dings of the National A c ademy of Scienc es of the U.S.A. 52 :1506–1511 . Reprinted as R eprints in The ory and Appli c ations of Cate gories 12:1–35, 2005. La wvere, F. W. and R . Rosebrugh, 2003. Sets for Mathematics. Cam bridge Universit y Press, C ambridge. Mac Lane, S . and I. Mo erdijk, 1994. Sheav es in Geometry and Logic. Springer, Berlin. McLart y , C., 2004. Ex ploring categorical structuralism. Philosophia Mathematic a 12 :37–53 . P ar´ e, R., 1974. Colimits in top oi. Bul letin of the A meric an Mathematic al So ciety 80 :556–5 61. Street, R ., 1981. Notions of top os. Bul letin of the Aust r alian Mathematic al So ciety 23 :199–2 08. 27