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Domenico Fiorenza, Elena Martinengo, A short note on -groupoids and the period map for projective manifolds, Publications of the nLab vol. 2 no. 1 (2012) arXiv:0911.3845
Dipartimento di Matematica - Sapienza, Università di Roma; P.le Aldo Moro 5, I-00185 Roma Italy - fiorenza@mat.uniroma1.it
Institut für Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany - elenamartinengo@gmail.com
We show how several classical results on the infinitesimal behaviour of the period map for smooth projective manifolds can be read in a natural and unified way within the framework of ∞-categories.
A common criticism of ∞-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naïve intuition of what an ∞-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths’ expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.
The use of the language of ∞-categories should not be looked at as providing new proofs for these results; namely, up to a change in language, our proofs verbatim reproduce arguments from the recent literature on the subject, particularly from the work of Marco Manetti and collaborators on dglas in deformation theory. Rather, by this change of language we change our point of view on the classical theorems above: in the perspective of ∞-sheaves from Lu09a, all these theorems have a very simple local nature which can be naturally expressed in terms of ∞-groupoids (or, equivalently, of dglas); their classical global counterparts are then obtained by taking derived global sections. It is worth remarking that, if one prefers proofs which do not rely on the abstract machinery of ∞-categories, one can rework the arguments of this note in purely classical terms. Namely, once the abstract -nonsense has suggested the “correct” local dglas, one can globalize them by means of an explicit model for the derived global sections, e.g., via resolutions by fine sheaves as in FM09, or by the Thom-Sullivan-Whitney model as in IM10.
Since most of the statements and constructions we recall in the paper are well known in the -categorical folklore, despite our efforts in giving credit, it is not unlikely we may have misattributed a few of the results; we sincerely apologize for this. We thank the referee for accurate remarks which helped us a lot in improving the present paper, and Ezra Getzler, Donatella Iacono, Marco Manetti, Jonathan Pridham, Carlos Simpson, Jim Stasheff, Bruno Vallette, Gabriele Vezzosi, and the Lab for suggestions and several inspiring conversations on the subject of this paper.
Through the whole paper, is a fixed characteristic zero field, all algebras are defined over and local algebras have as residue field. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required.
With any nilpotent dgla is naturally associated the simplicial set
where stands for the Maurer-Cartan functor mapping a dgla to the set of its Maurer-Cartan elements, and is the simplicial differential graded commutative associative algebra of polynomial differential forms on algebraic -simplexes, for . The importance of this construction, which can be dated back to Sullivan’s Su77, relies on the fact that, as shown by Hinich and Getzler Hi97, Ge09, the simplicial set is a Kan complex, or -to use a more evocative name- an ∞-groupoid. A convenient way to think of ∞-groupoids is as homotopy types of topological spaces; namely, it is well known1 that any ∞-groupoid can be realized as the ∞-Poincaré groupoid, i.e., as the simplicial set of singular simplices, of a topological space, unique up to weak equivalence. Therefore, the reader who prefers to can substitute homotopy types of topological spaces for equivalence classes of ∞-groupoids. To stress this point of view, we’ll denote the -truncation of an ∞-groupoid by the symbol . More explicitely, is the -groupoid whose -morphisms are the -morphisms of for , and are homotopy classes of -morphisms of for . In particular, if is the ? spring groupoid of a topological space , then is the set of path-connected components of , and is the usual Poincaré groupoid of .
The next step is to consider an -category, i.e., an ∞-category whose hom-spaces are ∞-groupoids. This can be thought as a formalization of the naïve idea of having objects, morphisms, homotopies between morphisms, homotopies between homotopies, et cetera. In this sense, endowing a category with a model structure should be thought as a first step towards defining an -category structure on it.
Turning back to dglas, an easy way to produce nilpotent dglas is the following: pick an arbitrary dgla ; then, for any differential graded local Artin algebra , take the tensor product , where is the maximal ideal of . Since both constructions
and
are functorial, their composition defines a functor
where, by definition, a formal ∞-groupoid is a functor . Note that is the usual set valued deformation functor associated with , i.e., the functor
where the gauge equivalence of Maurer-Cartan elements is induced by the gauge action
of on the subset of . However, due to the presence of nontrivial irrelevant stabilizers, the groupoid is not equivalent to the action groupoid , unless is concentrated in nonnegative degrees. We will come back to this later. Also note that the zero in gives a natural distinguished element in : the isomorphism class of the trivial deformation. Since this marking is natural, we will use the same symbol to denote both the set and the pointed set .
It is important to remark that the functors of the form are very special ones among all formal ∞-groupoids. To begin with, and so, in particular, is a homotopically trivial ∞-groupoid. Another characterzing property of the functors of the form among formal ∞-groupoids is that, under suitable assumptions, they commute with homotopy pullbacks; see Pr10, Lu11 for a precise statement. In other words, if we call “formal moduli problems” those formal ∞-groupoids which satisfy the two conditions we have just observed for , what we are saying is that is actually a functor
And a very good reason for working with ∞-groupoids valued deformation functors rather than with their apparently handier set-valued or groupoid-valued versions is the following remarkable result, which allows one to move homotopy constructions back and forth between dglas and formal moduli problems.
(Pridham-Lurie) The functor is an equivalence of -categories.
Here the -category structures involved are the most natural ones, and they are both induced by standard model category structures. Namely, on the category of dglas one takes surjective morphisms as fibrations and quasi-isomorphisms as weak equivalences, just as in the case of differential complexes, whereas the model category structure on the right hand side is induced by the standard model category structure on Kan complexes as a subcategory of simplicial sets. A proof of the above equivalence can be found in Pr10, Lu11.
We will often identify a dgla with the functor it defines by the rule . With this in mind, we will occasionally apply constructions that generally only make sense for nilpotent dglas (such as ) to arbitrary dglas. What we mean in these cases is that the construction is applied not to a single dgla, but to the functor from to nilpotent dglas it defines. The same kind of consideration applies to our somehow colloquial use of the expression “∞-groupoid” in the following sections; namely, by that we will occasionally mean “formal ∞-groupoid”, or even “formal stack in ∞-groupoids”. The precise meaning to be given to “∞-groupoid” will always be clear from the context.
If is a formal moduli problem, then the simplicial set has a natural structure of simplicial vector space, and so, via the Dold-Kan correspondence, it is equivalent to the datum of a chain complex: the tangent complex of . Passing from to the associated classical moduli problem , the only datum we read of the tangent complex is its homotopy class, i.e., since we are working on a field, its cohomology. In particular, we have a natural isomorphism
of functors between the tangent space to the classical moduli problem associated to a dgla and the first cohomology group of the dgla seen as a cochain complex. Let us rephrase this in a more explicit form. As we noticed in the previous section, is the functor of Artin rings , hence
This isomorphism is natural. Namely, given a morphism of dglas, let us write for the induced morphism of classical moduli problems,
Then the differential of ,
is naturally identified with
The second cohomology group defines a natural obstruction theory for , i.e., obstructions for the classical moduli problem are naturally identified with elements in , see Ma02. Note that this does not mean that each element in represents an obstruction: one can have dglas with nontrivial governing unobstructed deformation problems. The naturally of the obstruction theory given by the second cohomology groups means that, if is a morphism of of dglas, the induced morphism in cohomology,
maps obstructions for the classical moduli problem to obstructions for the classical moduli problem . In particular, if the moduli problem is unobstructed (e.g., if the functor is smooth), then
Let and be two dglas. The hom-space of morphisms between and in the -category of dglas is conveniently modelled as the simplicial set , where is the Chevalley-Eilenberg-type dgla associated with the pair . It is given as the total dgla of the bigraded dgla
endowed with the Lie bracket
defined by
with ranging in the set of -unshuffles and and standing for the Koszul sign, and with the differentials
and
given by
and
An explicit determination for the signs in the above formulas can be found, e.g, in LM95,Sc04. These operations are best seen pictorially:
At least in higher categories folklore. ↩
2011, 2012
Publications of the nLab, Volume 2 (2012)
Contents
vol. 2, no. 1 – Domenico Fiorenza and Elena Martinengo, A short note on -groupoids and the period map for projective manifolds
article type: research article
submission type: author submission of arXiv article arXiv:0911.3845
submitted: July 30th, 2011
refereeing: by expert anonymous referee chosen by the nLab steering committee
status: accepted for publication, 23rd August 2012, published 13th September 2012
referee reports: the entire refereeing process that this article went through is documented at FiorenzaMartinengo2012 - refereeing
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This page documents the refereeing process that the publication FiorenzaMartinengo2012 went through.
Originally article submission, July 30th, 2011
(submission letter, FiorenzaMartinengo2012Submission.tex, FiorenzaMartinengo2012Submission.pdf)
Steering committee selects and contacts anonymous expert referee, August 5, 2011
Submission of slightly revised article, December 24th, 2011
(re-submission letter, FiorenzaMartinengo2012SubmissionB.tex FiorenzaMartinengo2012SubmissionB.pdf).
Referee is notified of revision, December 28th, 2011
(post)
May 2nd, 2012
Referee notifies steering committee that report will be further delayed due to circumstances beyond the referee’s control.
First referee report received June 28, 2012
Domenico Fiorenza, Elena Martinengo,A short note on infinity-groupoids and the period map for projective manifolds
In this note the authors explain how the modern homotopy-theoretic formulation of deformation theory sheds light on some classical results in the subject, particularly the period mapping relating deformations of complex structure on a projective manifold to variations of the Hodge structure on its cohomology.
The role of differential graded Lie algebras (dgla’s) in deformation theory was first emphasized by Nijenhuis and Richardson in the 1960’s, and was further elaborated upon by Deligne, Drinfeld, Kontsevich and others. Classically, deformation theory is given by a functor associating to a dgla a formal space. This formal space (more precisely, a functor from the category of Artinian algebras to Sets) describes the formal neighborhood of a point in the moduli space of structures encoded by the dgla as the solutions of the Maurer-Cartan equation. The modern approach extends this formulation in two directions:
(1) the deformation functor of a dgla should take values not in formal spaces but formal stacks (in higher-dimensional groupoids), i.e. functors from Artinian algebras to simplicial sets, rather than just sets. This captures the higher-order symmetry information encoded in the components of the dgla of low (negative) degrees.
(2) (not discussed in the present note) these formal stacks should be derived, i.e. their domain should be extended from Artinian algebras to their homotopical version, chain dg algebras (or, to go beyond characteristic 0, to simplicial or E-infinity algebras). This captures the information about the singularities of the Maurer-Cartan variety encoded in the high (positive) degree components of the dgla, and in particular sheds light on the obstruction theory.
In addition to capturing new information, this extended formulation often provides conceptual explanation of the classical phenomena which are recovered as the 0-truncation of the resulting homotopy types.
The authors approach is to first construct a certain map of sheaves of dgla’s on a manifold X arising in a simple way from the contraction of differential forms by vector fields (they first explain this in the abstract setting of “Cartan homotopies” of dgla’s). Taking the (extended) deformation functors turns this into a map of infinity-sheaves of formal stacks. Taking derived global sections and 0-truncating produces the classical period map.
Throughout the note the authors keep their presentation informal, steering clear of the technicalities of infinity-categories. Since they are not claiming any new results but merely offering a conceptual explanation of some old ones, this is acceptable. My only complaint is that the presentation sometimes gets a bit too informal, to the point of confusing the reader. I have provided some comments on the text below, which I feel should be addressed before the note is published.
the statement of the Theorem on p. 3 is incorrect. First, the category Art should be replaced by dgArt or equivalent, otherwise the degree part of the dgla will stay invisible. Second, the essential image of Def is not all formal stacks but only those that can arise from formal moduli problems, and there are conditions describing those. The precise statement can be found in DAG X (http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf), Thm 0.0.13 (see also Warning 0.0.12). In fact, it seems this result, although generally of fundamental importance, plays no role in this note, as all the deformation problems considered explicitly arise from dgla’s.
On p.4 the authors introduce the “internal hom” dgla, Hom(g,h) and claim that MC(Hom(g,h) tensored with differential forms on simplices) is homotopy equivalent to Hom_oo(g,h). But since the authors do not exhibit any other model for Hom_oo besides MC(Hom(g,h)…), the statement seems meaningless. The authors should state explicitly what other model they are comparing MC(Hom..) with.
It is not clear what role is played by L-infinity morphisms in the note, in view of the fact that a Cartan homotopy is defined by conditions describing when a generally L-infinity morphism is actually a strict map of dgla’s. In fact, it is also unclear why one needs the “Lie derivative” l to be a strict map rather than L-infinity.
the authors occasionally apply constructions that generally only make sense for nilpotent dgla’s (such as exp,) to arbitrary dgla’s. The interpretation is probably that it is supposed to be applied not to a single dgla, but to the functor from Art to nilpotent dgla’s defined by this dgla. But it would be less confusing if this were spelled out explicitly.
Likewise, the authors often say “infinity-groupoid”, when they really mean a formal stack in infinity-groupoids, e.g. in footnote 4 on p.9.
It would help if the authors explained what they mean by “the differential of P” (since P is a map of formal spaces, its differential must be its value on the dual numbers modulo constants).
Reaction of the authors to the first referee report, June 29, 2012
Dear Referee,
Thanks a lot for your extremely careful report, and for the truly insightful remarks and suggestions it contains. We agree with all of them and are now planning to revise the note accordingly.
Below, we are sketching the kind of revision we have in mind for each of your comments, so that -should you enjoy having a look at it- you can eventually comment on this before we actually implement it into the new version of the note.
With our best regards,
Domenico and Elena
the statement of the Theorem on p. 3 is incorrect. First, the category Art should be replaced by dgArt or equivalent, otherwise the degree >1 part of the dgla will stay invisible. Second, the essential image of Def is not all formal stacks but only those that can arise from formal moduli problems, and there are conditions describing those. The precise statement can be found in DAG X (http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf), Thm 0.0.13 (see also Warning 0.0.12).
We absolutely agree. Actually, the Artin algebras involved in the note are differential graded ones, but writing “(differential graded) local Artin algebra” instead of “differential graded local Artin algebra” on page 2 was definitely not a good idea, while denoting their category as Art instead of dgArt was so bad to be evil.
Concerning the statement of the Theorem on page 3, again, we absolutely agree. At the time we wrote that in the first arXiv version of the note we were not aware of the rigorous description of formal moduli problems. Then, after Lurie’s ICM talk, we added a pointer to that but did not upgarde the statement from its “so informal to be false” version to the correct one. Luckily, we will be able to do this now.
In fact, it seems this result, although generally of fundamental importance, plays no role in this note, as all the deformation problems considered explicitly arise from dgla’s.
Right. We are accordingly planning to deemphasize the role played by this result in the note. Its original role was to make it not surprising for the reader that the hom-space of (L_oo) morphisms between dglas could be realized as the simplicial set of Maurer-Cartan elements of a suitable dgla. But this is precisely one of thise points where our being informal ends up with being confusing, so we are now planning to state the equivalence between dglas and formal moduli problems in its correct way, as a result of fundamental importane on its own, but not as something palying an actual specific role in the note.
On p.4 the authors introduce the “internal hom” dgla, Hom(g,h) and claim that MC(Hom(g,h) tensored with differential forms on simplices) is homotopy equivalent to Hom_oo(g,h). But since the authors do not exhibit any other model for Hom_oo besides MC(Hom(g,h)…), the statement seems meaningless. The authors should state explicitly what other model they are comparing MC(Hom..) with.
Again we agree. The correct way to express what we had in mind is to define Hom_oo(g,h) via its simplicial model MC(Hom(g,h)…) with no reference to other models. Namely, we are planning to reduce the whole first part of Section 2 to a single sentence like “The hom-space Hom_oo(g,h) of morphisms between G and h in the (oo,1)-category of dglas is conveniently modelled as the simplicial set MC(Hom(g,h)…), where Hom(g,h) is the Chevalley-Eilenberg-type dgla defined as follows…”, and revise accordingly the rest of the section.
It is not clear what role is played by L-infinity morphisms in the note, in view of the fact that a Cartan homotopy is defined by conditions describing when a generally L-infinity morphism is actually a strict map of dgla’s. In fact, it is also unclear why one needs the “Lie derivative” l to be a strict map rather than L-infinity.
Indeed having so much space for L_oo morphisms in the note is a reminescence of a time when we were less uded to homotopy invariant constructions and were on the other hand used to deal with explict L_oo morphisms as tools for explicit computations. At the “transition towards homotopy” period the note was written it was nice to us to see the definition of Cartan homotopy as a condition describing when a certain L_oo morphism was actually strict. But now that we don’t look at strict L_oo morphisms as something cool anymore (and indeed there is nothing intrinsic in them) we are going to deemphasize this. Namely, we are now planning to give the plain definition of Cartan homotopy at the beginning of the section as something motivated by classical Cartan identities and to show directly (ore just say, since it is a one line computation) that the Lie derivative associated to a Cartan homotopy is a dgla morphism. And next just to observe the gauge equivalence e^{-i}0=l expressing the fact that l is a dgla morphism homotopy equivalent to zero.
the authors occasionally apply constructions that generally only make sense for nilpotent dgla’s (such as exp,) to arbitrary dgla’s. The interpretation is probably that it is supposed to be applied not to a single dgla, but to the functor from Art to nilpotent dgla’s defined by this dgla. But it would be less confusing if this were spelled out explicitly. -Likewise, the authors often say “infinity-groupoid”, when they really mean a formal stack in infinity-groupoids, e.g. in footnote 4 on p.9.
Again, absolutely right. We are going to spell these out explicitly.
It would help if the authors explained what they mean by “the differential of P” (since P is a map of formal spaces, its differential must be its value on the dual numbers modulo constants).
Here we could add a few lines to Section 1 as follows: after having mentioned formal moduli problems we could recall that the tangent space to a formal moduli problem P is P(k[epsilon]/(epsilon^2)) and that the differential of a morphism \phi: P –> P’ is \phi(k[epsilon]/(epsilon^2)): P(k[epsilon]/(epsilon^2)) –> P’(k[epsilon]/(epsilon^2)). By the way, we are working with unitary local Artin algebras but of such an algebra we actually take only the maximal ideal m_A: should we better work directly with nilpotent Artin algebras (and so define the tangent space to P as P((epsilon)/(epsilon^2)) instead?
Reaction of the referee, June 30, 2012
There’s not much to say, really, except I’m glad we’re on the same page. I’m sorry for not noticing the parenthesized “(differential graded)” on page 2 and assuming the authors meant discrete Artinian algebras throughout. There’s also a question of what dgArt should really mean: dg algebras whose underlying superalgebra is Artinian, or any infinity-category equivalent to that, eg. of dg algebras whose homology is Artinian (one could even take those algebras to be quasi-free). Regarding the last question, I guess it’s largely a matter of taste. I’m used to having all my algebras unital. But in this case they are also augmented, so it amounts to the same thing.
Submission of revised version by the authors, August 17, 2012: this is what is now the final version
Final referee reaction, August 23, 2012: The referee had no further comments and accepted the revised version.
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Tom Leinster, An informal introduction to topos theory, Publications of the nLab vol. 1 no. 1 (2011)
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK; Tom.Leinster@glasgow.ac.uk. Supported by an EPSRC Advanced Research Fellowship.
This short text is for readers who are confident in basic category theory but know little or nothing about toposes. It is based on some impromptu talks given to a small group of category theorists. I am no expert on topos theory. These notes are for people even less expert than me.
In keeping with the spirit of the talks, what follows is light on both detail and references. For the reader wishing for more, almost everything here is presented in respectable form in Mac Lane and Moerdijk’s very pleasant introduction to topos theory (1994). Nothing here is new, not even the expository viewpoint (very loosely inspired by Johnstone (2003)).
As a rough indication of the level of knowledge assumed, I will take it that you are totally comfortable with the Yoneda Lemma and the concept of cartesian closed category, but I will not assume that you know the definition of subobject classifier or of topos.
Section 1 explains the definition of topos. The remaining three sections discuss some of the connections between topos theory and other subjects. There are many more such connections than I will mention; I hope it is abundantly clear that these notes are, by design, a quick sketch of a large subject.
Section 2 is on connections between topos theory and set theory. There are two themes here. One is that, using the language of toposes, we can write down an axiomatization of sets that sticks closely to how sets are actually used in mathematics. This provides an appealing alternative to ZFC. The other, related, theme is that
a topos is a generalized category of sets.
Section 3 is on connections with geometry (in a broad sense); there the thought is that
Section 4 is on connections with universal algebra:
What this means is that there is one topos embodying the concept of ‘ring’, another embodying the concept of ‘field’, and so on. This is the story of classifying toposes.
Sections 2–4 can be read in any order, except that ideally §3 (geometry) should come before §4 (universal algebra). You can read §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 seem rather mysterious.
Algebraic geometers beware: the word ‘topos’ is used by mathematicians in two slightly different senses, according to circumstance and culture. There are elementary toposes and Grothendieck toposes. Category theorists tend to use ‘topos’ to mean ‘elementary topos’ by default, although Grothendieck toposes are also important in category theory. But when an algebraic geometer says ‘topos’, they almost certainly mean ‘Grothendieck topos’ (what else?).
Grothendieck toposes are categories of sheaves. Elementary toposes are slightly more general, and the definition is simpler. They are what I will emphasize here. Grothendieck toposes are the subject of Section 3, and appear fleetingly elsewhere; but if you only want to learn about categories of sheaves, this is probably not the text for you.
I thank Andrei Akhvlediani, Eugenia Cheng, Richard Garner, Nick Gurski, Ignacio Lopez Franco and Emily Riehl for their participation and encouragement. Aspects of Section 4 draw on a vaguely similar presentation of vaguely similar material by Richard Garner. I thank the organizers of Category Theory 2010 for making the talks possible, even though they did not mean to: Francesca Cagliari, Eugenio Moggi, Marco Grandis, Sandra Mantovani, Pino Rosolini, and Bob Walters. I thank Filip Bár, Jon Phillips, Urs Schreiber, Mike Shulman, Alex Simpson, Danny Stevenson and Todd Wilson for suggestions and corrections. I am especially grateful to Todd Trimble for carefully reading an earlier version and suggesting many improvements.
The hardest part of the definition of topos is the concept of subobject classifier, so I will begin there. For motivation, I will speak of ‘the category of sets’ (and functions). What exactly this means will be discussed in Section 2, but for now we proceed informally.
In the category of sets, inverse images are a special case of pullbacks. That is, given a map of sets and a subset , we have a pullback square
In particular, this holds when is a 1-element subset of :
There is no virtue in distinguishing between one-element sets, so we might as well write instead of ; then the inclusion becomes the map picking out , and we have a pullback square
Next consider characteristic functions of subsets. Fix a two-element set (‘true’ and ‘false’). Then for any set , the subsets of are in bijective correspondence with the functions . In one direction, given a subset , the corresponding function is defined by
(). In the other, given a function , the corresponding subset of is . To say that this latter process is a bijection is to say that for all , there is a unique function such that . In other words: for all , there is a unique function such that
is a pullback square.
This property of sets can now be stated in purely categorical terms. We use to indicate a mono ( monomorphism monic).
Let be a category with finite limits. A subobject classifier in is an object together with a map such that for every mono in , there exists a unique map such that
is a pullback square.
So, we have just observed that has a subobject classifier, namely, the two-element set. In the general setting, we may write as (or properly, ) and call it the characteristic function of (or ).
To understand this further, we need two lemmas.
In any category, the pullback of a mono is a mono. That is, if
is a pullback square and is a mono, then so is .
In any category with a terminal object , every map out of is a mono.
So, pulling back along any map gives a mono into .
It will also help to know the result of the following little exercise (Johnstone (2003), A1.6.1). It says, roughly, that in the definition of subobject classifier, the fact that is terminal comes for free.
Let be a category and let be a mono in . Suppose that for every mono in , there is a unique map such that there is a pullback square
Then is terminal in .
This leads to a second description of subobject classifiers. Let be the category whose objects are monos in and whose maps are pullback squares. Then a subobject classifier is exactly a terminal object of .
Here is a third way of looking at subobject classifiers. Given a category and an object , a subobject of is officially an isomorphism class of monos (where isomorphism is taken in the slice category ). For example, when , two monos
are isomorphic if and only if they have the same image; so subobjects of correspond one-to-one with subsets of . I say ‘officially’ because half the time people use ‘subobject of ’ to mean simply ‘mono into ’, or slip between the two meanings without warning. It is a harmless abuse of language, which I will adopt.
For , let be the class of subobjects (in the official sense) of . Assume that is well-powered, that is, each is a set rather than a proper class. Assume also that has pullbacks. By Lemma 2, every map in induces a map of sets, by pullback. This defines a functor .
Third description: a subobject classifier is a representation of this functor .
This makes intuitive sense, since for to be representable means that there is an object satisfying
naturally in . In the motivating case of the category of sets, this directly captures the thought that subsets of a set correspond naturally to maps .
Now we show that this is equivalent to the original definition. By the Yoneda Lemma, a representation of amounts to an object together with an element that is ‘generic’ in the following sense:
for every object and element , there is a unique map such that .
In other words, a representation of is a mono in satisfying the condition in Fact 4. In other words, it is a subobject classifier.
A topos (or elementary topos) is a cartesian closed category with finite limits and a subobject classifier.
The primordial topos is . It has special properties not shared by most other toposes. This is the subject of Section 2.
For any set , the category of -indexed families of sets is a topos. Its subobject classifier is the constant family , where is a two-element set.
For any group , the category of left -sets is a topos. Its subobject classifier is the set with trivial -action.
Encompassing all the previous examples, if is any small category then the category of presheaves on is a topos. We can discover what its subobject classifier must be by a thought experiment: if is a subobject classifier then by the Yoneda Lemma,
for all . So must be the set of subfunctors of ; and one can check that defining in this way does indeed give a subobject classifier. A subfunctor of is called a sieve on ; it is a collection of maps into satisfying a certain condition.
For any topological space , the category of sheaves on is a topos. This is the subject of Section 3. Modulo a small lie that I will come back to there, the space can be recovered from the topos . Hence the class of spaces embeds into the class of toposes, and this is why toposes can be viewed as generalized spaces.
Sheaves will be defined and explained in Section 3. To give a brief sketch: denote by the poset of open subsets of ; then a presheaf on the space is a presheaf on the category , and a sheaf on is a presheaf with a further property. I will consistently use ‘sheaf’ to mean what some would call ‘sheaf of sets’. A sheaf of groups, rings, etc. is the same as an internal group, ring etc. in .
The category of finite sets is a topos. Similarly, can be replaced by in all of the previous examples, giving toposes of finite -sets, finite sheaves, etc.
You might ask ‘why is the definition of topos what it is? Why that particular collection of axioms? What’s the motivation?’ I will not attempt to answer, except by explaining several ways in which the definition has been found useful. It is also worth noting that the topos axioms have many non-obvious consequences, giving toposes a far richer structure than most categories. For example, every map in a topos factorizes, essentially uniquely, as an epi followed by a mono. More spectacularly, the axioms imply that every topos has finite colimits. This can be proved by the following very elegant strategy, due to Paré (1974). For every topos , we have the contravariant power set functor . It can be shown that is monadic. But monadic functors create limits, and has finite limits. Hence has finite limits; that is, has finite colimits.
Here I will describe what makes ‘the’ category of sets special among all toposes, and explain why I just put ‘the’ in quotation marks. This is the stuff of revolution: it can completely change your view of set theory. It also provides an invaluable insight into topos theory as a whole.
We begin by listing some special properties of the topos , using only the most commonplace assumptions about how sets and functions behave.
The terminal object is a separator (generator). That is, given maps in , if for all then .
It is worth dwelling on what this says. Maps correspond to elements of , and we make no notational distinction between the two. Moreover, given an element and a map , we can compose the maps
to obtain a map , and this is the map corresponding to the element . (We might harmlessly write both and as .) Thus, elements are a special case of functions, and evaluation is a special case of composition.
The property above says that if for all then . In other words, a function is determined by its effect on elements.
Write for the initial object of (the empty set). Then . Equivalently, is not equivalent to the terminal category .
A topos satisfying properties 1 and 2 is called well-pointed.
This property says, informally, that there is a set consisting of the natural numbers.
What are the ‘the natural numbers’, though? One way to get at an answer is to use the principle that sequences can be defined recursively. That is, given a set , an element , and a map , there is a unique sequence in such that
A sequence in is just a map , and if we write for the function (‘successor’), then (1) says exactly that the diagram
commutes.
Let be a category with a terminal object, . A natural numbers object in is a triple , with , , and , that is initial as such: for any triple of the same type, there is a unique map such that (2) commutes (with in place of ).
Property 3 is, then, that has a natural numbers object.
Epis split. That is, for any epimorphism (surjection) in , there exists a map such that . The splitting chooses for each an element of the nonempty set . The existence of such splittings is precisely the Axiom of Choice. Generally, a category is said to satisfy the Axiom of Choice (or to ‘have Choice’) if epis split.
In summary,
sets and functions form a well-pointed topos with natural numbers object and Choice.
The category of sets has many other elementary properties (such as the fact that the subobject classifier has exactly two elements), but they are all consequences of the properties just mentioned.
But what is this thing called ‘the category of sets’? What do we have to assume about sets in order to prove that these properties hold?
Many mathematicians do not like to be bothered with such questions, because they know that the standard answer will be something like ‘sets are anything satisfying the axioms of ZFC’—and they feel that ZFC is irrelevant to what they do, and prefer not to hear about it.
The standard answer is valid, in the sense that for every model of ZFC, there is a resulting category of sets satisfying the properties above. But it may seem irrelevant, because at no point in establishing the properties did it feel necessary to call on an axiom system: all the properties are suggested directly by the naive imagery of a set as a bag of dots.
There is, however, another type of answer—and this was Lawvere’s radical idea. It is this:
In other words, we do away with ZFC entirely, and ask instead that sets and functions form a well-pointed topos with natural numbers object and Choice. ‘The’ category of sets is any category satisfying these axioms. In fact we should say a category of sets, since there may be many different such categories, as we shall see.
This is Lawvere’s Elementary Theory of the Category of Sets (ETCS), stated in modern language. (See Lawvere (1964), or Lawvere and Rosebrugh (2003) for a good expository account.) It is nearly fifty years old, but still has not gained the currency it deserves, for reasons on which one can speculate.
You might be thinking that this is circular: that this axiomatization of sets depends on the notion of category, and the notion of category depends on some notion of collection or set. But in fact, ETCS does not depend on the general notion of category. It can be stated without using the word ‘category’ once.
To see this, we need to back up a bit. The ZFC axiomatization of sets looks, informally, like this:
there are some things called ‘sets’
there is a binary relation ‘’ on sets
some axioms hold.
People seeing this (or the formal version) often ask certain questions. What does ‘some things’ mean? Do you mean that there is a set of sets? (No.) What exactly is meant by ‘binary relation’? (It means that for each set and set , the statement ‘’ is deemed to be either true or false.) What do you mean, ‘deemed’? Etc. This is not a logic course, and I will not attempt to answer the questions except to say that there is an assumed common understanding of these terms. To hide behind jargon, ZFC is a first-order theory.
The ETCS axiomatization of sets looks like this:
there are some things called ‘sets’
for each set and set , there are some things called ‘functions from to ’
for each set , set and set , there is a binary operation assigning to each pair of functions
a function
some axioms hold.
You can ask the same kind of logical questions as for ZFC—what exactly is meant by ‘binary operation’? etc.—which again I will not attempt to answer. The difficulties are no worse than for ZFC, and again, in the jargon, ETCS is a first-order theory.
Stated in this way, the ETCS axioms begin by saying that composition is associative and has identities (so that sets, functions and composition of functions define a category); then they say that binary products and equalizers of sets exist, and there is a terminal set (so that the category of sets has finite limits); and so on, until we have said that sets and functions form a well-pointed topos with natural numbers object and Choice. You can do it in about ten axioms.
Here ends the digression.
ZFC axiomatizes sets and membership, whereas ETCS axiomatizes sets and functions. Anything that can be expressed in one language can be expressed in the other: in the usual implementation of ZFC, a function is defined as a suitable subset of , and in ETCS, an element of is defined as a function from the terminal set to . But an advantage of the categorical approach is that it avoids the chains of elements of elements of elements that are so important in traditional set theory, yet seem so distant from most of mathematics.
ZFC is slightly stronger than ETCS. ‘Stronger’ means that everything that can be deduced about sets from the ETCS axioms can also be deduced in ZFC, but not vice versa. ‘Slightly’ is meant in a sociological sense. I believe it has been said that the mathematics in an ordinary undergraduate syllabus (excluding, naturally, any course in ZFC) makes no more assumptions about sets than are made by ETCS. If that is so, it must also be the case that for many mathematicians, nothing in their entire research career requires more than ETCS.
The technical relationship between ZFC and ETCS is well understood. It is known exactly which fragment of ZFC is equivalent to ETCS (namely, ‘bounded’ or ‘restricted’ Zermelo with Choice; see Mac Lane and Moerdijk (1994)). It is also known what needs to be added to ETCS in order to obtain a system of equal strength to ZFC. This extra ingredient is an axiom scheme (a countably infinite family of axioms) that set theorists in the traditional mould would call Replacement, and category theorists would call a form of cocompleteness. It says, informally, that given any set and family of sets specified by a first-order formula, the coproduct exists. The existence of this coproduct is expressed by saying that there exist a set and a map (to be thought of as the projection ) such that for each , the inverse image is isomorphic to . See Section 8 of McLarty (2004) for details.
Topos theory therefore provides a different viewpoint on set theory. Let us take a brief look from this new viewpoint at a famous theorem of set theory: that the Continuum Hypothesis is independent of the usual set-theoretic axioms, as proved by Gödel and Cohen.
Temporarily, let us say that a ‘category of sets’ is a well-pointed topos with natural numbers object and Choice, satisfying the axiom scheme of Replacement. A category of sets is said to satisfy the Continuum Hypothesis if for all objects ,
(As usual, denotes the natural numbers object; is the subobject classifier.) Stated categorically, the theorem is this: given any category of sets, you can build one that satisfies the Continuum Hypothesis and one that does not. This is only a rephrasing of the standard statement, but if you are more at home with the term ‘category’ than with ‘model of a first-order theory’, you might find it less mysterious.
So far we have seen the benefits of viewing the/a category of sets as a special topos. But the other way round, there are great benefits to viewing a topos as a generalized category of sets. For example, we might view as the category of sets varying through (discrete) time. The set of human beings alive today is an object of : as the meaning of ‘today’ changes, the set changes. A sheaf can similarly be understood as a set varying through space.
People (especially Lawvere) sometimes refer to the category of sets as the (or a) topos of constant sets, to contrast it with toposes of variable sets. There are also toposes whose objects can informally be thought of as ‘cohesive’ sets, which means the following. In an ordinary set, the points have no relation or attachment to each other: they do not ‘cohere’. But a cohesive set carries something like a topology or smooth structure, so that the points are in some sense stuck together. For example, there are toposes of smooth spaces, which are the setting for synthetic differential geometry. From this point of view, the category of ordinary sets is extreme among all toposes: its objects are sets with no variation or cohesion at all.
Viewing the objects of a topos as generalized sets is much more than a useful mental technique. In fact, it is valid to use set-like language and reasoning in any topos, provided that we stick to certain rules. This language is called the ‘internal language’ of the topos.
Many of the central ideas of topos theory are simple, but that simplicity can easily be obscured by the richness of structure available in a topos. Such is the case for the internal language. I will therefore describe the idea in a much more basic setting.
First let be any category whatsoever, and let be an object of . A generalized element of is simply a map in with codomain . A generalized element may be said to be of shape , or to be an -element of . In the special case that is terminal, -elements are called global elements. (See Example 9(3) for a hint on the reason for the name.) In the category of sets, the global elements are the ordinary elements, but in other categories, the global elements might be very uninteresting: consider the category of groups, for instance.
Given a map in , any generalized element of gives rise to a generalized element of . This is the composite , but can also be thought of as ‘’: see the remarks on property 1 at the beginning of this section. For maps , we have
(Proof of : take .) This is emphatically not true if we replace ‘generalized’ by ‘global’: again, consider groups.
This language of generalized elements is the internal language of the category. It fits well with ordinary categorical terminology and notation?. For example, let be a category with finite products. In the internal language, the definition of product reads, informally: an -element of consists of an -element of together with an -element of . Apart from the ‘-’ prefixes, this is identical to the ordinary description of the cartesian product of sets and . And in standard categorical notation?, the map with components and is denoted by , thus extending the set-theoretic notation? for a (global) element of a cartesian product.
To see why the internal language is useful, consider, for instance, internal groups in a finite product category . A group in is an object together with maps
satisfying some axioms. Those axioms are usually expressed as commutative diagrams, which have been obtained by translating the classical axioms into diagrammatic form. But there is no need to translate them: the classical axioms can simply be repeated verbatim and interpreted as statements about generalized elements. This is equivalent. For example, it is easy to show that the commutative diagram for associativity is equivalent to the statement that
for all generalized elements of of the same shape. (They have to be the same shape in order for expressions such as to make sense.) And just as for ordinary elements in , there is no harm in writing instead of , and similarly instead of .
More valuably still, proofs written down in the classical set-theoretic scenario will actually be valid in an arbitrary finite product category , as long as whatever was said about elements in is also true for generalized elements in . For example, whenever is a group in and , we have
Proof:
We can immediately conclude that the implication (4) holds whenever is a group in an arbitrary finite product category and are generalized elements of of the same shape. Indeed, each step in the proof is an application of an axiom such as (3) valid in the general setting.
The internal language is a massively labour-saving device. To prove that an equation valid in ordinary groups is also valid for internal groups, you merely need to cast an eye over the proof and convince yourself that it holds for generalized elements too. In contrast, try proving the internal version of the equation
by diagrammatic methods. First it has to be stated diagrammatically. It says that the diagram
commutes. Then it has to be proved, by filling the inside of this diagram with instances of the diagrams encoding the group axioms. (It seems to need at least ten or so inner diagrams.) But once you have an elementwise proof, all this effort is unnecessary. And the example (5) chosen was very simple: for more complex statements, the benefits of the internal language become clearer still.
The internal language of toposes is similar to that of finite product categories, but much richer. As well as being able to form pairs of generalized elements, we can take generalized elements of exponentials (to be thought of as families of maps ), form subobjects such as
(the equalizer of ), and so on. Almost anything that can be expressed or proved in the category of sets can be reproduced in an arbitrary topos. The only sticking points are the law of the excluded middle and the axiom of choice. Any proof that avoids those—any constructive proof, in a sense that can be made precise—generalizes to an arbitrary topos.
Phrases with more or less the same meaning as ‘internal language’ are ‘Mitchell--Bénabou language’ and ‘internal logic’. See, for instance, Mac Lane and Moerdijk (1994) or Johnstone (2003). There you can also find more spectacular applications of topos theory to set theory, including topics such as forcing.
This section covers concepts such as sheaf, geometric morphism (map of toposes), Grothendieck topos, and locale. But the most important thing I want to explain is how and why geometry has inspired so much of topos theory.
Let be a topological space. (Following tradition, I will switch from my previous convention of using to denote an object of a topos.) Write for its poset of open subsets. A presheaf on is a functor . It assigns to each open subset a set , whose elements are called sections over (for reasons to be explained). It also assigns to each open a function , called restriction from to and denoted by . I will write for the category of presheaves on .
Examples 1 and 2 are qualitatively different: continuity is a local property, but boundedness is not. This difference can be captured by asking the following question. Let be a family of open subsets of , and take, for each , a section . Might there be some such that for all ?
For this to stand a chance of being true, functoriality demands that the sections must satisfy a ‘matching condition’: for all and . A sheaf is a presheaf such that for every family of open sets and every matching family , there is a unique such that for all .
The first example above, with continuous functions, is a sheaf. The proof can be split into two parts. Given and , there is certainly a unique function (continuous or not) such that for all . The question now is whether is continuous; and because continuity is a local property, it is.
The second example above, with bounded functions, is not a sheaf (for a general space ). This is because boundedness is not a local property.
The sheaf of continuous real-valued functions is rather floppy, in the sense that there are usually many ways to extend a continuous function from a smaller set to a larger one. Often people consider sheaves made up of holomorphic or rational functions, which are much more rigid: there are typically few or no ways to extend. It is quite normal for there to be no global sections (sections over ) at all.
Take any continuous map of topological spaces (which can be thought of as a kind of bundle over ). Then there arises a sheaf on , in which is the set of continuous maps such that the triangle on the left commutes:
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There is also an abstract categorical explanation of where the concept of sheaf comes from. Fix a space . We have a functor
where is the category of topological spaces, is the slice category, and . This functor embodies the simple thought that an open subset of a topological space can be treated as a space in its own right. We now apply to two very general categorical constructions, from which the sheaf concept will appear automatically.
First, purely because the domain of is small and the codomain has small colimits, there is an induced adjunction
The right adjoint is given by
where and . This is, in fact, the process described in Example 9(4): the sheaf defined there is . The left adjoint can be described as a coend or colimit: for ,
where the colimit is over all and , and the map from the colimit to is the canonical one.
Second, every adjunction restricts canonically to an equivalence between full subcategories: one consists of the objects at which the unit of the adjunction is an isomorphism, and the other of the objects at which the counit is an isomorphism. Write the equivalence obtained from the adjunction above as
It can be shown that this is the same category of sheaves as before. In this way, the notion of sheaf arises canonically from the very simple functor . The notion of étale bundle also arises canonically: étale bundles over are (by definition, if you like) the objects of . Among other things, this equivalence shows that every sheaf is of the form described in Example 9(4). See Mac Lane and Moerdijk (1994) for details.
One way or another, we have the category of sheaves on . It is a topos. Its subobject classifier is given by
The crucial fact about is that—modulo a small lie that I will repair later—
can be recovered from .
So the class of topological spaces embeds into the class of toposes. We can think of toposes as generalized spaces.
A common technique in topos theory is to take a concept from topology or geometry and extend it to toposes. For example, suppose you hear someone talking about ‘connected toposes’. You may have no idea what one is, but you can bet that the definition has been obtained by determining what property of the topos corresponds to connectedness of the space , then taking that as the definition of connectedness for all toposes.
The next few subsections are all examples of this generalization process.
So far I have said nothing about maps between toposes. There is an obvious candidate for what a map of toposes should be: a functor preserving finite limits, exponentials, and subobject classifiers. Such a functor is called a logical morphism. They have a part to play, but there is another notion of map of toposes that has been found much more useful. It can be derived by generalizing from topology.
Every map in induces an adjunction
This is not obvious. The right adjoint is easy to construct—
(, )—but the left adjoint is harder. It can be made easy by invoking the equivalence between sheaves and étale bundles; but I will not go into that, or give any other description of .
It is a fact that preserves finite limits. It is also a fact (modulo the usual small lie) that there is a natural correspondence between continuous maps and adjunctions (6) in which the left adjoint preserves finite limits. So now we know what continuous maps look like in topos-theoretic terms. We duly generalize:
Let and be toposes. A geometric morphism is an adjunction
in which the left adjoint preserves finite limits. (People often say ‘left exact left adjoint’.) The right adjoint is called the direct image part of , and is the inverse image part.
I will write for the category of toposes and geometric morphisms. (Really it’s a 2-category, in an obvious way.) By construction, we have a functor
which is (2-categorically) full and faithful, modulo the usual small lie.
Every functor induces a string of adjoint functors
between presheaf categories. Here , and and are left and right Kan extension along , respectively. Since has a left adjoint, it preserves limits. Hence is a geometric morphism .
It turns out that, for any topological space , the inclusion has a finite-limit-preserving left adjoint. It is called sheafification or the associated sheaf functor. So the inclusion of sheaves into presheaves is a geometric morphism.
Since is a full subcategory, the inclusion is full and faithful; and for totally general reasons, this is equivalent to the counit of the adjunction being an isomorphism. In other words, sheafifying a sheaf does not change it.
Let us generalize another concept of topology. The points of a topological space correspond to the maps (where is the one-point space), which correspond to the geometric morphisms . But , so we make the following definition.
A point of a topos is a geometric morphism .
For any subspace of a space , the inclusion is an embedding, that is, a homeomorphism to its image. It can be shown that a map of spaces is an embedding if and only if the direct image part of the corresponding geometric morphism is full and faithful. So, as usual, we generalize:
A geometric morphism is an embedding (or inclusion) if the direct image functor is full and faithful.
We then say that is a subtopos of . At least, this is the right thing to say up to equivalence. Perhaps we should reserve that word for when is actually a (full) subcategory of and is the inclusion , rather than allowing to be any old full and faithful functor. But a full and faithful functor induces an equivalence to its image, so it makes no real difference.
Probably the easiest toposes are the presheaf toposes: those equivalent to for some small category . So maybe subtoposes of presheaf toposes are relatively easy too. They have a special name:
A topos is Grothendieck if it is (equivalent to) a subtopos of some presheaf topos.
For instance, we saw in Example 11(2) that is a subtopos of , for any topological space . Hence is a Grothendieck topos.
Being Grothendieck is generally thought of as a mild condition on a topos. A Grothendieck topos has all small limits, which immediately disqualifies toposes such as , , etc. But other than toposes arising from finite sets (or sets subject to some other cardinality bound), most of the toposes that people have worked with are Grothendieck. A notable exception is the effective topos, the maps in which can be thought of as computable functions. Other non-Grothendieck toposes occur in the topos-theoretic approach to non-standard analysis.
There is a theorem of Giraud giving a list of conditions on a category equivalent to it being a Grothendieck topos. It includes non-elementary axioms such as ‘there is a small generating set’. (‘Non-elementary’ means that it refers to a pre-existing notion of set.) The Grothendieck toposes are sometimes regarded as the nice toposes, but perhaps the definition of Grothendieck topos is not as nice as the definition of elementary topos.
Definition 14 is not the definition of Grothendieck topos that you will find in most books. I will now give a brief indication of what the standard definition is and why it is equivalent to the one above.
Fix a small category . There is a one-to-one correspondence between the subtoposes of and the Grothendieck topologies on . A Grothendieck topology is a kind of explicit, combinatorial structure; it specifies which diagrams
in are to be thought of as ‘covering families’ and which are not. (There are axioms.) The motivating example is that given a topological space , there is a canonical Grothendieck topology on : a family of subsets of is covering if and only if .
The bijection
is written
A pair , consisting of a small category equipped with a Grothendieck topology , is called a site, and is the category of sheaves on that site. For example, let be a topological space, take , and take to be the Grothendieck topology mentioned above; then . Most books proceed as follows: define Grothendieck topology, define site, define the category of sheaves on a site, then define a Grothendieck topos to be a category equivalent to the category of sheaves on some site.
I do not know a short way to explain why the subtoposes of correspond to the Grothendieck topologies on . The following two paragraphs may make it seem easier, or harder.
First, there is an explicit classification of the subtoposes of any topos . Indeed, it can be shown that the subtoposes of correspond to the maps satisfying certain equations. (Such a is called a Lawvere--Tierney topology on , although this is so distant from the original usage of the word ‘topology’ that some people object; Peter Johnstone, for instance, uses local operator instead.) By definition of subobject classifier, it is equivalent to say that a subtopos of amounts to a subobject of satisfying certain axioms.
Second, take . We know (Example 6(4)) that is given by . Hence a subtopos of corresponds to a collection of sieves in , satisfying certain axioms. Calling these the ‘covering sieves’ gives the notion of Grothendieck topology.
Here I will explain the ‘small lie’ mentioned several times above, and make amends. I will also explain why topos theorists are fond of jokes about pointless topology.
The definition of sheaf on a topological space does not mention the points of . It mentions only the open sets and inclusions between them, and uses the fact that it is possible to take arbitrary unions and finite intersections of open sets. Having observed this, you can see why the space cannot always be recovered from the topos . For instance, if is indiscrete (has no open sets except and ) and nonempty, then is the same no matter how many points has.
The idea now is to split the process into two steps. First, we forget the points of , leaving just the set of open sets, ordered by inclusion. Then, we form the category of ‘sheaves’ on that ordered set (defined as for topological spaces, almost verbatim).
A frame is a partially ordered set such that every subset has a join ( least upper bound sup), every finite subset has a meet ( greatest lower bound inf), and finite meets distribute over joins. A map of frames is a map preserving order, joins and finite meets.
A topological space has a frame of open subsets, and a continuous map induces a map of frames. This gives a functor
We now perform a linguistic manoeuvre. is the desired category of ‘pointless spaces’. But we cannot wholeheartedly say that a frame is a pointless space, because the maps of frames are the wrong way round. So we introduce a new word—locale—and define the category of locales by . We can wholeheartedly say that a locale is a pointless space.
There is a functor , defined just as for topological spaces except that unions become joins and intersections become meets. The functor factorizes as
This is the two-step process mentioned above.
Whenever I have said ‘modulo a small lie’, you can interpret that as ‘use locales instead of topological spaces’. For example, really is full and faithful, in a suitably up-to-isomorphism sense: locale maps correspond one-to-one with isomorphism classes of geometric morphisms . This means that is equivalent to a full subcategory of . (Actually it is an equivalence of 2-categories, but I will gloss over that point.)
Every locale gives rise to a topos—but the converse is also true. Given a topos , the subobjects of form a poset . Assuming that has enough colimits, is a frame. This process defines a functor
I am now quietly changing to mean the toposes with small colimits; this includes all Grothendieck toposes.
You might think that could have no interesting subobjects, since that is the case in the most obvious topos, . But there are toposes that are nearly as obvious in which is not trivial. For instance, take for any set : then is the power set of .
Now a wonderful thing is true. The functor just defined is left adjoint to the inclusion . This means that is (equivalent to) a reflective subcategory of . Hence the counit is an isomorphism:
for any locale . This is how you recover a locale from its topos of sheaves.
So sits inside as a subcategory of the best kind: full and reflective, like abelian groups in groups. It is reasonable to say that a locale is a special sort of topos. More formally, a topos is localic if it is of the form for some locale . Localic toposes are easy to work with; if you were having trouble proving something for arbitrary toposes, you might start by trying to prove it in this special case.
Since every locale is of the form for some topos , locale theory can be regarded as the fragment of topos theory concerning subobjects of . A subobject of is a map , which can reasonably called a truth value. In that sense, locale theory is the study of truth values.
The notion of locale can also be seen as a decategorification of the notion of Grothendieck topos. A poset is a category enriched in the two-element totally ordered set . There is a Yoneda embedding , which has a finite-meet-preserving left adjoint if and only if is a frame. Analogously, it is almost true that for a category , the Yoneda embedding has a finite-limit-preserving left adjoint if and only if is a Grothendieck topos. (This result is due to Street (1981). ‘Almost’ refers to a set-theoretic size condition.) A map of frames is a function preserving joins and finite meets, and the inverse image part of a geometric morphism is a functor preserving colimits and finite limits. Thus, locales play roughly the same role among 2-enriched categories as Grothendieck toposes play among -enriched categories.
How much has been lost by passing from topological spaces to locales? In most people’s view, not much. For example, we observed that all nonempty indiscrete spaces give rise to the same locale; but many mathematicians regard indiscrete spaces with points as ‘pathological’ and would be positively happy to see them go.
In fact, some things are gained. For example, a subgroup of a topological group need not be closed, and non-closed subgroups are often regarded as pathological (since the corresponding quotients are non-Hausdorff). But it is a theorem that every subgroup of a localic group is closed. See for instance Section C5.3 of Johnstone (2003).
The functor has a right adjoint, which I will not describe. As mentioned above, every adjunction restricts canonically to an equivalence between full subcategories. In this case, this gives an equivalence between:
a full subcategory of , whose objects are called the sober spaces
a full subcategory of , whose objects are called the spatial locales.
Another way of interpreting the phrase ‘modulo a small lie’ is ‘true for sober spaces’. Sobriety amounts to a rather mild separation condition. For example, every Hausdorff space is sober. So in passing from a Hausdorff space to a locale, or to a topos, nothing whatsoever is lost.
There is a kind of attitudinal paradox here. Many algebraic topologists think only about Hausdorff spaces, and regard non-Hausdorff spaces as pathological. But these are often the same people who feel strongly that topological spaces are not really about open sets; they think in terms of points and paths and homotopies. So it is perhaps paradoxical that the Hausdorff condition guarantees that a space can be understood in terms of its open sets alone: the topos of sheaves depends on nothing else, and contains all the information about the original space.
The point of this section is to explain what people mean when they talk about the classifying topos of a theory. Another way to look at it is this: I will explain how toposes can be viewed as cousins of operads and Lawvere theories.
In classical universal algebra, an algebraic theory (or strictly, a presentation of an algebraic theory) consists of a bunch of operation symbols of specified arities, together with a bunch of equations. To take the standard example, the (usual presentation of the) theory of groups consists of
together with the usual equations. You can speak of ‘models’ of an algebraic theory in any category with finite products. In our example, they are the internal groups in .
But there are other ways of looking at such theories.
Consider the free finite product category equipped with an internal group. (There are general reasons why such a thing must exist.) Its universal property is that for any finite product category , the finite-product-preserving functors correspond to the internal groups in .
Concretely, looks something like this. It must contain an object , the underlying object of the internal group. Since has finite products, it must also contain an object for each . There is no reason for it to have any other objects, and since it is free, it does not. A map is (by definition of product) an -tuple of maps ; and the maps are (by freeness) whatever maps must exist for any internal group in any finite product category. That is, they are the -ary operations in the theory of groups: the words in letters.
This category is called the Lawvere theory of groups. The same goes for rings, lattices, etc. In all these cases, is a finite product category with the further property that the objects are in bijection with the natural numbers, the product of objects corresponding to addition of numbers. This further property holds because the theories described so far have been single-sorted: a model is a single object equipped with some structure.
But there are also many-sorted theories, such as the two-sorted theory of pairs in which is a ring and an -module. So we can widen the notion of algebraic theory to include all (small) finite product categories. Some people say that an algebraic theory is just a finite product category. Others say that algebraic theories correspond to finite product categories. Others still, more traditionally, say that algebraic theories correspond to only certain finite product categories.
Terminology aside, we can play the same game for other classes of limit. For example, it makes no sense to talk about internal categories in an arbitrary finite product category, because the definition of internal category needs pullbacks. (Composition in an internal category is a map .) But we can talk about internal categories in a finite limit category; and as before, there is a free finite limit category equipped with an internal category. This means that for any finite limit category , the finite-limit-preserving functors correspond to the internal categories in . A small finite limit category is called (or corresponds to) an essentially algebraic theory.
In a category with finite products you can talk about internal groups but not, in general, internal categories. In a category with finite limits you can talk about both. By extending the list of properties that the category is assumed to satisfy, you can accommodate more and more sophisticated kinds of theory. (The theory of internal categories is more ‘sophisticated’ than that of groups in the sense that composition is only defined for some pairs of maps, whereas classical universal algebra can only handle operations defined on all pairs.) The properties need not be of the form ‘limits of such-and-such a type exist’. For example, it is sometimes useful to assume epi-mono factorization, as we shall see.
There is a trade-off here. As you allow more sophisticated language, you widen the class of theories that can be expressed, but you narrow the class of categories in which it makes sense to take models. (You also make more work for yourself.) In the same way, if you trade in your motorbike for a double-decker bus, you increase the number of passengers you can carry, but you restrict where you can carry them: no low bridges or tight alleyways. (You also increase your fuel costs.) It is sensible, then, to use the smallest class of theories containing the ones you are interested in. For example, you could treat groups as an essentially algebraic theory, but that would mean you could only take models in categories with all finite limits, when in fact just products would do.
Before I get onto toposes, I want to point out a slightly different direction that you can take things in. Rather than just altering the properties that the categories are assumed to have, you can also alter the structure with which they are equipped.
Take monoidal categories, for instance. We can speak of internal monoids in any monoidal category. Hence, the theory of monoids can be regarded as the free monoidal category containing an internal monoid. (This is in fact the category of finite ordinals.) Similarly, it makes sense to speak of algebras for an operad in any monoidal category, and we can associate to the free monoidal category containing a -algebra. Thus, for any monoidal category , monoidal functors correspond to -algebras in .
We might define a monoidal theory to be a small monoidal category. This gets us into the territory of PROPs, where there are nontrivial theorems such as the classification of 2-dimensional topological quantum field theories: the symmetric monoidal theory of (or, ‘PROP for’) commutative Frobenius algebras is the category of smooth 1-manifolds and diffeomorphism classes of cobordisms.
All of this is to give an impression of how far-reaching these ideas are. It is a sketch of the context in which classifying toposes can be understood.
You will have guessed that the same kind of thing can be said for toposes as for categories with finite products, finite limits, etc. Since toposes have very rich structure (much more than just finite limits), they correspond to a very wide class of theories indeed.
An example of the kind of theory that can be interpreted in a topos is the theory of fields. (This is rather a feeble example, but I want to keep it simple.) A field is, of course, a commutative ring satisfying the axioms
and
By a mechanical process, this definition can be turned into a definition of ‘internal field in a topos’. As compensation for the imprecision of the rest of this section, I will give the definition in detail; but if you want to skip it, the point to retain is that it is a mechanical process.
Let be a topos. We certainly know how to define ‘commutative ring in ’: that makes sense in any category with finite products. Let be a commutative ring in . The nontriviality axiom, , is expressed by saying that the equalizer of
is the initial object . For the other axiom, let us first define the subobject consisting of the units (invertible elements). The ‘set’ is the pullback
Now we want to define the ‘set’ of units as the image of the composite map
We can talk about images in a topos, since every map in a topos factorizes essentially uniquely as an epi followed by a mono. So, define by the factorization
The second field axiom states that every element of lies in either the subobject or the subobject . In other words, it states that the map
is epi. Here we have used the fact that every topos has coproducts, written .
If you have read Section 2, you will recognize that the informal talk of ‘sets’ (really, objects of ) and the use of set-theoretic notation? are something to do with the internal language of a topos. This gives a hint of how the process can be mechanized.
(There are actually several possible theories of fields, depending on exactly how you write down the axioms. They all have the same models in —namely, fields—but they do not have the same models in other toposes. For example, a genuinely different theory is obtained by changing axiom (8) to ‘, ’. But this does not affect the main point: given a list of formally-expressed axioms such as (7) and (8), there is an automatic process converting it into a definition that makes sense in an arbitrary topos.)
You now have the choice between a short story and a long story.
The short story is that what we did for finite product and finite limit categories can also be done for toposes. The theories corresponding to toposes are called the geometric theories, and the topos corresponding to a particular geometric theory is called its classifying topos.
The long story is longer because there are two different notions of map of toposes—and you need to decide what a map of toposes is in order to state the universal property of the topos resulting from a theory.
The more obvious but less used notion of map of toposes is a functor preserving all the structure in sight: finite limits, exponentials, and the subobject classifier. These are called logical morphisms. Now in a topos, you can interpret a really vast range of theories: any ‘higher-order theory’, in fact. (First order means that you can only quantify over elements of a set; in a second order theory you can also quantify over subsets of a set; and so on.) Models of any such theory get along well with logical morphisms, because logical morphisms preserve everything. So you can tell a similar story for toposes, logical morphisms and higher order theories as for finite product categories, finite-product-preserving functors and algebraic theories.
The more popular notion of map of toposes is that of geometric morphism. (Here it helps to have read Section 3, where the definition is motivated.) A geometric morphism between toposes is a functor with a finite-limit-preserving left adjoint. The corresponding theories are the geometric theories. I will not give the definition, but it is not too bad an approximation to say that they are the same as the first-order theories: every geometric theory is first-order, and almost every first-order theory that one encounters is geometric.
Given a geometric theory, a classifying topos for the theory is a cocomplete topos with the property that for any cocomplete topos , models of the theory in correspond naturally to geometric morphisms . Every geometric theory has a classifying topos.
There are two surprises here. One is the appearance of the word ‘cocomplete’, which I will not explain and will not bother inserting below. It is generally thought of as a mild condition (satisfied by any Grothendieck topos, for instance).
The bigger surprise is the reversal of direction. The previous cases lead us to expect models in to correspond to maps . However, since a geometric morphism is a pair of adjoint functors, the choice of direction is a matter of convention. As the name suggests, the choice that society made was motivated by geometry. Perhaps if the motivation had been universal algebra, it would have been the other way round. (This is an aspect of the thought that geometry is dual to algebra.) A map of toposes would then have been a finite-limit-preserving functor with a right adjoint, which is more or less the same thing as a functor preserving finite limits and small colimits.
If a topos is thought of as a generalized space (as in Section 3) then the classifying topos of a theory can be thought of as its space of models. Indeed, a point of the classifying topos is (by Definition 12) a geometric morphism , which is exactly a model of the theory in . Some familiar topological spaces can be construed as classifying toposes. For example, there is a ‘theory of Dedekind cuts’ whose classifying topos is , that is, regarded as a topos.
Given how much structure a topos contains, it is surprising how many classifying toposes can be described simply. I will now describe the classifying topos of any algebraic theory, by the venerable expository device of doing it just for groups.
We will need the notion of finite presentability. A group (in ) is finitely presentable if it admits a presentation by a finite set of generators subject to a finite set of relations. The category of finitely presentable groups and all homomorphisms between them will be written .
Finite presentability is a more categorical concept than it might seem. Writing for the free group monad, a relation (equation) in a set of generators is an element of . So, a family of relations is a map , or equivalently a diagram
in , or equivalently a diagram
in , where is the free group functor. The group presented by these generators and relations is the coequalizer of this diagram in . Hence a group is finitely presentable precisely when it is the coequalizer of some diagram in which and are finite sets.
This formulation of finite presentability in uses the free group functor . But in fact, there is a general definition of finite presentability of an object of any category. I will not go into this.
As promised, the classifying topos for groups is easy to describe:
The classifying topos for groups is .
In other words, for any topos , a group in is the same thing as a geometric morphism .
The same goes for other algebraic theories. This yields something interesting even for very trivial theories. Take the theory of objects, whose models in a category are simply objects of . A finitely presentable set is just a finite set. Hence for any topos , objects of correspond to geometric morphisms . The topos is therefore called the object classifier.
We have been asking, for a given theory, ‘what topos classifies it?’ But we can turn the question round and ask, for a given topos , ‘what does classify?’ In other words, what are the geometric morphisms from an arbitrary topos into ? It is a fact that every topos is the classifying topos of some geometric theory—although given how wide a class of theories that is, perhaps this does not say very much.
There are clean answers to this reversed question for many toposes . In particular, this is so when is the topos of sheaves on a site (Section 3). Here I will just tell you the answer for a smaller class of toposes.
Let be a category with finite limits. Then the presheaf topos classifies finite-limit-preserving functors out of .
In other words, for any topos , a geometric morphism is the same thing as a finite-limit-preserving functor .
(If you know about flat functors, you can drop the assumption that has finite limits: for any small category , the presheaf topos classifies flat functors out of . This is one version of Diaconescu's Theorem.)
So there is a back-and-forth translation between geometric theories and the toposes that classify them. In many cases, this translation is surprisingly straightforward.
Johnstone, P. T., 2003. Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides. Oxford University Press.
Lawvere, F. W., 1964. An elementary theory of the category of sets. Proceedings of the National Academy of Sciences of the U.S.A. 52:1506–1511. Reprinted as Reprints in Theory and Applications of Categories 12:1–35, 2005.
Lawvere, F. W. and Rosebrugh R., 2003. Sets for Mathematics. Cambridge University Press, Cambridge.
MacLane, S. and Moerdijk I., 1994. Sheaves in Geometry and Logic. Springer, Berlin.
McLarty, C., 2004. Exploring categorical structuralism. Philosophia Mathematica 12:37–53.
Paré, R., 1974. Colimits in topoi. Bulletin of the American Mathematical Society 80:556–561.
Street, R., 1981. Notions of topos. Bulletin of the Australian Mathematical Society 23:199–208.
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2011, 2012
Publications of the nLab, Volume 1 (2011)
Contents
vol. 1, no. 1 – Tom Leinster, An informal introduction to topos theory
article type: expository article
submission type: author submission of arXiv article
submitted: Jan, 2011
refereeing: by expert anonymous referee chosen by the nLab steering committee
status: accepted for publication, 27th June, 2011
referee reports: see here
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