topos

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Category Theory
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[[!include category theory - contents]]
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# Contents #
* table of contents
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## Idea
 {#Idea}

There are various different perspectives on the notion of _topos_.   One is that a topos is a [[category]] that looks like a category of [[space]]s that sit by [[local homeomorphism]]s over a given base [[space]]: all spaces that are _locally modeled on_ a given base space.

The archetypical class of examples are [[sheaf topos]]es $Sh(X) = Et(X)$ over a [[topological space]] $X$: these are the categories of [[étale space]]s over $X$, topological spaces $Y$ that are equipped with a [[local homeomorphisms]] $Y \to X$.

When $X = *$ is the [[point]], this is just the category [[Set]] of all [[sets]]: spaces that are _modeled on the point_. This is the archetypical topos itself.

What makes the notion of topos powerful is the following fact: even though the general topos contains objects that are considerably different from and possibly considerably richer than plain sets and even richer than étale spaces over a topological space, the general abstract [[category theory|category theoretic]] properties of every topos are essentially the same as those of [[Set]]. For instance in every topos all small [[limits]] and [[colimits]] exist and it is [[cartesian closed category|cartesian closed]] (even [[locally cartesian closed category|locally]]). This means that a large number of constructions in [[Set]] have immediate analogs [[internalization|internal to]] every topos, and the analogs of the statements about these constructions that are true in $Set$ are true in _every_ topos.

On the one hand this may be thought of as saying that toposes are _very nice categories of spaces_ in that whatever construction on spaces one thinks of
-- for instance formation of [[quotient]]s or of [[fiber product]]s or of [[mapping space]]s --
the resulting space with the expected general abstract properties will exist in the topos. In this sense toposes are _convenient categories_ for geometry -- as in: [[convenient category of topological spaces]], but even more convenient than that.

On the other hand, by de-emphasizing the geometric interpretation of their objects and just using their good abstract properties, this means that toposes are contexts with a powerful [[internal logic]]. The internal logic of toposes is [[intuitionistic logic|intuitionistic]] [[higher order logic]]. This means that, while the [[law of excluded middle]] and the [[axiom of choice]] may fail, apart from that, every logical statement not depending on these does hold [[internalization|internal to]] _every_ topos.

For this reason toposes are often studied as abstract contexts "in which one can do mathematics", independently of their interpretation as categories of spaces. These two points of views on toposes, as being about geometry and about logic at the same time, is part of the richness of topos theory.

On a third hand, however, we can de-emphasize the role of the objects of the topos and instead treat the topos itself as a "generalized space" (and in particular, a [[vertical categorification|categorified]] space).  We then consider the topos $Sh(X)$ as a representative of $X$ itself, while toposes not of this form are "honestly generalized" spaces.  This point of view is supported by the fact that the assignment $X\mapsto Sh(X)$ is a full embedding of (sufficiently nice) topological spaces into toposes, and that many topological properties of a space $X$ can be detected at the level of $Sh(X)$.  (This is even more true once we pass to [[(∞,1)-toposes]].)

From this point of view, the objects of a topos (regarded as a category) should be thought of instead as *sheaves on* that topos (regarded as a generalized space).  And just as sheaves on a topological space can be identified with local homeomorphisms over it, such "sheaves on a topos" (i.e. objects of the topos *qua* category) can be identified with other *toposes* that sit over the given topos via a [[étale geometric morphism|local homeomorphism of toposes]].

Finally, mixing this point of view with the second one, we can regard toposes over a given topos $E$ instead as "toposes in the $E$-world of mathematics."  For this reason, the theory of toposes over a given base is formally quite similar to that of arbitrary toposes.  And coming full circle, this fact allows the use of "base change arguments" as a very useful technical tool, even if our interest is only in one or two particular toposes *qua* categories.

### 'What a topos is like:'

 (i) 'A topos is a category of sheaves on a site'

 (ii) 'A topos is a category with finite limits and power-objects'

 (iii) 'A topos is (the embodiment of) an intuitionistic higher-order theory'

 (iv) 'A topos is (the extensional essence of) a first-order (infinitary) geometric theory'

 (v) 'A topos is a [[total category|totally cocomplete]] object in the meta-2-category CART of cartesian (i.e. , finitely complete) categories'

 (vi) 'A topos is a generalized space'

 (vii) 'A topos is a semantics for intuitionistic formal systems'

 (viii) 'A topos is a Morita equivalence class of continuous groupoids'

 (ix) 'A topos is the category of maps of a power allegory'

 (x) 'A topos is a category whose canonical indexing over itself is complete and well-powered'

 (xi) 'A topos is the spatial manifestation of a giraud frame'

 (xii) 'A topos is a setting for synthetic differential geometry'

 (xiii) 'A topos is a setting for synthetic domain theory',

And so on. But the important thing about the elephant is that 'however you approach it, it is still the same animal'. [[Elephant]]

## Definitions

The general notion of _topos_ is that of

* [Elementary toposes](#ElementaryTopos).

A specialization of this which is important enough that much of the literature implicitly takes it to be the general definition is the notion of

* [Grothendieck/sheaf toposes](#SheafToposes)

This is the notion relevant for applications in [[geometry]] and [[geometric logic]], whereas the notion of elementary toposes is relevant for more general applications in [[logic]].

For standard notions of mathematics to be available [[internal logic|inside]] a given topos one typically at least needs a [[natural numbers object]]. Its existence is guaranteed by the axioms of a sheaf topos, but not by the more general axioms of an elementary topos. Adding the existence of a natural numbers object to the axioms of an elementary topos yields the notion of a

* [W-topos](#WToposes).

### Elementary toposes
 {#ElementaryTopos}

A quick formal definition is that an **elementary topos** is a [[category]] which

1. has [[finitely complete category|finite limits]],

1. is [[cartesian closed category|cartesian closed]], and

1. has a [[subobject classifier]].

There are alternative ways to state the definition; for instance,

1. has [[finite limit]]s and
2. has [[power objects]].

In a way, however, these concise definitions can be misleading, because a topos has a great deal of other structure, which plays a very important role but just happens to follow automatically from these basic axioms.  Most importantly, an elementary topos is all of the following:

*  [[locally cartesian closed category|locally cartesian closed]],
*  [[finitely cocomplete category|finitely cocomplete]],
*  a [[Heyting category]],
*  a [[pretopos]].

The last two imply that it has an [[internal logic]] that resembles ordinary mathematical reasoning, and the presence of exponentials and power objects means that this logic is [[higher order logic|higher order]].

### Grothendieck/sheaf toposes
 {#SheafToposes}

The above is the definition of an *elementary* topos.  We also have the (historically earlier) notion of [[Grothendieck topos]]: a __Grothendieck topos__ is a topos that is neither too small nor too large, in that it is:

*  [[cocomplete category|cocomplete]] (not too small), and
*  has a [[small set|small]] [[generating set]] (not too large).

Equivalently, a Grothendieck topos is any category [[equivalence of categories|equivalent]] to the [[category of sheaves]] on some
[[small category|small]] [[site]].

### $W$-toposes
 {#WToposes}

There is a further elementary property of [[Set]] that might have gone into the definition of elementary topos but historically did not: the existence of a [[natural numbers object]].  Any topos with this property is called a __topos with NNO__ or a __$W$-topos__.  The latter term comes from the result that any such topos must have (not only an NNO but also) all [[W-types]].

### Toposes over a base

* [[base topos]]

### Morphisms of toposes

There are two kinds of [[homomorphisms]] between toposes that one considers:

* [[geometric morphism]] -- this is the kind of morphism that regards a topos as a generalized [[topological space]].

* [[logical morphism]] -- this is the kind of morphism that regards a topos in terms of its [[internal logic]].

Accordingly there is a [[2-category]] [[Topos]] of toposes, whose

* [[object]]s are toposes;

* [[morphism]]s are [[geometric morphism]]s;

* [[2-morphism]]s are [[natural transformation]]s between the [[functor]]s underlying the geometric morphisms.

## Properties

### Extensivity

Every topos is an [[extensive category]].  For [[Grothendieck toposes]], infinitary extensivity is part of the characterizing [[Giraud's theorem]].  For elementary toposes, see [[toposes are extensive]].

### Adhesiveness

Every topos is an [[adhesive category]]. For [[Grothendieck toposes]] this follows immediately from the adhesion of [[Set]], for elementary toposes it is due to ([Lack-Sobocinski](#LackSobocinski)).

### Epimorphisms

In a topos [[epimorphism]]s are stable under [[pullback]] and hence the [[(epi, mono) factorization system]] in a topos is a [[stable factorization system]].

### Relation to abelian categories

While crucially different from [[abelian categories]], there is some intimate relation between toposes and abelian categories. For more on that see [[AT category]].

### Reasoning in a topos

Any result in [[ordinary mathematics]] whose proof is [[finite mathematics|finitist]] and [[constructive mathematics|constructive]] automatically holds in any topos.  If you remove the restriction that the proof be finitist, then the result holds in any topos with a [[natural numbers object]]; if you remove the restrictions that the proof be constructive, then the result holds in any [[boolean topos]].  On the other hand, if you add the restriction that the proof be predicative in the weaker sense used by constructivists, then the result may fail in some toposes but holds in any $\Pi$-[[Pi-pretopos|pretopos]]; if you add the restriction that the proof be predicative in a stronger sense, then the result holds in any [[Heyting pretopos]].

Therefore, one can prove results in toposes and similar categories by reasoning, not about the [[objects]] and [[morphisms]] in the topos themselves, but instead about [[sets]] and [[functions]] in the normal language of structural [[set theory]], which is more familiar to most mathematicians.  One merely has to be careful about what axioms one uses to get results of sufficient generality.

The [[internal logic|internal language]] of a topos is called [[Mitchell-Bénabou language]].

## Examples {#Examples}

### Specific examples and key results

* The archetypical topos is [[Set]]. Notice that this happens to be a [[Grothendieck topos]]: it is the [[category of sheaves]] on the [[point]].

  The [[full subcategory]] [[FinSet]] is also an elementary topos, and the inclusion functor $FinSet \hookrightarrow Set$ is a [[logical morphism]]. This is not a Grothendieck topos.

  More generally, for $\kappa$ a [[cardinal|strong limit cardinal]] the full subcategory $Set_\kappa$ of sets or [[cardinality]] less than $\kappa$ is a topos.

* For $C$ any (small) [[site]], the [[category of sheaves]] $Sh(C)$ is a [[Grothendieck topos]].  Either by definition or by [[Giraud's theorem]], every Grothendieck topos arises in this way.  Important examples include:

  * The case where the [[Grothendieck topology]] is the trivial one, so that also all categories of [[presheaf|presheaves]] (on small categories) are (Grothendieck) toposes.

  * The case of sheaves on (the [[site]] given by the [[category of open subsets]] of) a [[topological space]], or more generally a [[locale]].

  * The category $G Set$ of [[set]]s equipped with the [[action]] of a [[group]] $G$: this is the topos of presheaves on the category $\mathbf{B}G$ which is the [[delooping]] [[groupoid]] of $G$.

 * Continuing the last example above, if $G$ is a topological group, the category $G Set$ of [[category of G-sets|sets with a continuous action of G]] (that is, the action map $G\times X\to X$ is [[continuous map|continuous]], where $X$ has the [[discrete topology]]) is a topos, and in fact a Grothendieck topos (though this may not be obvious).

 * More generally, $G$ may be a [[topological groupoid]], or even a [[localic groupoid]].  In fact, it is a theorem that every Grothendieck topos can be presented as the topos of "continuous sheaves" on a localic groupoid.

* Again if $G$ is a topological group, the category $Unif(G)$ of *[[uniformly continuous map|uniformly continuous]]* $G$-sets is also a topos, but not (in general) one of Grothendieck's.  For example, if $G$ is the [[profinite completion]] of $\mathbb{Z}$, then a continuous $G$-set is a $\mathbb{Z}$-set all of whose orbits are finite, while a uniformly continuous one is a $\mathbb{Z}$-set with a finite upper bound on the size of all its orbits.

* For $E$ a topos and $X \in E$ any [[object]], also the [[overcategory]] or [[slice category]] $E/X$ is again a topos. ([[Elephant]], A.2.3.2). See also [[over-topos]].

* If $E$ is a topos and $j \colon E \to E$ is a [[exact functor|lex]] idempotent monad, the category of $j$-algebras is a topos. Each such $j$ corresponds to a [[Lawvere-Tierney topology]] in $E$, and the category of $j$-algebras is equivalent to the category of sheaves for this topology. An example is the [[double negation|double-negation topology]].

* An obvious example of an elementary topos that is not a Grothendieck topos is the category [[FinSet]] of finite sets.  More generally, for any [[strong limit cardinal]] $\kappa$, the category of sets of cardinality $\lt\kappa$ is an elementary topos, and as long as $\kappa \gt\omega$ it has a [[NNO]]. $Set_{\lt \kappa}$ doesn't even admit a geometric morphism to $Set$.

* Since the definition of elementary topos is "algebraic," there exist [[free topos]]es generated by various kinds of data. The category of toposes (and [[logical functor]]s) is [[2-monadic]] over the 2-category of categories, functors, and natural isomorphisms. It has an [[initial object]] which is sometimes called *the free topos*.  More generally, any [[higher-order logic|higher-order]] [[type theory]] (of the sort which can be interpreted in the [[internal logic]] of a topos) generates a free topos containing a model of that theory.  Such toposes (for a consistent theory) are never Grothendieck's.

* If $G$ is a [[large category|large]] [[groupoid]] with a small set of objects, then the category $[G,Set]$ (which makes sense if we define "large" and "small" relative to a [[Grothendieck universe]]) is a topos, but not in general a Grothendieck topos.  Note that this topos is in fact [[complete category|complete]] and cocomplete, but it does not have a small generating set, and so is an [[unbounded topos]].

* If $\mathcal{F}$ is a [[filter]] of [[subterminal objects]] in any topos $E$, then there is a [[filterquotient]] topos $E//\mathcal{F}$.  Grothendieck-ness (and completeness and cocompleteness) are not in general preserved by this construction.

* If $C$ and $D$ are toposes and $F\colon C\to D$ is a [[lex functor]], then there is a topos $Gl(F)$ called the [[Artin gluing]] of $C$ and $D$ along $F$, and defined to be the [[comma category]] $(D/F)$.  If $C$ and $D$ are Grothendieck toposes then $Gl(F)$ is a $Set$-topos. If $F$ is [[accessible functor|accessible]], then $Gl(F)$ is again Grothendieck (hence [[bounded geometric morphism|bounded]]), but in general it may not be. (Note, though, that it is not clear whether it can be proven in ZFC that there exist any inaccessible lex functors between Grothendieck toposes, although from a proper class of [[measurable cardinal]]s one can construct an inaccessible lex endofunctor of $Set$.)

* The [[Eilenberg-Moore category|category of coalgebras]] for any [[exact functor|lex]] [[comonad]] on a topos is again a topos: a [[topos of algebras over a monad|topos of coalgebras]], and if the comonad is [[accessible functor|accessible]], this construction preserves Grothendieck-ness.  If the comonad is not accessible, then this topos is unbounded.

  For instance the [[Artin gluing]] $Gl(F)$ is equivalent to the category of coalgebras for the comonad on the topos $C\times D$ defined by $(c,d) \mapsto (c, d\times F(c))$.

* More generally, the category of coalgebras for any [[preserved limit|pullback-preserving]] comonad on a topos is again a topos. This covers both the preceding result and also the overcategory (slice category) result above, since $E/X$ is the category of coalgebras for the pullback-preserving comonad given by $X \times - \colon E \to E$. It also covers Artin gluing along a pullback-preserving functor.

* More generally still, [[Todd Trimble]] has a notion called a "modal operator" on a topos, from which one can construct a new topos of "$G$-structures": see [[toddtrimble:Three topos theorems in one]]. Special cases include the pullback-preserving comonad result just mentioned, and the result that the category of algebras for a lex [[idempotent monad]] is a topos. A related idea is Toby Kenney's notion of lex distributive [[diad]], from which one can also construct a topos.

* From any [[partial combinatory algebra]] one can construct a [[realizability topos]], whose [[internal logic]] is "computable" or "effective" mathematics relative to that PCA.  In particular, for [[Kleene's first algebra]], one obtains the [[effective topos]], the most-studied of realizability toposes.  Realizability toposes are generally not Grothendieck toposes.

* A topos can also be constructed from any [[tripos]].  This includes realizability toposes as well as toposes of sheaves on locales.

### Classes of examples

For various applications one uses toposes that have [[stuff, structure, property|extra structure or properties]].

* In the [[foundations]] of mathematics, one often studies [[well-pointed toposes]], especially models of [[ETCS]] as potential replacements for the category [[Set]].

* In [[synthetic differential geometry]] one studies [[smooth topos]]es as a context for axiomatic [[differential geometry]].

## Related concepts

*  [[category of sheaves]], [[Giraud's theorem]]

* [[base topos]], [[indexed topos]]

* [[slice topos]], [[fundamental theorem of topos theory]]

* [[predicative topos]]

* [[boolean topos]]

* [[local topos]], [[locally connected topos]], [[connected topos]], [[cohesive topos]]

* [[predicative topos]]

* [[pretopos]], [[Π-pretopos]], [[ΠW-pretopos]], [[list-arithmetic pretopos]]

* [[smooth structure on a topos]]

* [[monoidal topos]]

* [[isotropy group of a topos]]

* [[test topos]]

* [[classifying topos]]

[[!include flavors of higher toposes -- list]]

[[!include locally presentable categories - table]]

## References
  {#References}

### Introductions
 {#Introductions}

Brief expositions:

* [[Ieke Moerdijk]], *Background in topos theory*, chapter I in: _Classifying Spaces and Classifying Topoi_, Lecture Notes in Mathematics **1616**, Springer (1995) [[doi:10.1007/BFb0094441](https://doi.org/10.1007/BFb0094441)]

* [[Luc Illusie]], *What is... a Topos?*, Notices of the AMS **51** 9 (2004) [[pdf](https://www.ams.org/notices/200409/what-is-illusie.pdf), full volume:[pdf](https://www.ams.org/notices/200409/200409FullIssue.pdf)]

* [[Tom Leinster]], _[[Leinster2010|An informal introduction to topos theory]]_ (2010) [[arXiv:1012.5647](https://arxiv.org/abs/1012.5647)]

* [[John Baez]], *Topos Theory in a Nutshell* (2021) [[web](http://math.ucr.edu/home/baez/topos.html)]

* MathProofsable, Category Theory - Toposes [video playlist] (https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k)

Lecture notes:

* [[Jean Bénabou]], *Problèmes dans les topos*, Séminaire de mathématique pure **34**, Université de Louvain (1973) [[[Benabou-ProblemesDansLesTopos.pdf:file]]]

* [[Ross Street]], _A survey of topos theory_, lecture notes (1978) [[pdf](http://www.math.mq.edu.au/~street/ToposSurvey.pdf), [[Street-SurveyToposTheory.pdf:file]]]

* {#Wyler} [[Oswald Wyler]], *Lecture Notes on Topoi and Quasitopoi*, World Scientific (1991) [[doi:10.1142/1047](https://doi.org/10.1142/1047)]

* {#Joyal15} [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), Paris (November 2015)

* [[André Joyal]], _Geometric aspects of topos theory in relation with logical doctrines_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris 2015 ([video recording](https://www.youtube.com/watch?v=kaZpOEOAUzE))

* [[Ieke Moerdijk]], [[Jaap van Oosten]], _Topos Theory_, Master class notes (2007) [[pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf)]

* [[Jaap van Oosten]], *Topos Theory* (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), [[vanOosten-ToposTheory.pdf:file]]]

* [[Francis Borceux]], *Some glances at topos theory*, lecture notes, Como (2018) [[pdf](https://tcsc.lakecomoschool.org/files/2018/01/Borceux.pdf), [[Borceux-ToposTheory.pdf:file]], [video playlist](https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11)]

A monograph that aims to be more expository, focusing on [[presheaf toposes]]:

* [[Marie La Palme Reyes]], [[Gonzalo E. Reyes]], [[Houman Zolfaghari]], _Generic figures and their glueings.  A constructive approach to functor categories_, Polimetrica (2008) [[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf), ISBN:8876990046]

### Monographs
 {#ReferencesMonographs}

* [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp. (Dover reprint 2014)

* [[Michael Barr]], [[Charles Wells]], _[[Toposes, Triples, and Theories]]_, Springer Heidelberg 1985. ([TAC reprint](http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf))

* [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, 1992

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]] 3 - Categories of Sheaves_, Cambridge UP 1994 ([ISBN:9780521061247](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/handbook-categorical-algebra-volume-3?format=PB))

* [[Peter Johnstone]], *[[Elephant|Sketches of an elephant: a topos theory compendium]]*  Oxford University Press 2002, Volume 1 ([ISBN:9780198534259](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198534259)), Volume 2 ([ISBN:9780198515982](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198515982))

### Classifying toposes

Specifically on [[classifying toposes]]:

* [[Ieke Moerdijk]], _Classifying Spaces and Classifying Topoi_, Lecture Notes in Mathematics 1616, Springer 1995 (ISBN: 978-3-540-60319-1, [doi:10.1007/BFb0094441](https://doi.org/10.1007/BFb0094441))

* [[Olivia Caramello]], _Theories, Sites, Toposes_, Oxford University Press, 2017.
[doi](https://doi.org/10.1007/BFb0094441)

### Categorical logic and elementary toposes
 {#CategoricalLogicAndElementaryToposes}

On [[categorical logic]] via toposes:

* [[Colin McLarty]], _Elementary Categories, Elementary Toposes_, Oxford University Press 1992 (reprinted in 2005) ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732))

* Jonathan Chapman, Frederick Rowbottom, _Relative Category Theory and Geometric Morphisms.  A Logical Approach_, Oxford University Press 1992 ([ISBN:9780198534341](https://global.oup.com/academic/product/relative-category-theory-and-geometric-morphisms-9780198534341))

* J. L. Bell, _Toposes and Local Set Theories.  An Introduction_, Oxford University Press 1988 ([doi:10.2307/2274680](https://doi.org/10.2307/2274680))

* [[Robert Goldblatt]], _Topoi. The categorial analysis of logic_, Studies in Logic and the Foundations of Math. __98__, North-Holland Publ. Co., Amsterdam, 1979, 1984; (Rus. transl. Mir Publ., Moscow 1983).

* {#LambekScott86} [[Joachim Lambek]], [[Philip J. Scott]], _Introduction to higher order categorical logic_, Cambridge Studies in Advanced Mathematics 7 (1986) ([ISBN: 0-521-24665-2](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534))

* [[Bart Jacobs]], _Categorical logic and type theory_, Studies in Logic and the Foundations of Mathematics 141 (1999).  North-Holland Publishing Co.
ISBN: 0-444-50170-3

### Algebra, ringed toposes, algebraic geometry

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete 64 (1972) ([ISBN:9783540055730](https://www.springer.com/gp/book/9783540055730))

* [[Francis Borceux]], Gilberte van den Bossche, _Algebra in a localic topos with applications to ring theory_, Lecture Notes in Mathematics 1038 (1983), ISBN: 3-540-12711-9 ([ISBN:9783540127116](https://www.springer.com/gp/book/9783540127116))

### Quantum theory

* [[Cecilia Flori]], _A first course in topos quantum theory_. Lecture Notes in Physics 868 (2013).
ISBN: 978-3-642-35712-1; 978-3-642-35713-8

* [[Cecilia Flori]], _A second course in topos quantum theory_. Lecture Notes in Physics 944 (2018).
ISBN: 978-3-319-71107-2; 978-3-319-71108-9

### Original articles

Original source of (Grothendieck) topoi:

* [[SGA4]] Exposé IV by Grothendieck and Verdier (and Exposé V for cohomology in a (Grothendieck) topos)

That every topos is an [[adhesive category]] is discussed in

* [[Stephen Lack]], Pawel Sobociński, _Toposes are adhesive_ ([pdf](http://users.ecs.soton.ac.uk/ps/papers/toposesAdhesive.pdf))
 {#LackSobocinski}

### History

* [[Colin McLarty]], _The Uses and Abuses of the History of Topos Theory_ , Brit. J. Phil. Sci., 41 (1990) ([JSTOR](http://www.jstor.org/stable/687825)) [PDF](http://bjps.oxfordjournals.org/content/41/3/351.full.pdf) ([direct link](http://www.cwru.edu/artsci/phil/UsesandAbuses%20HistoryToposTheory.pdf))

* [[John W. Gray]], _Fragments of the history of sheaf theory_,
in: Applications of Sheaves, pp. 1–79,
Lecture Notes in Mathematics 753,
ISBN: 978-3-540-09564-4,
[doi](https://doi.org/10.1007/BFb0061812).

According to appendix C.1 in

* {#LawvereRosebrugh03} [[William Lawvere]], [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , Cambridge UP  2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))

> "Topos" is a Greek term intended to describe the objects studied by
"[analysis situs](http://en.wikipedia.org/wiki/Analysis_Situs_%28paper%29)," the Latin term previously established by Poincaré to signify the science of place [or situation]; in keeping with those ideas, a $\mathcal{U}$-topos was shown to have presentations in various "sites".

A historical analysis of Grothendieck's 1973 Buffalo lecture series on toposes and their precedents is in

* {#McLarty18} [[Colin McLarty]], _Grothendieck's 1973 topos lectures_, Séminaire Lectures grothendieckiennes, 3 May (2018) ([YouTube video](https://www.youtube.com/watch?v=hhWT5V0oaSI))

### General theory

On the application of [[diads]] to constructing toposes:

* Toby Kenney, _Diads and their Application to Topoi_, Applied Categorical Structures 17 (2009): 567-590.

[[!redirects topos]]
[[!redirects topoi]]
[[!redirects toposes]]
[[!redirects elementary topos]]
[[!redirects elementary topoi]]
[[!redirects elementary toposes]]
[[!redirects Lawvere-Tierney topos]]
[[!redirects Lawvere-Tierney topoi]]
[[!redirects Lawvere-Tierney toposes]]
[[!redirects Lawvere–Tierney topos]]
[[!redirects Lawvere–Tierney topoi]]
[[!redirects Lawvere–Tierney toposes]]
[[!redirects Lawvere--Tierney topos]]
[[!redirects Lawvere--Tierney topoi]]
[[!redirects Lawvere--Tierney toposes]]

[[!redirects W-topos]]
[[!redirects W-topoi]]
[[!redirects W-toposes]]
[[!redirects topos with NNO]]
[[!redirects topoi with NNO]]
[[!redirects toposes with NNO]]
[[!redirects topos with an NNO]]
[[!redirects topoi with NNOs]]
[[!redirects toposes with NNOs]]


topos approach to quantum mechanics

> More details on the content of this entry are at _[[Bohr topos]]_.

***

There are several approaches using [[topos theory]] to formulate and study foundations of [[quantum mechanics]]. Some people say "quantum topos theory", which is however a bit unfortunate as it is not about some quantum analogue of a topos (while such an object may be desired, say in noncommutative geometry), but about using the usual topoi to model quantum mechanics.

The main proposal is from [[Chris Isham]] and his school, see [[higher category theory and physics]].

* Andread Doering, C.J. Isham, _A Topos Foundation for Theories of Physics_  I-IV [quant-ph/0703060](http://arxiv.org/abs/quant-ph/0703060)

* Kunji Nakayama, _Sheaves in quantum topos induced by quantization_, [arxiv/1109.1192](http://arxiv.org/abs/1109.1192)

See also [[Kochen-Specker theorem]] for the topos theoretic interpretation summarizing

* [[Jeremy Butterfield]],  John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_, _I. quantum states as generalized valuations_, Internat. J. Theoret. Phys. 37(11):2669--2733, 1998, [MR2000c:81027](http://www.ams.org/mathscinet-getitem?mr=1669557), [doi](http://dx.doi.org/10.1023/A:1026680806775); _II. conceptual aspects and classical analogues_ Int. J. of Theor. Phys. 38(3):827--859, 1999, [MR2000f:81012](http://www.ams.org/mathscinet-getitem?mr=1697983), [doi](http://dx.doi.org/10.1023/A:1026652817988); _III. Von Neumann algebras as the base category_, Int. J. of Theor. Phys. 39(6):1413--1436, 2000, [arXiv:quant-ph/9911020](http://arxiv.org/abs/quant-ph/9911020), [MR2001k:81016](http://www.ams.org/mathscinet-getitem?mr=1788498),[doi](http://dx.doi.org/10.1023/A:1003667607842); _IV. Interval valuations_, Internat. J. Theoret. Phys. __41__ (2002), no. 4, 613–639, [MR2003g:81009](http://www.ams.org/mathscinet-getitem?mr=1902067), [doi](http://dx.doi.org/10.1023/A:1015276209768) {#ButterfieldHamiltonIsham}

Another, more recent variant of using topos theory in quantum foundations is due Nijmengen school, see [[Bohr topos]] for detail.

* Chris Heunen, [[Nicolaas Landsman]], Bas Spitters, A topos for algebraic quantum theory [arXiv:0709.4364](http://arxiv.org/abs/0709.4364)

[[!redirects quantum topos theory]]
[[!redirects topos approaches to quantum mechanics]]


topos of coalgebras over a comonad

[[!redirects topos of algebras over a monad]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

If a [[monad]] or [[comonad]] $T$ on a [[topos]] $\mathcal{E}$ is sufficiently well behaved, then the [[Eilenberg-Moore category|category of (co)algebras]] $T Alg(C)$ over the (co)monad is itself an ([[elementary topos|elementary]]) topos.

## Properties

### General

+-- {: .num_prop #ToposProperty}
###### Proposition

Let $\mathcal{E}$ be a [[topos]]. Then

* if a [[comonad]] $T : \mathcal{E} \to \mathcal{E}$ is [[exact functor|left exact]], then the [[Eilenberg-Moore category|category of coalgebras]] $T CoAlg(\mathcal{E})$ is itself an ([[elementary topos|elementary]]) topos.

  Moreover,

  * the [[free functor|cofree/forgetful adjunction]]

    $$
      (U \dashv F)
      :
      \mathcal{E} \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}}  T CoAlg(\mathcal{E})
    $$

    is a [[geometric morphism]].

  * If $T$ is furthermore [[accessible monad|accessible]] and $\mathcal{E}$ is a [[sheaf topos]], then also $T CoAlg(\mathcal{C})$ is a sheaf topos.

  * Even if $T$ is merely [[pullback]]-preserving, the category of coalgebras is a topos.

* Therefore, if a [[monad]] $T : \mathcal{E} \to \mathcal{E}$ has a [[right adjoint]], then the [[Eilenberg-Moore category|category of algebras]] $T Alg(\mathcal{E})$ is itself an ([[elementary topos|elementary]]) topos. (Because the right adjoint of a monad carries a comonad structure, evidently a left exact comonad, and there is a canonical equivalence between the category of algebras over the monad and the category of coalgebras over the comonad.)

* If a monad on a topos is pullback-preserving and [[idempotent monad|idempotent]], then the category of algebras is a subtopos (hence the category of sheaves for some [[Lawvere-Tierney topology]]).

=--

The result for left exact comonads appears for instance as ([MacLaneMoerdijk, V 8. theorem 4](#MacLaneMoerdijk)); the result for monads possessing a right adjoint appears in _op. cit._ as corollary 7. The statement on pullback-preserving comonads is given in The [[Elephant]], A.4.2.3. For [[(∞,1)-toposes]] see [this MO discussion](http://mathoverflow.net/a/206695/381).

### Image factorization of toposes

+-- {: .num_prop}
###### Proposition

The [[geometric morphism]]s of the form $ p = (U \dashv F) : \mathcal{E} \to T CoAlg(\mathcal{E})$ from prop. \ref{ToposProperty} are precisely, up to [[equivalence of categories|equivalence]], the [[geometric surjection]]s.

=--

This appears as ([MacLaneMoerdijk, VII 4. prop. 4](#MacLaneMoerdijk)).

This way the [[geometric surjection/embedding factorization]] in [[Topos]] is constructed. See there for more.

## Examples

+-- {: .num_remark}
###### Observation

For $(f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ any [[geometric morphism]], the induced [[comonad]]

$$
  f^* f_* : \mathcal{E} \to \mathcal{E}
$$

is evidently left exact, hence $(f^* f_*) CoAlg(\mathcal{E})$ is a topos of coalgebras. See also at _[[monadic descent]]_.

=--

+-- {: .num_remark}
###### Observation
The so-called "[[over-topos|fundamental theorem of topos theory]]", that an [[overcategory]] of a topos is a topos, is a corollary of the result that the category of coalgebras of a pullback-preserving comonad on a topos is a topos (the slice $\mathcal{E}/X$ being the category of coalgebras of the comonad $X \times - \colon \mathcal{E} \to \mathcal{E}$).
=--

## Related concepts

* Also a category of [[algebra over an algebraic theory|algebras over]] a *[[commutative theory|commutative]]* [[finitary algebraic theory]] in [[Set]] has properties very close to the properties of a Grothendieck topos, in fact only one axiom has to be modified. This is one of the themes of the theory of [[vectoids]] of [[Nikolai Durov]].

## References

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], section V 8. of _[[Sheaves in Geometry and Logic]]_

* [[Peter Johnstone]], _When is a variety a topos?_, Algebra Universalis, 21 (1985) 198-212

* [[Peter Johnstone]], _Collapsed toposes and cartesian closed varieties_, [link](http://www.sciencedirect.com/science/article/pii/002186939090230L)

* [[Peter Johnstone]], _Cartesian monads on toposes_, [link](http://www.sciencedirect.com/science/article/pii/S002240499600165X)

[[!redirects topos of coalgebras over a comonad]]

[[!redirects toposes of algebras over a monad]]
[[!redirects toposes of coalgebras over a comonad]]

[[!redirects topoi of algebras over a monad]]
[[!redirects topoi of coalgebras over a comonad]]

[[!redirects topos of coalgebras]]


topos of laws of motion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In ([Lawvere 97](#Lawvere97)) it was observed that [[equations of motion]] in [[physics]] can (almost, see below) be formalized in [[synthetic differential geometry]] as follows.

Let $\mathbf{H}$ be an ambient [[smooth topos|synthetic differential]] [[topos]] (such as the [[Cahiers topos]] of [[smooth spaces]] and [[formal smooth manifolds]]).

The canonical [[line object]] $\mathbb{A}^1 = \mathbb{R}$ of this models the [[continuum]] line, the abstract [[worldline]]. Let

$$
  D \hookrightarrow \mathbb{R}
$$

be the inclusion of the first order [[infinitesimally thickened point|infinitesimal]] neighbourhood of the origin of $\mathbb{R}$ -- in the [[internal logic]] this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the [[spectrum of a commutative ring|spectrum]] of the [[ring of dual numbers]] over $\mathbb{R}$.

Then for $X \in \mathbf{H}$ any [[object]] which we are going to think of as a [[configuration space]] of a [[physical system]]. For instance if the system is a [[particle]] propagating on a [[spacetime]], then $X$ is that spacetime. Or $X$ may be the [[phase space]] of the system.

Accordingly the [[mapping space]] $[\mathbb{R}, X] \in \mathbf{H}$ is the [[smooth space|smooth]] [[path space]] of $X$. This is the space of potential [[trajectories]] of the [[physical system]].

If $X$ is thought of as [[phase space]], then every point in there determines a unique [[trajectory]] starting at that point. This means that time evolution is then an [[action]] of $\mathbb{R}$ on $X$. As $X$ here might be any space, we have the collection

$$
  \mathbb{R}Act(\mathbf{H}) \in Topos
$$

of all $\mathbb{R}$-[[actions]] on objects in $\mathbf{H}$. This is again a [[topos]], and hence this is a first version of what one might call a _[[topos of laws of motion]]_.

On the other hand, if we think of $X$ as [[configuration space]], then it is (in the simplest but common case of [[physical systems]]) a [[tangent vector]] in $X$ that determines a [[trajectory]], hence a point in $[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth [[path space]] to the [[tangent bundle]], which sends each path to its [[tangent vector]]/[[derivative]] at $0 \in \mathbb{R}$. A [[section]] of this map is hence an assignment that sends each tangent vector to a [[trajectory]] which starts out with this tangent. Specifying such a section is hence part of what it means to have [[equations of motion]] in [[physics]]. Accordingly in _[[Toposes of laws of motion]]_ [[Lawvere]] called the collection of such data a Galilian [[topos of laws of motion]].

Of course this is not quite yet what is actually used and needed in physics.  On p. 9 of ([Lawvere 97](#Lawvere97)) this problem is briefly mentioned:

> But what about actual dynamical systems in the spirit of [[Galileo]], for example, second-order ODE's? (Of course, the [[symplectic geometry|symplectic]] or [[Hamiltonian mechanics|Hamiltonian systems]] that are also much studied do address this question of states of [[becoming|Becoming]] versus locations of [[being|Being]], but in a special way which it may not be possible to construe as a topos;

It turns out that it does exist as a "[[higher topos theory|higher topos]]".

In _[[schreiber:Classical field theory via Cohesive homotopy types]]_ it is observed that if $\mathbf{H}$ to be not just a [[topos]] but an [[(infinity,1)-topos|infnity-topos]], then genuine [[classical mechanics]] governed by [[Hamilton's equations]] and [[Hamilton-Jacobi theory]] arises by actions of $\mathbb{R}$ on objects in the [[slice (infinity,1)-topos]] $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$, where $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ is the [[moduli stack]] of [[circle group]]-[[principal connections]]. So

$$
  \mathbb{R}Act\left(
    \mathbf{H}_{/\mathbf{B}U(1)_{conn}}
  \right)
$$

is actually a "topos of laws of motion" in the sense of Hamilton-Lagrange-Jacobi [[classical mechanics]]. For more on this see _[[Higher toposes of laws of motion]]_.

## Related concepts

* [[Higher toposes of laws of motion]]

## References

The notion originates around

* [[William Lawvere]] _[[Categorical dynamics]]_, 1967 Chicago lectures ([pdf](http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf))

and was made more explicit in

* [[William Lawvere]], _[[Toposes of laws of motion]]_, 1997
 {#Lawvere97}

[[!redirects toposes of laws of motion]]


topos of pointed objects

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The category of [[pointed object|pointed objects]] $1\backslash \mathcal{E}$ of a topos $\mathcal{E}$ has [[zero object|zero objects]] hence can be the degenerate topos at best. By altering the notion of morphism it is nevertheless possible to obtain a topos $^\bullet\mathcal{E}$ with objects $1\to X$, called the **topos of pointed objects**.

## Definition

Let $\mathcal{E}$ be a topos. The _topos $^\bullet\mathcal{E}$ of pointed objects_ has objects the morphisms $1\to X$ and morphisms pullback squares:

$$
\begin{array}{cccc}1& \to  & X   \\
\downarrow     &                   & \downarrow \\
1       & \to & Y
\end{array}

$$

## Properties

* Foremost, $^\bullet\mathcal{E}$ is a topos (cf. [Freyd 1987](#freyd87)).

## Related entries

* [[pointed object]]

* [[point]]

* [[fibration of points]]

## Reference

* {#freyd87}[[Peter Freyd]], _Choice and Well-ordering_ , APAL **35** (1987) pp.149-166. (section 5)


topos of recursive sets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
The **topos of recursive sets**, or, for short, the _recursive topos_ , is a [[Grothendieck topos]] $\mathcal{R}$ that provides a [[cartesian closed category|cartesian closed]] context to do higher order recursion theory in. It can be viewed as a discrete analogue to the [[Johnstone's topological topos|topological topos]].

## Definition
Let $\mathbf{R}$ be the monoid of all total [[recursive function|recursive functions]] $\mathbb{N}\to\mathbb{N}$ under concatenation viewed as a category with one object $\mathbb{N}$. Consider the [[coverage]] $J(\mathbb{N})$ whose covers consist of finite families of maps $(f_i:\mathbb{N}\to\mathbb{N} {}|{} 1\leq i \leq n)$ such that $\cup_i Im(f_i)=\mathbb{N}$.

The _recursive topos_ is defined as $\mathcal{R}=Sh(\mathbf{R}, J)$.

## Properties

+-- {: .num_prop}
###### Proposition
$\mathcal{R}$ is a [[local topos]] i.e. the [[global section functor]] $\Gamma:\mathcal{R}\to Set$ has a right adjoint: $\Gamma\dashv B: Set\to \mathcal{R}$.
=--

This appears as prop.1.5 in [Mulry (1982)](#Mulry82). $B$ maps a set $X$ to the sheaf $X^{\mathbb{N}}$ of sequences of elements in $X$.

+-- {: .num_prop #RNO}
###### Proposition
The object $\mathbb{R}_D$ of [[real numbers object|Dedekind real numbers]] corresponds to the sheaf of sequences of real numbers $\langle r_n\rangle$ satisfying:

For all $n\in \mathbb{N}^+$ there exists a finite recursive cover $A_1,\dots A_k$ such that for all $m_1, m_2$ in $A_i$ $(i=1,\dots,k)$, $|r_{m_2}-r_{m_1}|\lt 1/n$.
=--

This is prop.4.1 in [Mulry (1982)](#Mulry82).

## Related entries

* [[Johnstone's topological topos]]

* [[bornological topos]]

* [[effective topos]]

* [[recursive function]]

## References

The recursive topos was introduced in

* {#Mulry82}[[Philip Mulry|P. S. Mulry]], _Generalized Banach-Mazur functionals in the topos of recursive sets_ , JPAA **26** (1982) pp.71-83.

A comparison with the [[effective topos]] is in chapter 6 of

* [[Giuseppe Rosolini]], _Continuity and Effectiveness in Topoi_ , PhD thesis University of Oxford 1986. ([ps](ftp://ftp.disi.unige.it/pub/person/RosoliniG/papers/coneit.ps.gz))

See also

* [[Peter Johnstone]], _Sketches of an Elephant vol. I_, Oxford UP 2002. (example A2.1.11(k), p.81)

* [[William Lawvere|F. William Lawvere]], _Diagonal arguments and cartesian closed categories_ , pp.134-145 in LNM **92** Springer Heidelberg 1969. (Reprinted with author's commentary as [TAC Reprint **15** (2006)](http://tac.mta.ca/tac/reprints/articles/15/tr15abs.html))

* [[Philip Mulry|P. S. Mulry]], _Adjointness in recursion_ ,
APAL **32** (1986), pp.281–289.

* [[Philip Mulry|P. S. Mulry]], _A categorical approach to the theory of computation_ ,
APAL **43** (1989), pp.293–305.

* [[Philip Mulry|P. S. Mulry]], _Partial map classifiers and partial cartesian closed categories_ , Theor. Comp. Sci. **136** (1994) pp.109–123.

[[!redirects Mulry topos]]
[[!redirects Mulry-Lawvere topos]]
[[!redirects Lawvere-Mulry topos]]
[[!redirects recursive topos]]


topos of trees

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The **[[topos]] of trees** is the category of [[presheaves]] over the [[ordinal]] of [[natural numbers]], $\omega$. More properly, this is the category of trees of  _height_ bounded by $\omega$, in that every path from the root has length $\omega$ or less (when considered as ordinals). These trees can, however, be arbitrarily branching at every level.

An object is then a family of sets $X_i$ for each $i \in \omega$ with restriction functions $X_i \leftarrow X_{i+1}$. We can visualize this as a (potentially infinite) tree (really, forest) where an element of any $X_i$ is a node of the tree, and the restriction functions map each node to its parent node.

## Related pages

* [[synthetic guarded domain theory]]
* [[guarded recursion]]


topos of types

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $C$ be a [[coherent category]].  Write $\tau C$ for the category whose objects are pairs $(X,F)$ where $X$ is an object of $C$ and $F$ is a [[prime filter]] on $Sub_C(X)$.  This is the full subcategory of the [[category of filters]] of $C$ spanned by the prime filters.  Jet $J_p$ be the [[Grothendieck topology]] on $\tau C$ induced from the [[coherent topology]] on the category of filters.

As defined by [Makkai](#Makkai), the **topos of types** $\mathbf{T}(C)$ of $C$ is the [[sheaf topos]]

$$
  \mathbf{T}(C) \coloneqq Sh_{J_p}(\tau C)
  \,.
$$

**Note:** The word "types" here has nothing to do with [[type theory]].  Instead, refers to the notion of [[type (in model theory)|type in model theory]].

## Properties

### In terms of the coherent hyperdoctrine

Let $C$ be a [[coherent category]]. The [[canonical extension]] $\mathcal{S}_C^\delta$ of its [[coherent hyperdoctrine]] $\mathcal{S}_C$  is naturally an [[internal locale]] in the [[presheaf topos]] $[C^{op}, Set]$ (as well as in the [[sheaf topos]] $Sh_{coh}(C)$ for the [[coherent topology]] on $C$).

The topos of types is equivalent to the topos of [[internal sheaves]] over this [[internal locale]].

$$
  \mathbf{T}(C)
  \simeq
  Sh(\mathcal{S}_C^\delta)
  \,.
$$

This is due to ([Coumans, theorem 25](#Coumans)).

In particular this implies that there is a [[localic geometric morphism]]

$$
 \mathbf{T}(C) \to Sh_{coh}(C)
 \,.
$$

Let $Mod(C) \simeq Func_{coh}(C,Set)$ be the category of [[models]] of the [[coherent theory]] given by $C$ in [[Set]].

There is a [[full subcategory]] $\mathcal{K} \hookrightarrow Mod(C)$ such that the [[Yoneda extension]] of the [[evaluation map]] $ev : C \to Func(\mathcal{K}, Set)$ is the [[inverse image]] of a [[geometric morphism]]

$$
  \phi_{ev} : Set^{\mathcal{K}} \to Sh_{coh}(C)
  \,.
$$

The [[(hyperconnected, localic) factorization system|hyperconnected / localic factorization]] of this morphism is through the topos of types of $C$:

$$
  \phi_{ev}
   :
  Set^{\mathcal{K}}
  \stackrel{hyperconn.}{\to}
  \mathbf{T}(C)
  \stackrel{localic}{\to}
  Sh_{coh}(C)
  \,.
$$

This is ([Coumans, theorem 39](#Coumans)).

## Related concepts

* [[canonical extension]]

## References

The notion was introduced in

* [[Michael Makkai]], _The topos of types_, Logic Year 1979-80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80), Lecture Notes in Math. __859__, Springer, Berlin (1981) pp. 157-201, [doi](http://dx.doi.org/10.1007/BFb0090947).
 {#Makkai}

The relation to [[canonical extension]] is made explicit in

* Dion Coumans, _Generalizing canonical extensions to the categorical setting_, Annals of Pure and Applied Logic
December 2012, Vol.163(12):1940–1961  doi:[10.1016/j.apal.2012.07.002](http://dx.doi.org/10.1016/j.apal.2012.07.002), [pdf](http://www.math.ru.nl/~coumans/Generalisingcanonicalextensiontothecategoricalsetting-DionCoumans-vs2.pdf)
{#Coumans}

[[!redirects toposes of types]]
[[!redirects topoi of types]]


Topos Theory

[[!redirects Topos Theory 1977]]

> Incidentally, the reaction of one Cambridge colleague on learning
that I'd appeared as a Dover reprint was 'I didn't know you were dead'.  Peter Johnstone[^mail]

This page compiles material related to

* [[Peter Johnstone]]:

  **Topos Theory**,

  Academic Press (1977)

  Dover reprint (2014)

  xxiii + 367 pages

on [[topos theory]].

Peter Johnstone's classic text on [[topos theory]] from 1977 served as the standard reference in the field until the publication of MacLane-Moerdijk's *[[Sheaves in Geometry and Logic]]* in 1992 and the monumental first two volumes of *[[Sketches of an elephant]]* in 2002, also by Johnstone. _Topos Theory_ reportedly contains almost all results in topos theory known in the mid 1970s.

(Johnstone also wrote *[[Stone Spaces]]*.)

Although the author takes no prisoners the text is not really as unreadable as the introduction to the [[Elephant]] boasts but contains in a concise form what still constitutes the core of the subject.

It has also several unique features in comparison with its successor volumes in that it offers (somewhat more opiniated) remarks on the _historical development of topos theory_ and an emphasis of the intuitionistic algebra and theory of spectrum prominent in the early 1970s, a chapter on _cohomology theory_ and due to the recent paperback reprint an affordable price.

## Table of contents

* chap. 0. _Preliminaries_
* chap. 1. _Elementary Toposes_
* chap. 2. _Internal Category Theory_
* chap. 3. _Topologies and Sheaves_
* chap. 4. _Geometric Morphisms_
* chap. 5. _Logical Aspects of Topos Theory_
* chap. 6. _Natural Number Objects_
* chap. 7. _Theorems of Deligne and Barr_
* chap. 8. _Cohomology_
* chap. 9. _Topos Theory and Set Theory_
* Appendix _Locally Internal Categories_

## References

* [[Robert Seely]], _Review, P. T. Johnstone, Topos Theory_ , JSL **47** no.2 (1982) pp.448-450. ([draft](www.math.mcgill.ca/rags/JSL/PTJ.pdf))

[^mail]: [mail to catlist March 2014](http://comments.gmane.io/gmane.science.mathematics.categories/8044).

[[!redirects Johnstone Topos Theory]]
[[!redirects Johnstone topos theory]]

category:reference


topos theory - contents

**[[topos theory]]**

* [[Toposes]]

## Background

* [[category theory]]

  * [[category]]

  * [[functor]]

## Toposes ##

* [[(0,1)-topos]], [[Heyting algebra]], [[locale]]

* [[pretopos]]

* [[topos]]

  * [[Grothendieck topos]]

    * [[category of presheaves]]

      * [[presheaf]]

      * [[representable functor|representable presheaf]]

    * [[category of sheaves]]

      * [[site]]

        * [[sieve]]

        * [[coverage]], [[Grothendieck pretopology|pretopology]], [[Grothendieck topology|topology]]

      * [[sheaf]]

      * [[sheafification]]

* [[quasitopos]]

* [[base topos]], [[indexed topos]]

## Internal Logic

* [[categorical semantics]]

* [[internal logic]]

  * [[subobject classifier]]

  * [[natural numbers object]]

## Topos morphisms

* [[logical morphism]]

* [[geometric morphism]]

  * [[direct image]]/[[inverse image]]

  * [[global section|global sections]]

  * [[geometric embedding]]

  * [[surjective geometric morphism]]

  * [[essential geometric morphism]]

    * [[locally connected geometric morphism]]

    * [[connected geometric morphism]]

    * [[totally connected geometric morphism]]

  * [[étale geometric morphism]]

  * [[open geometric morphism]]

  * [[proper geometric morphism]], [[compact topos]]

  * [[separated geometric morphism]], [[Hausdorff topos]]

  * [[local geometric morphism]]

  * [[bounded geometric morphism]]

  * [[base change]]

  * [[localic geometric morphism]]

  * [[hyperconnected geometric morphism]]

  * [[atomic geometric morphism]]

## Extra stuff, structure, properties

* [[topological locale]]

* [[localic topos]]

* [[petit topos|petit topos/gros topos]]

* [[locally connected topos]], [[connected topos]], [[totally connected topos]], [[strongly connected topos]]

* [[local topos]]

* [[cohesive topos]]

* [[classifying topos]]

* [[smooth topos]]

## Cohomology and homotopy

* [[cohomology]]

* [[homotopy groups in an (infinity,1)-topos|homotopy]]

* [[abelian sheaf cohomology]]

* [[model structure on simplicial presheaves]]

## In higher category theory

* [[higher topos theory]]

* [[(0,1)-topos]]

  * [[(0,1)-site]]

* [[2-topos]]

  * [[2-site]]

  * [[2-sheaf]], [[stack]]

* [[(∞,1)-topos]]

  * [[(∞,1)-site]]

  * [[(∞,1)-sheaf]], [[∞-stack]], [[derived stack]]

## Theorems

* [[Diaconescu's theorem]]

* [[Barr's theorem]]

[Edit this sidebar](/nlab/edit/topos+theory+-+contents)


toposes are extensive

# Toposes are extensive

* table of contents
{: toc}

## Statement

Every [[elementary topos]] is a finitary [[extensive category]].  In fact, more is true: if an elementary topos has [[coproducts]] indexed by any [[arity class]] $\kappa$, then it is $\kappa$-extensive.  In particular, a cocomplete elementary topos is infinitary extensive.

## Proofs

There are several possible proofs.  In all cases, stability of coproducts under pullback is automatic since a topos is locally cartesian closed, so it suffices to consider [[disjoint coproduct|disjointness]].

### From finitary to infinitary

As shown at [[extensive category]], if we assume [[classical logic]] then $\coprod_{j\in J} A_j \cong A_1 \sqcup A_2 \sqcup \coprod_{j\neq 1,2} A_j$, so that in a finitary-extensive category, any existing coproducts are stable and disjoint.  Thus, in this case it suffices to deal with finite coproducts.  However, in topos theory we are often interested in toposes "over an arbitrary base", in which case the ambient logic is [[constructive logic]] and so this argument does not apply.

Nevertheless, we mention a couple of proofs that apparently apply only to the finitary case in addition to the fully general version.

### Via pushouts of monos

A proof applying to finite coproducts is given in [Johnstone, A2.4.4](#Johnstone): since $0\to X$ and $0\to Y$ are monic, it suffices to show that the pushout of a mono is a mono and also a pullback.

To see this, let $f:A\rightarrowtail B$ be monic and $g:A\to C$ any morphism, and let $h:B\to P C$ be the name of $(f,g):A\to B\times C$, i.e. $(f,g)$ is the pullback $\in_C \rightarrowtail P C \times C$ along $(h,1)$.  In other words, $(k,l):X\to B\times C$ factors through $(f,g)$ iff $(h k,l)$ factors through $\in_C$ (i.e. $l(x) \in h k(x)$ for all $x:X$ in the [[internal logic]]).  This is in particular the case if $h k = \{\} \circ l$ (i.e. $h k(x) = \{l(x)\}$ for all $x:X$ in the internal logic), which implies that the square
$$\array{A & \xrightarrow{g} & C \\ ^f\downarrow && \downarrow^{\{\}} \\ B & \xrightarrow{h} & P C }$$
is a pullback.

Now given a pushout
$$\array{A & \xrightarrow{g} & C \\ ^f\downarrow && \downarrow^{d} \\ B & \xrightarrow{c} & D }$$
with $f$ monic, with $h$ as above we have an induced $m:D \to P C$ by the universal property of the pushout, with $m d = \{\}$ and $m c = h$.  Since $\{\}$ is monic, $d$ must also be monic, and since the previous square is a pullback, so must this one be.

### Via quasi-disjointness

A different proof applying to finite coproducts can be obtained by combining A1.5.14 and A1.6.2 of [Johnstone](#Johnstone).  Define $R$ and $S$ to make the following squares pullbacks:
$$\array{ R & \xrightarrow{r} & A & \xleftarrow{s} & S\\
  ^{r'}\downarrow && \downarrow^i && \downarrow\\
  A & \xrightarrow{i} & A+B & \xleftarrow{j} & B}$$
Specifically, $(r,r')$ is the [[kernel pair]] of $i$.  In particular, there is an induced diagonal $\triangle : A \to R$ such that $r \triangle = 1_A$.  On the other hand, since pullback preserves colimits, $(r,s)$ is a coproduct diagram.  Thus, the pair of morphisms $1_R:R\to R$ and $\triangle \circ s : S\to R$ factor (uniquely) through some $h:A\to R$, so that in particular $h r = 1_R$.  Thus, $r:R\to A$ has both a left and a right inverse, so it is an isomorphism, and hence $i$ is monic.  Dually, $j$ is monic.

Now we claim that any two morphisms $f,g:S\to X$ with domain $S$ are equal.  For the pair of morphisms $R \xrightarrow{r} A \to A+X$ and $S \xrightarrow{f} X \to A+X$ factor uniquely through $A$ (which, recall, is the coproduct $R+S$).  But since $r$ is an isomorphism, this factorization must be the left coprojection $A\to A+X$.  Therefore, the composite $S \xrightarrow{f} X \to A+X$ is equal to the composite $S \xrightarrow{s} A \to A+X$, and similarly so is the composite $S \xrightarrow{g} X \to A+X$.  Since $X\to A+X$ is monic by the previous paragraph, $f= g$.

Finally, since $0$ is a strict initial object, $0\to S$ is monic.  And of course $1_S : S\to S$ is also monic, and these two monomorphisms are classified by two maps $f,g:S\to \Omega$ into the subobject classifier.  By the previous paragraph, $f= g$, hence $S\cong 0$ and is initial.

### Via logical functors

Finally, we give a proof that applies constructively to arbitrary coproducts.  This proof is due to Jibladze and can be found (in fibered-topos language) in [Streicher, Appendix A](#Streicher). (Moens himself always just assumed that internal sums are stable and disjoint whereas Jibladze proved that for every fibered topos with internal sums these are actually stable and disjoint.)

Suppose $\mathcal{E}$ a topos with $J$-indexed coproducts for all $J$ belonging to some [[arity class]] $\kappa$.  Then $\Delta : \mathcal{E} \to \mathcal{E}^J$ is a [[logical functor]] with a left adjoint $\coprod_J$, so (by A2.3.8 of [Johnstone](#Johnstone), see also [[logical functor]]) to show that $\coprod_J$ induces equivalences $\mathcal{E}^J/\{A_j\} \to \mathcal{E}/\coprod_j A_j$ (which is one characterization of $\kappa$-extensivity) it suffices for $\coprod_J$ to be faithful.  As with any adjunction, to show the left adjoint to be faithful it suffices to show that the unit $\eta : Id \to \Delta \coprod_J$ is monic, which means in this case that the injections $A_j \to \coprod_j A_j$ are all monic.

Now since $\kappa$ is an arity class, $J\times J$ also belongs to it, and hence so do the fibers of the diagonal $J\to J\times J$.  Thus, $\mathcal{E}$ has coproducts indexed by the [[subsingletons]] determined by the equalities $(j=k)$ for any $j,k\in J$.  Moreover, for any $j$ we can write $A_j = \coprod_{k\in J} \coprod_{(j=k)} A_{k}$, and under this identification the injection $A_j \to \coprod_{k\in J} A_{k}$ is obtained by applying the functor $\coprod_J$ to the evident family of maps $\coprod_{(j=k)} A_{k}\to A_k$.  But since $\coprod_J$ is the left adjoint of a logical functor, it preserves monomorphisms (see A2.4.8 of [Johnstone](#Johnstone), also [[logical functor]]).  Thus, it suffices to prove that each $\coprod_{(j=k)} A_{k}\to A_k$ is monic.

Now, the functor $\coprod_{(j=k)} : \mathcal{E}^{(j=k)} \to \mathcal{E}$ has a logical right adjoint sending $B\in\mathcal{E}$ to the subsingleton-indexed family $\{B\}_{(j=k)}$, and the units of its adjunction $\{A\}_{(j=k)} \to \{ \coprod_{(j=k)} A \}_{(j=k)}$ are isomorphisms (to define an inverse, it suffices to consider the object $\coprod_{(j=k)} A$ under the assumption $j=k$, in which case this object is isomorphic to $A$).  Thus, $\coprod_{(j=k)}$ is fully faithful, and in particular faithful, and since its right adjoint is also logical, it induces an equivalence $\mathcal{E}^{(j=k)} \simeq \mathcal{E}/\coprod_{(j=k)} 1$.

However, the unit of a pullback adjunction $\mathcal{E}/I \rightleftarrows \mathcal{E}$ is $A\to A\times I$, and if this is an isomorphism then in particular $I\cong I\times I$ and thus $I$ is subterminal.  Moreover, in this case the counits of the adjunction are the projections $A\times I \to A$, which are pullbacks of the mono $I\to 1$ and therefore also monic.  Finally, in our case where $I = \coprod_{(j=k)} 1$, the counit associated to $A_k$ is precisely the map $\coprod_{(j=k)} A_{k} \to A_k$ that we wanted to prove to be monic.

## References

* {#Johnstone} [[Peter Johnstone]], [[Sketches of an Elephant]]

* {#Streicher} [[Thomas Streicher]], *Fibered categories a la Jean Benabou*, [citeseer](http://citeseerx.ist.psu.edu/viewdoc/citations;jsessionid=76DB3E2F57E81C74E05109B8F7E11E44?doi=10.1.1.678.919)


Toposes of laws of motion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

This page collects links related to the lecture note

* [[William Lawvere]],

  _Toposes of laws of motion_ ,

  transcript of a talk in Montreal, Sept. 1997

  ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))

on the formulation of [[differential equations]]/[[continuum mechanics]] in [[synthetic differential geometry]] and the notion of [[toposes of laws of motion]].

Another early text in this direction is Lawvere's _[[Categorical dynamics]]_. Related texts on the [[foundations]] of [[physics]] in [[topos theory]] include the collection _[[Categories in Continuum Physics]]_.

An open question concerning the characterization of "Toposes of laws of motion" is raised as question 7 "The algebra of time" in

* _Open problems in topos theory_, April 2009 ([pdf](http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf))

Entries with related discussion include _[[geometry of physics]]_ and _[[higher category theory and physics]]_. Refinement to [[higher topos theory]] is discussed at _[[Higher toposes of laws of motion]]_.

## Related entries

* _[[Some Thoughts on the Future of Category Theory]]_

category: reference


Toposes, Triples, and Theories

This page hosts material related to the textbook

* [[Michael Barr]], and  [[Charles Wells]]

  \linebreak

  **Toposes, Triples, and Theories**

  \linebreak

  Grundlehren der math. Wissenschaften **278**

  Springer (1985)

  \linebreak

  republished in:

  [[TAC reprints series|Reprints in Theory and Applications of Categories]]

  **12** (2005) 1-287

  [tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)

on [[toposes]], [[monads]] ("triples"), and [[universal algebra]], [[categorical algebra]] and [[categorical semantics]] via [[sketches]].

\linebreak

See also the following, which was a review of the first edition of the book. Some, but not all, of the issues were addressed in the second edition:

* {#Lair86} [[Christian Lair]], _À propos de «Toposes, triples and theories» de messieurs M. Barr et C. Wells_, Diagrammes **15** 4 (1986) L1-L20 [[numdam:DIA_1986__15__A4_0](http://www.numdam.org/item/DIA_1986__15__A4_0)]

\linebreak

## Contents

## Preface
###1. Categories

#### 1 Definition of category . . . . . . . . . . . . . . . . . . . . . . . . . 1

####2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....10

####3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . 14

####4 Elements and Subob jects . . . . . . . . . . . . . . . . . . . . . . . 17

####5 The Yoneda Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 22

####6 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

####7 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

####8 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

####9 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

####10 Filtered colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

####11 Notes to Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

###2. Toposes 62

####1 Basic Ideas about Toposes . . . . . . . . . . . . . . . . . . . . . . 62

####2 Sheaves on a Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

####3 Properties of Toposes . . . . . . . . . . . . . . . . . . . . . . . . . 72

####4 The Beck Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 77

####5 Notes to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 80

###3. Triples 82

####1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . 82

####2 The Kleisli and Eilenberg-Moore Categories . . . . . . . . . . . . 87

####3 Tripleability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

####4 Properties of Tripleable Functors . . . . . . . . . . . . . . . . . . 103

####5 Sufficient Conditions for Tripleability . . . . . . . . . . . . . . . . 108

####6 Morphisms of Triples . . . . . . . . . . . . . . . . . . . . . . . . . 110

####7 Adjoint Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

####8 Historical Notes on Triples . . . . . . . . . . . . . . . . . . . . . . 120

###4. Theories 122

* [[algebraic theory]]

####1 Sketches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

* [[sketch]]

####2 The Ehresmann-Kennison Theorem . . . . . . . . . . . . . . . . . 127

####3 Finite-Product Theories . . . . . . . . . . . . . . . . . . . . . . . . 129

* [[Lawvere theory]]

####4 Left Exact Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 135

####5 Notes on Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

###5. Properties of Toposes 147

####1 Tripleability of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

####2 Slices of Toposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

####3 Logical Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

####4 Toposes are Cartesian Closed . . . . . . . . . . . . . . . . . . . . 156

####5 Exactness Properties of Toposes . . . . . . . . . . . . . . . . . . . 158

####6 The Heyting Algebra Structure on Ω . . . . . . . . . . . . . . . . 165

###6. Permanence Properties of Toposes 169

####1 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

####2 Sheaves for a Topology . . . . . . . . . . . . . . . . . . . . . . . . 174

####3 Sheaves form a topos . . . . . . . . . . . . . . . . . . . . . . . . . 179

####4 Left exact cotriples . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

####5 Left exact triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

####6 Categories in a Topos . . . . . . . . . . . . . . . . . . . . . . . . . 188

####7 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . 194

####8 Giraud's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

###7. Representation Theorems 206

####1 Freyd's Representation Theorems . . . . . . . . . . . . . . . . . . 206

####2 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . . 210

####3 Morphisms of Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

####4 Deligne's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

####5 Natural Number Objects . . . . . . . . . . . . . . . . . . . . . . . 221

####6 Countable Toposes and Separable Toposes . . . . . . . . . . . . . 229

####7 Barr's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

####8 Notes to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 236

###8. Cocone Theories 238

####1 Regular Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

####2 Finite Sum Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 241

####3 Geometric Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 242

####4 Properties of Model Categories . . . . . . . . . . . . . . . . . . . 244

###9. More on Triples 250

####1 Duskin's Tripleability Theorem . . . . . . . . . . . . . . . . . . . 250

####2 Distributive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

####3 Colimits of Triple Algebras . . . . . . . . . . . . . . . . . . . . . . 262

####4 Free Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

###Bibliography 273

###Index of exercises 278

###Index


Topospaces

A wiki with basics of [[topology]].

* [Webpage](http://topospaces.subwiki.org/wiki/Main_Page)

category: reference


TopVect

Given a [[field]] $k$, the [[category]] **$TopVect_k$** is the [[category]] whose [[object|objects]] are [[topological vector space]]s and whose [[morphism]]s are [[continuous linear map]]s.


Tor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[homological algebra]], the _$Tor$-functor_ is the [[derived tensor product]]: the [[left derived functor]] of the [[tensor product of modules|tensor product of]] $R$-[[modules]], for $R$ a [[commutative ring]].

Together with the [[Ext-functor]] it constitutes one of the central operations of interest in [[homological algebra]].

## Definition

Given a [[ring]] $R$ the [[bifunctor]] $\otimes_R : Mod_{R} \times {}_{R}Mod\to Ab$
from two copies of $R$-[[Mod]] to [[Ab]] is a [[right exact functor]]. Its
[[left derived functors]] are the **Tor-functors

$$
  Tor(-,B)
  :
  Mod_R \to Ab
$$

and

$$
  Tor(A,-)
  :
  {}_{R}Mod \to Ab
$$

with respect to one argument with fixed another, if they exist, are parts of a bifunctor

$$
  Tor : Mod_{R}\times {}_{R}Mod\to Ab
  \,.
$$

## Properties

### Existence and balancing
 {#ExistenceAndBalancing}

Given a right $R$-module

$$
  A \in Mod_R
$$

and a left $R$-module

$$
  B \in {}_R Mod
$$

there are in principle three different ways to compute their derived tensor product $Tor_\bullet(A,B)$:

1. keeping $B$ fixed and deriving the functor

   $$
     (-) \otimes_R B : Mod_R \to Ab
   $$

1. keeping $A$ fixed and deriving the functor

   $$
     A \otimes_R (-) : {}_R Mod \to Ab
   $$

1. deriving the functor

   $$
     (-) \otimes_R (-) : Mod_R \times {}_R Mod \to Ab
   $$

   in both arguments

+-- {: .num_theorem}
###### Theorem

If both $Mod_{R}$ and $_{R}Mod$ have [[projective object|enough projectives]], then all these three derived functors exist and all give the same result.
=--

+-- {: .proof}
###### Proof

Existence is clear from the very definition of [[derived functor in homological algebra]]. So we show that deriving in the left argument gives the same result as deriving in the right argument.

Let $Q^A_\bullet \stackrel{\simeq_{qi}}{\to} A$ and $Q^B_\bullet \stackrel{\simeq_{qi}}{\to} B$ be [[projective resolutions]] of $A$ and $B$, respectively. The corresponding [[tensor product of chain complexes]]
$Tot (Q^A_\bullet\otimes Q^B_\bullet)$, hence the [[total complex]] of the degreewise [[tensor product of modules]] [[double complex]] carries the [[filtered chain complex|filtration]] by horizontal degree as well as that by vertical degree.

Accordingly there are the corresponding two [[spectral sequences of a double complex]], to be denoted here $\{{}^{A}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $A$-degree) and $\{{}^{B}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $B$-degree). By the discussion there, both converge to the chain homology of the total complex.

We find the value of both spectral sequences on low degree pages according to the general discussion at _[spectral sequence of a double complex - low degree pages](spectral+sequence+of+a+double%20complex#LowDegreePages)_.

The 0th page for both is

$$
  {}^A E^0_{p,q} = {}^B E^0_{p,q} \coloneqq Q^A_p \otimes_R Q^B_q
  \,.
$$

For the first page we have

$$
  \begin{aligned}
    {}^A E^1_{p,q}
    & \simeq H_q(C_{p,\bullet})
    \\
    & \simeq  H_q( Q^A_p \otimes Q^B_\bullet )
  \end{aligned}
$$

and

$$
  \begin{aligned}
    {}^B E^1_{p,q}
    & \simeq H_q(C_{\bullet,p})
    \\
    & \simeq  H_q( Q^A_\bullet \otimes Q^B_p )
  \end{aligned}
  \,.
$$

Now using the [[universal coefficient theorem]] [in homology](universal%20coefficient%20theorem#InHomology) and the fact that $Q^A_\bullet$ and $Q^B_\bullet$ is a [[resolution]] by [[projective objects]], by construction, hence of tensor [[acyclic objects]] for which all [[Tor]]-modules vanish, this simplifies to

$$
  \begin{aligned}
     {}^A E^1_{p,q}
     & \simeq Q^A_p \otimes H_q(Q^B_\bullet)
     \\
     & \simeq \left\{
       \array{
           Q^A_p \otimes_R B & if\; q = 0
           \\
           0 & otherwise
       }
     \right.
  \end{aligned}
$$

and similarly

$$
  \begin{aligned}
     {}^B E^1_{p,q}
     & \simeq H_q(Q^A_\bullet) \otimes_R Q^B_p
     \\
     & \simeq \left\{
       \array{
           A \otimes_R Q^B_p & if\; q = 0
           \\
           0 & otherwise
       }
     \right.
  \end{aligned}
  \,.
$$

It follows for the second pages that

$$
  \begin{aligned}
    {}^A E^2_{p,q}
    & \simeq
    H_p(H^{vert}_q(Q^A_\bullet \otimes Q^B_\bullet))
    \\
   & \simeq
   \left\{
     \array{
     (L_p( (-)\otimes_R B ))(A) & if \; q = 0
     \\
     0 & otherwise
     }
  \right.
  \end{aligned}
$$

and

$$
  \begin{aligned}
    {}^B E^2_{p,q} & \simeq
    H_p(H^{hor}_q(Q^A_\bullet \otimes Q^B_\bullet))
    \\
   & \simeq
   \left\{
     \array{
       (L_p ( A \otimes_R (-) ))(B) & if \; q = 0
       \\
       0 \; otherwise
     }
   \right.
  \end{aligned}
  \,.
$$

Now both of these second pages are concentrated in a single row and hence have converged on that page already. Therefore, since they both converge to the same value:

$$
  L_p((-)\otimes_R B)(A)
  \simeq
  {}^A E^2_{p,0}
  \simeq
  {}^A E^\infty_{p,0}
  \simeq
  {}^B E^2_{p,0}
  \simeq
  L_p(A \otimes_R (-))(B)
  \,.
$$

=--

### Respect for direct sums and filtered colimits
 {#RespectForDirectSumsAndFilteredColimits}

+-- {: .num_prop #Tor1RespectsDirectSum}
###### Proposition

Each $Tor_n^R(-,N)$ respects [[direct sums]].

=--

+-- {: .proof}
###### Proof

Let $S \in$ [[Set]] and let $\{N_s\}_{s \in S}$ be an $S$-family of $R$-[[modules]]. Observe that

1. if $\{(F_s)_\bullet\}_{s \in S}$ is an family of [[projective resolutions]], then their degreewise [[direct sum]] $(\oplus_{s \in S} F)_\bullet$ is a projective resolution of $\oplus_{s \in S} N_s$.

1. the tensor product functor distributes over direct sums (this is discussed at _[tensor product of modules -- monoidal category structure](module#MonoidalCategoryStructure)_)

1. the [[chain homology]] functor preserves direct sums (this is discussed at _[chain homology - respect for direct sums](http://ncatlab.org/nlab/show/chain+homology+and+cohomology#RespectForDirectSum)_).

Using this we have

$$
  \begin{aligned}
    Tor_n^R(\oplus_{s \in S} N_s, N)
    & \simeq
    H_n\left( \left(\oplus_{s \in S} F\right) \otimes  N  \right)
    \\
    & \simeq
    H_n\left( \oplus_{s \in S} \left(F_s \otimes N \right)  \right)
    \\
    & \simeq \oplus_{s \in S} H_n( F_s \otimes N )
    \\
    & \simeq \oplus_{s \in S} Tor_n(N_s, N)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #TorPreservesFilteredColimits}
###### Proposition

Each $Tor_n^R(-,N)$ respects [[filtered colimits]].

=--

+-- {: .proof}
###### Proof

Let hence $A \colon I \to R Mod$ be a [[filtered category|filtered]] [[diagram]] of modules. For each $A_i$, $i \in I$ we may find a [[nLab:projective resolution]] and in fact a [[nLab:free resolution]] $(Y_i)_\bullet \stackrel{\simeq_{qi}}{\to} A$. Since [[chain homology]] commutes with filtered colimits (this is discussed at _[chain homology - respect for filtered colimits](http://ncatlab.org/nlab/show/chain+homology+and+cohomology#RespectForDirectSum)_), this means that

$$
  (\underset{\to_i}{\lim} Y_i)_\bullet \to A
$$

is still a [[quasi-isomorphism]]. Moreover, by [[Lazard's criterion]] the degreewise filtered colimits of [[free modules]] $\underset{\to_i}{\lim} (Y_i)_n$ for each $n \in \mathbb{N}$ are [[flat modules]]. This means that $\underset{\to_i}{\lim} (Y_i)_\bullet \to A$ is [[flat resolution]] of $A$. By the very definition or else by the basic properties of [[flat modules]], this means that it is a $(-)\otimes N$-[[acyclic resolution]]. By the discussion there it follows that

$$
  Tor_n^\mathbb{Z}(A,N)
  \simeq
  H_n( (\underset{\to_i}{\lim} Y_i) \otimes N )
  \,.
$$

Now the [[tensor product of modules]] is a [[left adjoint]] [[functor]] (the [[right adjoint]] being the [[internal hom]] of modules) and so it commutes over the filtered colimit to yield, using again that [[chain homology]] commutes with filtered colimits,

$$
  \begin{aligned}
  \cdots
    &
    \simeq
    H_n( \underset{\to_i}{\lim} (Y_i \otimes N) )
    \\
    & \simeq
   \underset{\to_i}{\lim} H_n( Y_i \otimes N )
    \\
    & \simeq
    \underset{\to_i}{\lim} Tor_n( A_i, N)
 \end{aligned}
  \,.
$$

=--

### Symmetry in the two arguments
 {#SymmetryInTheTwoArguments}

+-- {: .num_prop}
###### Proposition

For $N_1, N_2 \in R Mod$ and $n \in \mathbb{N}$ there is a [[natural isomorphism]]

$$
  Tor_n(A,B) \simeq Tor_n(B,A)
  \,.
$$

=--

We first give a proof for $R$ a [[principal ideal domain]] such as $\mathbb{Z}$.

+-- {: .proof}
###### Proof

Let $R$ be a [[principal ideal domain]] such as $\mathbb{Z}$ (in the latter case $R$[[Mod]]$\simeq$ [[Ab]]). Then by the discussion at _[projective resolution -- length-1 resolutions](projective+resolution#Lenght1ResolutionsOfAbelianGroups)_ there is always a [[short exact sequence]]

$$
  0 \to F_1 \to F_0 \to N \to 0
$$

exhibiting a [[projective resolution]] of any module $N$. It follows that $Tor_{n \geq 2}(-,-) = 0$.

Let then $0 \to F_1 \to F_2 \to N_2 \to 0$ be such a short resolution for $N_2$. Then by the [long exact sequence of a derived functor](derived+functor+in+homological+algebra#LongExactSequence) this induces an [[exact sequence]] of the form

$$
  0 \to Tor_1(N_1, F_1) \to Tor_1(N_1, F_0) \to Tor_1(N_1, N_2) \to N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0
  \,.
$$

Since by construction $F_0$ and $F_1$ are already [[projective modules]] themselves this collapses to an exact sequence

$$
  0 \to Tor_1(N_1, N_2) \hookrightarrow N_1 \otimes F_1 \to N_1 \otimes F_0 \to N_1 \otimes N_2 \to 0
  \,.
$$

To the last three terms we apply the natural [[braided monoidal category|symmetric braiding]] [[isomorphism]] in $(R Mod, \otimes_R)$  to get

$$
  \array{
    0 &\to& Tor_1(N_1, N_2) &\hookrightarrow& N_1 \otimes F_1 &\to& N_1 \otimes F_0 &\to& N_1 \otimes N_2 &\to& 0
    \\
    && \downarrow && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
  &&
    \\
    0 &\to& Tor_1(N_2, N_1) &\hookrightarrow& F_1 \otimes N_1 &\to& F_0 \otimes N_1 &\to& N_2 \otimes N_1 &\to& 0
  }
  \,.
$$

This exhibits a morphism $Tor_1(N_1,N_2) \to Tor_1(N_2, N_1)$ as the morphism induced on [[kernels]] from an isomorphism between two morphisms. Hence this is itself an isomorphism. (This is just by the [[universal property]] of the [[kernel]], but one may also think of it as a simple application of the the [[four lemma]]/[[five lemma]].)

=--

### Localization
 {#Localization}

(...)

For instance ([Weibel, cor. 3.2.13](#Weibel)).

## Explicit computations for $Tor_1$

Over a commutative ring $R$, the computation of the Tor functor can be
reduced to the computation of each $Tor(M, R/I)$ where $I$ is a finitely generated ideal of $R$.

### $Tor_1$ and the torsion modules
{#RelationToTorsionGroups}

We shall start looking at the case where $I$ is a principal ideal generated by a [[regular element]] $r \in R$.

\begin{definition}
For $M$ and $R$-module and $r$ a [[regular element]] of $R$, we write

$$
  {}_r M \coloneqq \{ x \in M | r \cdot x = 0 \}
$$

for the **$r$-[[torsion submodule]]**.
\end{definition}

The following proposition explains why Tor functors are called this way.

\begin{proposition}
\label{TorOutOfCyclicGroup}
Let $R$ be a commutative ring, let $M$ be an $R$-module and let $r \in R$ be a [[regular element]] of $R$, then
$$
  Tor_1^R(R/(r), M)
  =
  {}_r M
$$
\end{proposition}

\begin{proof}
Since $r$ is regular, one has a short exact sequence
$$
  0 \to R \stackrel{\cdot r}{\to} R \stackrel{mod\, r}{\to}
  R/(r) \to 0
$$
which once tensored with $M$ gives us the long exact sequence
$$
  0 \to Tor_1(R/(r), M) \to M \stackrel{\cdot r}{\to} M \to M/rM \to 0
$$
One can then identify $Tor_1(R/(r), M)$ with the kernel of the map $x \mapsto rx$, that is ${}_r M$.
\end{proof}

This is no longer the case when $r$ is not regular.

\begin{proposition}
Let $\pi$ be an idempotent element of $R$, then the $R$-module $R/(\pi)$ is flat, that is
$$
  Tor_1^R(R/(\pi), X) = 0
$$
for every $R$-module $X$.
\end{proposition}

\begin{proof}
Since $\pi$ is idempotent, $R/(\pi)$ can be identified with the direct summand $(1-\pi)R$ of $R$. It is then a projective $R$-module and is then flat.
\end{proof}

### The case of abelian groups

For $n \in \mathbb{N}$ with $n \geq 1$, write $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$ for the [[cyclic group]] of [[order]] $n$, as usual.

In the ring $\mathbb{Z}$ all non-zero elements are regular, we have thus
that for any abelian group $A$
$$
  Tor_1^\mathbb{Z}(\mathbb{Z}_n, A)
  \simeq
  {}_n A
$$
for every $n \neq 0$.

One can now leverage the knowledge of the structure of finite abelian groups to deduce the following proposition.

+-- {: .num_prop }
###### Proposition

Let $A$ be a [[finite abelian group]] and $B$ any abelian group. Then
$Tor_1(A,B)$ is a [[torsion group]].
Specifically, $Tor_1(A,B)$ is a [[direct sum]] of [[torsion subgroups]] of $B$.

=--

+-- {: .proof}
###### Proof

By a fundamental fact about [[finite abelian groups]] (see _[this theorem](finite+abelian+group#FiniteAbelianGroupIsDirectSumOfCyclics)_), $A$ is a [[direct sum of abelian groups|direct sum]] of [[cyclic group]] $A \simeq \oplus_k \mathbb{Z}_{p_k}$. By prop. \ref{Tor1RespectsDirectSum} $Tor_1$ respects this direct sum, so that

$$
  Tor_1(A,B) \simeq \oplus_k Tor_1(\mathbb{Z}_{p_k}, B)
  \,.
$$

By prop. \ref{TorOutOfCyclicGroup} every direct summand on the right is a torsion group and hence so is the whole direct sum.

=--

More generally we have:

+-- {: .num_prop}
###### Proposition

Let $A$ and $B$ be [[abelian groups]]. Write $Tor^\mathbb{Z}$ for the [[left derived functor]] of tensoring over $R = \mathbb{Z}$.  Then

1. $Tor^\mathbb{Z}_1(A,B)$ is a [[torsion group]]. Specifically it is a [[filtered colimit]] of torsion subgroups of $B$.

1. $Tor^{\mathbb{Z}}_1(\mathbb{Q}/\mathbb{Z}, A)$ is the [[torsion subgroup]] of $A$.

1. $A$ is a [[torsion subgroup|torsion-free group]] precisely if $Tor^\mathbb{Z}_1(A,-) = 0$, equivalently if $Tor^\mathbb{Z}_1(-,A) = 0$.

=--

For instance ([Weibel, prop. 3.1.2, prop. 3.1.3, cor. 3.1.5](#Weibel)).

+-- {: .proof}
###### Proof

The group $A$ may be expressed as a [[filtered colimit]]

$$
  A \simeq \underset{\to_i}{\lim} A_i
$$

of finitely generated [[subgroups]] (this is discussed at _[Mod - Limits and colimits](http://ncatlab.org/nlab/show/Mod#LimitsAndColimits)_). Each of these is a [[direct sum]] of [[cyclic groups]].

By prop. \ref{TorPreservesFilteredColimits} $Tor_1^\mathbb{Z}(-,B)$ preserves these colimits. By prop. \ref{TorOutOfCyclicGroup} every cyclic group is sent to a torsion group (of either $A$ or $B$).  Therefore by prop. \ref{TorOutOfCyclicGroup} $Tor_1(A,B)$ is a filtered colimit of direct sums of torsion groups. This is itself a torsion group.

=--

+-- {: .num_remark}
###### Remark

Analogous results fail, in general, for $\mathbb{Z}$ replaced by another ring $R$.

=--

+-- {: .num_cor}
###### Corollary

An [[abelian group]] is [[torsion subgroup|torsion free]] precisely if regarded as a $\mathbb{Z}$-[[module]] it is a [[flat module]].

=--

See at _[flat module - Examples](flat+module#Examles)_ for more.

### $Tor_1$ of two cyclic modules

\begin{proposition}
Let $R$ be a commutative ring and let $I \subset R$ and $J \subset R$ be two ideals. Then
$$
Tor^R_1(R/I, R/J) \simeq (I \cap J) /IJ
$$
\end{proposition}

\begin{proof}
One has a short exact sequence $0 \to I \to R \to R/I \to 0$. Tensoring it
with $R/J$, one gets the long exact sequence
$$
0 \to Tor_1(R/I, R/J) \to I \otimes R/J \to R/J \to R/I \otimes R/J \to 0
$$
Hence $Tor_1(R/I, R/J)$ can be identified with the kernel of the map
$I \otimes R/J \to R/J$ which is isomorphic to $(I \cap J)/IJ$.
\end{proof}

## Related concepts

* [[Ext]]

* [[Cotor]]

* [[flat resolution lemma]]

* [[universal coefficient theorem]]

## References

Standard textbook accounts include the following:

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994
 {#Weibel}

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press 1956.

* M. Kashiwara and P. Schapira, _[[Categories and Sheaves]]_, Springer (2000)

* S. I . Gelfand, Yu. I. Manin, _Methods of homological algebra_

Lecture notes include

* Daniel Murfet, _Tor_  ([pdf](http://therisingsea.org/notes/Tor.pdf))

section 3 of

* [[Peter May]], _Notes on Tor and Ext_ ([pdf](http://www.math.uchicago.edu/~may/MISC/TorExt.pdf))
 {#May}

and specifically for [[symmetric smash product of spectra|symmetric]] [[model categories of spectra]]

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 7 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

Original articles include

* Patrick Keef, _On the Tor functor and some classes of abelian groups_,  Pacific J. Math. Volume 132, Number 1 (1988), 63-84. ([Euclid](http://projecteuclid.org/euclid.pjm/1102689795))

[[!redirects Tor-functor]]
[[!redirects Tor functor]]
[[!redirects Tor-functors]]
[[!redirects Tor functors]]


Tore Kro

* [webpage](http://www.hiof.no/nor/avdeling-for-ingeniorfag/om-avdelingen/ansatte/?&displayitem=12871&module=admin)

category: people


toric variety

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[algebraic variety]] generalizing a [[torus]] with its [[abelian group]] structure.

## Combinatorial Aspects

A fan $\Delta$ is a collection of cones closed under the operations of taking faces and intersections. Each cone gives rise to an affine variety. The result of gluing these along intersections gives the toric variety of this fan $X_\Delta$.

This correspondence extends functorially. Fan morphisms between a fan $\Delta_1$ in $N_1$ to $\Delta_2$ in $N_2$ is a linear map $f$ from $N_1$ to $N_2$ such that every cone $\sigma \in \Delta_1$ goes to $f (\sigma) \subset \sigma_2$ where $\sigma_2$ is a cone in $\Delta_2$.

### Orbit-Cone Correspondence

## References

* Ezra Miller, _What is... a toric variety?_, Notices of the AMS, volume 55, number 5 ([pdf](http://www.ams.org/notices/200805/tx080500586p.pdf?q=toric))

* Pavel Dimitrov, _Toric varieties, a short introduction_ ([pdf](http://www.cim.mcgill.ca/~pdimit/math707-talk.pdf))

* Stephan Fischli, _On Toric Varieties_ ([pdf](http://www.sws.bfh.ch/~fischli/thesis.pdf))

* Helena Verrill, David Joyner, _Notes on toric varieties_ (2002) ([pdf](http://www.wdjoyner.org/papers/toric7.pdf))

[[!redirects toric varieties]]


Tornike Kadeishvili

__Tornike Kadeishvili__ is a Georgian topologist and one of the pioneers of using $A_\infty$-[[A-infinity-algebra|algebra]]s in algebraic topology. See also [[Kadeishvili's theorem]].

* T. V. Kadeishvili, _On the theory of homology of fiber spaces_, (Russian) International Topology Conference (Moscow State Univ., Moscow, 1979). Uspekhi Mat. Nauk 35 (1980), no. 3 (213), 183--188. Translated in Russ. Math. Surv. 35 (1980), no. 3 (238) 231--211

His work also discovered [[brace algebra]] structures.

* Tornike V. Kadeishvili, _The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies_, (Russian) Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988) 19--27. MR91a:18016

category: people

[[!redirects T. V. Kadeishvili]]


torque

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea ##

The [[moment]] of a [[force]].

## Definition ##

### For point particles ###

Given a point particle $P$ in an Euclidean space $\mathbb{R}^n$, an associated lever arm vector $\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n$ and a force $\overrightarrow{f}:\mathbb{R} \to \mathbb{R}^n$ acting on $P$, the **torque** or **moment of a force** $M$ is the wedge product of the lever arm vector and the force:

$$M(t) = \overrightarrow{r}(t) \wedge \overrightarrow{f}(t)$$

Thus, torque is a [[bivector]].

## See also

* [[areal velocity]]

* [[angular momentum]]

* [[moment of inertia]]

## References

See the references at *[[differential geometry of curves and surfaces]]*.

See also:

* Wikipedia, _[Torque](https://en.wikipedia.org/wiki/Torque)_

[[!redirects torques]]
[[!redirects moment of a force]]
[[!redirects moments of forces]]


torsion

The term _torsion_ can denote very different concepts:

* In [[algebra]], the [[torsion subgroup]] of a group $G$ is the group of elements of finite order (meaning: elements $g \in G$ such that there is $n \in \mathbb{N}$ such that $g \cdot g \cdot \cdots \cdot g = 1$ (with $n$ factors in the product)); similarly in ring theory an element of a module over a ring is a torsion element if it is annihilated by a nonzero element of the ring. A module is torsion (resp. torsion-free) if all its elements are torsion (resp. not torsion, except for zero). Classes of torsion and torsion-free modules are examples of pairs of classes of objects in [[abelian category|abelian categories]] which make a so-called [[torsion theory]] (introduced by Dickson), which is one of the approaches to the [[localization]] of abelian categories.

  For torsion in this sense see also _[[torsion module]]_, _[[torsion approximation]]_.

* In differential geometry of [[curve]]s, the [[torsion of a curve]] is a measure for how the curve tends to spirals out of the plane spanned by its [[tangent vector]] and the first [[derivative]] of that.

* In [[differential geometry]], the [[torsion of a metric connection]] on a [[tangent bundle]] is a measure for how the [[covariant derivative]] differs from the [[Lie bracket]].

  More generally for [[Cartan connections]] there is [[torsion of a Cartan connection]]. This expresses the [[obstruction]] to [[integrability of G-structures]].

* There is an invariant in [[homotopy theory]] called [[Reidemeister torsion]]; it is related to [[Whitehead torsion]] used in [[surgery theory]]; it is a topological invariant of a manifold which is a sort of nonabelian class, nowadays understood to relate to things like quantum dilogarithm, scissors congruences and geometry of hyperbolic 3-manifolds. Index theory relates in Riemannian geometry, Reidemeister torsion to the [[analytic torsion|analytic]] or [[Ray-Singer torsion]], more recently studied also by Witten by means of Feynman integral methods.

Usually there is no risk of confusion, since these terms are used in very different areas of mathematics. _Except_ maybe for the following situation:

* when a $U(1)$-[[gerbe]] on a manifold $X$ is used as the background field of a [[sigma-model]] for [[conformal field theory]], its curvature 3-form characterizes the torsion -- in the differential geometric sense -- of a connection induced on the tangent space of $X$ from the coresponding [[2-spectral triple]]. At the same time, it is of interest whether the [[characteristic class]] of the gerbe, which is an element in $H^3(X, \mathbb{Z})$ is a torsion element in the sense of group theory. Authors in this area will often say sentences like "The gerbe is torsion." which may sound ambiguous for this reason. But if the author knows what a gerbe is, he or she almost surely means the group-theoretic sense.

category: disambiguation


torsion approximation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[stable homotopy theory]] the [[rational homotopy theory|rational]] [[Bousfield localization of spectra]], hence $\mathbb{Q}$-localization $L_{S\mathbb{Q}}$, is accompanied dually by "$\mathbb{Q}$-acyclification" $G_{S\mathbb{Q}}$, forming a natural [[homotopy fiber sequence]]

$$
  G_{S \mathbb{Q}} X \longrightarrow X \longrightarrow L_{S\mathbb{Q}} X
$$

(by [this proposition](Bousfield+localization+of+spectra#LocalizationCofiber)).

This $G_{S\mathbb{Q}}$ is the operation of _universal torsion approximation_.
In the special case of [[chain complexes]] it corresponds to the [[derived functor]] that forms [[torsion subgroups]] (see the discussion at _[fracture theorem -- Arithmetic fracturing of chain complexes](fracture+theorem#CompletionAndTorsionOnDerivedCategories)_).

Similarly, if one already looks at [[p-local spectra]] then torsion approximation is the homotopy fiber of $S\mathbb{Z}[p^{-1}]$-localization.

## Definition

Let $A$  be an [[E-∞ ring]] and $\mathfrak{a} \subset \pi_0 A$ a [[generators and relations|finitely generated]] ideal of its underlying [[commutative ring]].

+-- {: .num_defn #TorsionInfinityModule}
###### Definition

An $A$-[[∞-module]] $N$ is an _$\mathfrak{a}$-[[torsion module]]_ if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

=--

([Lurie "Completions", def. 4.1.3](#LurieCompletions)).

+-- {: .num_prop #TorsionApproximation}
###### Proposition

The [[full sub-(∞,1)-category]]

$$
  A Mod_{\mathfrak{a}tor}
  \hookrightarrow
  A Mod
$$

is [[reflective sub-(∞,1)-category|co-reflective]] and the co-reflector $ʃ_{\mathfrak{a}}$ -- the _[[torsion approximation]]_ -- is [[smashing localization|smashing]].

=--

([Lurie "Completions", prop. 4.1.12](#LurieCompletions)).

+-- {: .num_prop}
###### Proposition

For $N \in A Mod_{\leq 0}$ then [[torsion approximation]], prop. \ref{TorsionApproximation}, intuced a [[monomorphism]] on $\pi_0$

$$
  \pi_0 ʃ_{\mathfrak{a}} N \hookrightarrow \pi_0 N
$$

including the $\mathfrak{a}$-nilpotent elements of $\pi_0 N$.

=--

([Lurie "Completions", prop. 4.1.18](#LurieCompletions)).

## Properties

### Relation to localization

+-- {: .num_prop}
###### Proposition

There is a [[natural transformation|natural]] [[homotopy fiber sequence]]

$$
  ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}
$$

relating $\mathfrak{a}$-torsion approximation on the left with $\mathfrak{a}$-[[localization of a module|localization]] on the right.

=--

([Lurie "Completions", remark 4.1.20](#LurieCompletions))

### As a modality in arithmetic cohesion

Under suitable conditions, torsion approximation forms an [[adjoint modality]] with [[adic completion]].

[[!include arithmetic cohesion -- table]]

## Related concepts

* [[Pontryagin duality for torsion abelian groups]]

## References

Discussion for [[chain complexes]] is in

* {#DwyerGreenlees99} [[William Dwyer]], [[John Greenlees]], section 4.1 of _Complete modules and torsion modules_, Amer. J. Math, 1999  ([pdf](https://www3.nd.edu/~wgd/Dvi/Complete.And.Torsion.pdf))

* [[Carles Casacuberta]] et al., _Models for torsion homotopy types_ ([pdf](http://atlas.mat.ub.es/personals/casac/articles/crhp.pdf))

Discussion in the generality of [[E-∞ rings]] and [[∞-modules]] is in

* {#LurieCompletions} [[Jacob Lurie]], section 4.1 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

[[!redirects torsion approximations]]