elementary (infinity,1)-topos

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#### $(\infty,1)$-Category theory
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#### $(\infty,1)$-Topos Theory
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## Idea

The notion of an _elementary (∞,1)-topos_ is an analogue of the notion of _[[elementary topos]]_ in [[(∞,1)-category theory]].  This is in contrast to the notion of a _Grothendieck_ _[[(∞,1)-topos]] [[equivalence of (∞,1)-categories|equivalent]] to an [[(∞,1)-category of (∞,1)-sheaves]]_, the analogue of a [[sheaf topos]], which is more specific (see [[geometric homotopy type theory]]).

{#AsCategoricalSemanticsForHoTT} Note that every _Grothendieck_ [[(∞,1)-topos]] provides [[categorical semantics]] for [[homotopy type theory]] with [[univalence|univalent]] [[types of types]] and also [[higher inductive types]] (HITs) -- see [here](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence).  Elementary $(\infty,1)$-toposes ought to be roughly the larger class of *all* $(\infty,1)$-categories that provide [[categorical semantics]] for such homotopy type theory.  More precisely, we will define an elementary $(\infty,1)$-topos to have the $(\infty,1)$-categorical structure that ought to correspond to that type-theoretic structure; a [[coherence theorem]] making it *actually* model the corresponding type theory is still unknown (but see at [[initiality conjecture]]).

In general, the problem with "elementary-izing" the notion of $(\infty,1)$-topos is that Grothendieck $(\infty,1)$-toposes have many properties that are not reflected in type theory due to the finitary nature of type theory; the question is to find appropriate "finitary shadows" of them.  For instance, one can construct [[initial algebras for endofunctors]] using a [[transfinite induction]] argument; thus in type theory we postulate the existence of [[inductive types]].  Deciding what class of "infinitary constructions" that are available in Grothendieck $(\infty,1)$-toposes can be described finitarily by something that deserves the name "[[higher inductive type]]" is overall an open question, but we can obtain reasonable definitions by restricting the class of [[HITs]] we ask for.

## Definition (impredicative case)

There are at least two ambiguities in the above sketch of a definition based on type theory: what exactly do we assume of the [[universes]] ([[object classifiers]]), and what general class of [[HITs]] do we ask for?  One reasonable answer to the universe question is that every object should be contained in a universe closed under all the relevant operations, and one reasonable answer to the HITs question is just the non-recursive ones.  This leads to the following definition.

+--{: .num_defn}
###### Definition
An **elementary $(\infty,1)$-topos** is an $(\infty,1)$-category $\mathbf{E}$ such that

* $\mathbf{E}$ has finite [[(∞,1)-limit|limits and colimits]]

* $\mathbf{E}$ is [[locally cartesian closed (∞,1)-category|locally cartesian closed]]

* There exists a [[subobject classifier]] (an [[object classifier]] that classifies the collection of all [[monomorphisms in an (∞,1)-category]])

* For any morphism $f:Y\to X$ in $\mathbf{E}$, there exists an [[object classifier]] in $\mathbf{E}$ classifying a class of morphisms that (1) includes $f$ and (2) is closed under fiberwise finite limits and colimits, composition (i.e. dependent sums), and dependent products.
=--

Note that this definition is "impredicative" just like an [[elementary topos]], in that it has a classifier for *all* subobjects.  We can't categorify this to a classifier for all objects due to size paradoxes; the existence of "enough" object classifiers closed under all the basic operations is the "next best thing".  We will refer to an object classifier satisfying (2) above as a **universe**.

## Properties

It is reasonable to hope for a coherence theorem showing that any elementary $(\infty,1)$-topos in the above sense admits a model of homotopy type theory with non-recursive [[higher inductive type|HITs]] (corresponding to the finite colimits), univalent [[type of types|universes]] (perhaps only weakly Tarski ones), and [[propositional resizing]].  See, for instance, _[[homotopytypetheory:model of type theory in an (infinity,1)-topos]]_ and _[relation between type theory and category theory -- Univalent homotopy type theory and infinity-toposes](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence)_.

In an elementary 1-topos we can construct finite colimits and dependent exponentials from non-dependent exponentials and the subobject classifier (or equivalently from [[power objects]]).  In the $\infty$-case this seems less likely to be true, since subobjects tell us less about objects that are not 0-truncated.  However, there is still a good deal of extra structure that "comes for free" from the above definition.

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###### Theorem
The initial object $0$ of $\mathbf{E}$ is [[strict initial object|strict]], i.e. any morphism $A\to 0$ is an equivalence.
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###### Proof
Just as for 1-categories, this is true in any cartesian closed $(\infty,1)$-category.  For if there is a morphism $f:A\to 0$ then the projection $A\times 0 \to A$ has a section $(1_A,f)$, so that $A$ is a retract of $A\times 0$; but $A\times 0 \cong 0$ by cartesian closedness since $(A\times -)$ has a right adjoint and hence preserves colimits.
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###### Theorem
Binary coproducts in $\mathbf{E}$ are [[disjoint coproducts|disjoint]], i.e. the injections $i:A\to A+B$ and $j:B\to A+B$ are monic, and their pullback is initial.
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###### Proof
This is also just like a corresponding proof for 1-toposes (see [[toposes are extensive]]).  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 $\mathbf{E}$ is locally cartesian closed, pullback preserves colimits, so $(r,s)$ 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 equivalence, 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 equivalent.  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 equivalence, this factorization must be (equivalent to) the left coprojection $A\to A+X$.  Therefore, the composite $S \xrightarrow{f} X \to A+X$ is equivalent 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, $f\simeq g$.

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, $f\simeq g$, hence $S\cong 0$ and is initial.
=--

In fact, disjoint binary coproducts can actually be constructed from local cartesian closure and the subobject classifier; see [FreyRasekh](#FreyRasekh).  This is similar to the fact that the finite colimits in an [[elementary topos]] can be constructed from the rest of the structure, but it is unclear whether it can be generalized beyond coproducts in the infinite-dimensional case.  In particular, it is shown in [FreyRasekh](#FreyRasekh) that local cartesian closure and the subobject classifier do *not* suffice to construct pushouts; the $\infty$-category of truncated spaces is a counterexample.

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###### Theorem
For any finite family of morphisms $\{f_i : Y_i \to X_i\}_{i\in I}$, there exists a universe classifying all the $f_i$.
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###### Proof
Since $f_i$ is the pullback of $\sum_i f_i$ along the injection $X_i \to \sum_i X_i$ (this follows from extensivity), a universe classifying $\sum_i f_i$ must also classify each $f_i$.
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In particular, we have "universe cumulativity":

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###### Corollary
For any universe $p:V\to U$, there is a universe $p':V'\to U'$ that classifies both $p$ and $U\to 1$.
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###### Theorem
All finite colimits are [[van Kampen colimit|van Kampen]].
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###### Proof
Since $\mathcal{E}$ is locally cartesian closed, all colimits are pullback-stable, so it suffices to show descent.  Suppose $D$ is a finite [[simplicial set]] and $F^\rhd:G^\rhd:D^\rhd \to \mathcal{E}$ are colimiting cones under diagrams $F,G:D\to \mathcal{E}$, and let $\alpha^\rhd : F\rhd \to G^\rhd$ be a natural transformation whose restriction $\alpha:F\to G : D\to \mathcal{E}$ is [[equifibered natural transformation|equifibered]].  Let $p:V\to U$ be a universe classifying $\alpha_d : F_d \to G_d$ for each vertex $d\in D_0$.  Then $\alpha$ is the pullback of $p$ along a cocone $q:G\to U$, yielding a cocone $r : \alpha \to p$ in $\mathcal{E}^\to$ which is equifibered.

Since $\alpha^\rhd$ is colimiting under $\alpha:D\to \mathcal{E}^\to$, we have an induced map $\alpha_\star \to p$, where $\star\in D^\rhd$ is the cocone vertex.  But since colimits are pullback-stable, the pullback of $G^\rhd$ along $p$ is a colimiting cocone under $F$, and hence isomorphic to $F^\rhd$.  Thus, $\alpha^\rhd$ is equifibered.
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###### Theorem
There exists a [[natural numbers object]].
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Intuitively, it seems as if $\Omega(S^1)$ (which can be constructed using finite limits and colimits) ought to be the [[integers]] and we should be able to construct the natural numbers from it; but it is unclear to the authors of this page how to do that.  Instead, we can (using the impredicative subobject classifier) define the smallest subobject of an object classifier $p:V\to U$ that contains the initial object and is closed under taking coproducts with the terminal object.  This is almost right, but it is not 0-truncated.  We could 0-truncate it, but to get a universal property relative to all objects rather than only 0-truncated ones, we can instead impose structure making the objects rigid, such as a partial order or a graph.  This argument is formalized in type theory [here](#ConstructingNat); to be precise either it would have to be translated manually into category-theoretic language, or the above conjecture about interpreting type theory in an elementary $(\infty,1)$-topos would have to be proven.

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###### Theorem
There is an [[(n-connected, n-truncated) factorization system]].
=--

The case $n=-1$ can be constructed impredicatively using the subobject classifier: internally, $\Vert A \Vert$ is the smallest [[truth value]] implied by $A$.  Then we can induct on *external* natural numbers, defining $\Vert A \Vert_{n+1}$ to be the $(-1)$-image of the composite $A \to U^A \to (U')^A$, where the second map is the $n$-truncation.  Note that in general it "goes up a universe level", so that the $n$-truncation "goes up $n+2$ universe levels".  However, unlike type theory, an elementary $(\infty,1)$-topos has no globally chosen tower of universes, so this doesn't literally make sense to say.  What we can say is that with this construction, it is not clear whether we can find object classifiers that are *closed* under the $n$-truncation.

However, [(Rijke17)](#Rijke17) shows (again in type theory) that using finite colimits and the natural numbers, the $(-1)$-truncation can be constructed without raising the universe level.  Assuming (as always) that this can be translated into $(\infty,1)$-category theory, it implies that we *can* find object classifiers closed under the $(-1)$-truncation, and hence (by induction) under the $n$-truncation for all $n$.  Moreover, with a natural numbers object this induction can be performed internally, obtaining in particular object classifiers closed under the $n$-truncation for all $n$ at once.

It seems likely that the construction of [[W-types]] in any elementary 1-topos can be repeated in the $(\infty,1)$-case.

There is some structure that the definition probably does not imply, namely a semantic counterpart of arbitrary *recursive* [[higher inductive types]], such as arbitrary [[localization]].  It is certain that these cannot always be constructed in an elementary 1-topos: for instance, HITs imply that all (even infinitary) [[algebraic theories]] have initial algebras, whereas this cannot be proven in [[ZF]] without the [[axiom of choice]] (under appropriate [[large cardinal]] hypotheses).  Thus, it seems very unlikely that they can be constructed in an elementary $(\infty,1)$-topos.  Moreover, it is even unclear exactly what $(\infty,1)$-categorical structure should correspond to arbitrary [[higher inductive types]], not least because we lack an accepted general definition of an "arbitrary HIT".

However, although this is an open question, it is arguably not a question about the definition of "elementary $(\infty,1)$-topos", but some stronger notion, analogous to a stronger notion of 1-topos that lacks an accepted name.  Moreover, a great deal of [[homotopy type theory]] (including synthetic homotopy theory) can be done with only non-recursive HITs, truncations, and a natural numbers type, and even in the absence of a general coherence theorem any particular result of HoTT could probably be explicitly written down in $(\infty,1)$-categorical language.

## Definition (predicative case)

Since there is no consensus even on the definition of [[predicative topos|predicative 1-topos]], it seems less likely that there is a definitive answer for a predicative $(\infty,1)$-topos.  However, it should probably include *at least* the following axioms (which as we have seen above, are all true in the impredicative case although not included in the definition):

* $E$ has finite [[(∞,1)-limit|limits and colimits]]
* $E$ is [[locally cartesian closed (∞,1)-category|locally cartesian closed]]
* For any finite family of morphisms $\{f_i:Y_i\to X_i\}_{i\in I}$, there exists a universe classifying each $f_i$.
* $E$ has a [[natural numbers object]], and perhaps also [[W-types]].

Note that the stronger universe axiom implies, as above, that finite colimits are van Kampen, and in particular therefore that finite coproducts are disjoint.  Instead of strengthening this axiom, we could assume explicitly that finite coproducts are disjoint, and then repeat the proof of descent given above.

## Examples
 {#Examples}
* [ (Rasekh21) ] (#Rasekh21) gives a construction using filter products using which we get a large class of elementary $(\infty,1)$-toposes that aren't Grothendieck $(\infty,1)$ toposes. To be precise Corollary 2.31 in the paper gives the following construction: Suppose we've an elementary $(\infty,1)$-topos $\mathcal{C}$, and a [[filter]] $\zeta$ on the [[subobject classifier]] of $\mathcal{C}$ which is product-closed, such that the [[filter quotient]] of the elementary topos of 0-truncated objects $(\tau_{0}(\mathcal{C}))_{\zeta}$ is not a Grothendieck topos, then the filter quotient $\mathcal{C}_{\zeta}$ is an elementary $(\infty,1)$-topos that isn't a Grothendieck $(\infty,1)$-topos. Further, examples for the specific case of  filter products are constructed.

* Other ideas for the construction of an  [[elementary (∞,1)-topos]] which is not a [[Grothendieck (∞,1)-topos]] also exist in the literature. (cf. [Anel 21](#Anel21)).

* [[Peter Scholze]] speculates in [this comment on the nCafé](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057721) that the [[(∞,1)-category]] $\mathrm{Cond}\infty\mathrm{Grpd}$ of [[condensed infinity-groupoids|condensed $\infty$-groupoids]] might form a [[predicative elementary (∞,1)-topos]] which is not a [[Grothendieck (∞,1)-topos]].

## Related pages

* [[elementary topos]]

* [[predicative topos]]

* [[Grothendieck (∞,1)-topos]], [[(∞,1)-pretopos]]

* [[homotopy type theory]]

* [[Awodey's conjecture]]

## References

A vague version of the above proposed definition is on the very last slide of

* {#Shulman12} [[Mike Shulman]], _Inductive and higher inductive types_ (2012) ([pdf](https://home.sandiego.edu/~shulman/hottminicourse2012/04induction.pdf))

It was elaborated further in

* [[Mike Shulman]], [Elementary (∞,1)-Topoi](https://golem.ph.utexas.edu/category/2017/04/elementary_1topoi.html) (blog post, 2017)

See also

* [[Michael Shulman]], _Towards elementary infinity-toposes_, Vladimir Voevodsky Memorial Conference, IAS, September 2018, [talk](https://www.youtube.com/watch?v=ld4YL787dAk).

* [[Michael Shulman]], [[Peter Lumsdaine]], 2016, _Semantics and syntax of higher inductive types_, ([slides](http://home.sandiego.edu/~shulman/papers/stthits.pdf))

* {#Joyal14} [[André Joyal]], _What is an elementary higher topos?_, talk at _[Reimagining The Foundations Of Algebraic Topology](https://www.msri.org/workshops/689)_ April 07, 2014 - April 11, 2014_ [web video](https://www.msri.org/workshops/689/schedules/18227) [PDF](https://www.msri.org/workshops/689/schedules/18227/documents/2046/assets/20468) (slides 57-end).  This lecture contains a proposed definition that is not an $(\infty,1)$-category but a presentation of one by a [[model category]]-like structure; this is closer to the type theory, but further from the intended examples.  In particular, there are unresolved coherence questions even as to whether every Grothendieck $(\infty,1)$-topos can be presented by a model in Joyal's sense (in particular, how strict can a universe be made, and can the [[natural numbers object]] be made fibrant).  Other than that, the main difference is that Joyal assumes only one fixed universe.

* [[Georg Hegel]], [book 2](Science+of+Logic#LehreVomWesen) of _[[Science of Logic]]_

For an alternative definition proven (in Theorem 3.16) to be equivalent to the one given above, see

* [[Nima Rasekh]], _A Theory of Elementary Higher Toposes_, ([arXiv:1805.03805](https://arxiv.org/abs/1805.03805), [ideals:2142/101508](https://www.ideals.illinois.edu/handle/2142/101508))

The construction of $n$-truncations without recursive HITs is in

* {#Rijke17} [[Egbert Rijke]], *The join construction*, [arxiv:1701.07538](https://arxiv.org/abs/1701.07538)

The construction of natural numbers from propositional resizing and univalence is in

* {#ConstructingNat} HoTT/Coq library, <https://github.com/HoTT/HoTT/blob/master/theories/PropResizing/Nat.v>

The construction of coproducts from local cartesian closure and the subobject classifier is in

* {#FreyRasekh} [[Jonas Frey]] and [[Nima Rasekh]], _Constructing Coproducts in locally Cartesian closed ∞-categories_, [arxiv](https://arxiv.org/abs/2108.11304), 2021

Results holding for elementary (∞,1)-toposes:

* [[Nima Rasekh]], _Yoneda Lemma for Elementary Higher Toposes_, ([arXiv:1809.01736](https://arxiv.org/abs/1809.01736))

* [[Nima Rasekh]], _Every Elementary Higher Topos has a Natural Number Object_, Theory and Applications of Categories **37** 13 (2021) 337-377.  ([arXiv:1809.01734](https://arxiv.org/abs/1809.01734), [tac:37-13](http://www.tac.mta.ca/tac/volumes/37/13/37-13abs.html))

* [[Nima Rasekh]], _Truncations and Blakers-Massey in an Elementary Higher Topos_, ([arXiv:1812.10527](https://arxiv.org/abs/1812.10527))

Using [[comprehension schemes]] to characterize categorical properties of elementary ∞-toposes:

* [[Raffael Stenzel]], _(∞,1)-Categorical Comprehension Schemes_ [[arXiv:2010.09663](https://arxiv.org/abs/2010.09663), talk: [YT](https://www.youtube.com/watch?v=qVhRruh_1jI)]

On [[homotopy type with finite homotopy groups|π-finite homotopy types]] as a partial analogue of the elementary 1-topos [[FinSet]]:

* {#Anel21} [[Mathieu Anel]], *The elementary infinity-topos of truncated coherent spaces*, 2021 ([arxiv:2107.02082](https://arxiv.org/abs/2107.02082))

The filter product construction is used to construct elementary (∞,1)-toposes which aren't Grothendieck (∞,1)-toposes in:

* {#Rasekh21} [[Nima Rasekh]], *Filter Quotients and Non-Presentable (∞,1)-Toposes*, ([arXiv:2001.10088](https://arxiv.org/abs/2001.10088))

[[!redirects elementary (∞,1)-topos]]
[[!redirects elementary (infinity,1)-topos]]
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[[!redirects elementary infinity-toposes]]

[[!redirects predicative elementary (∞,1)-topos]]
[[!redirects predicative elementary (infinity,1)-topos]]
[[!redirects predicative elementary (∞,1)-toposes]]
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[[!redirects predicative elementary infinity-toposes]]


This page hosts material related to the textbook:

* [[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&)

on [[category theory]] and ([[elementary topos|elementary]]) [[topos]] [[sheaf and topos theory|theory]].

The terminology "[[fundamental theorem of topos theory]]" may originate here, where it is proven in the generality of [[elementary toposes]].

category: reference


elementary class of structures

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### Context
#### Model theory
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#Contents#
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## Definition

A class $K$ of [[structure in model theory|structures]] of a given signature is an __elementary class__ if there is a [[first-order theory]] $T$ such that $K$ consists precisely of all models of $T$.

There is a vast generalizations for higher-order theories (and more) [[abstract elementary class]], [[metric abstract elementary class]].

## References

* wikipedia [elementary class](http://en.wikipedia.org/wiki/Elementary_class)

category: model theory

[[!redirects elementary classes of structures]]

[[!redirects elementary class of models]]
[[!redirects elementary class]]
[[!redirects elementary classes]]


elementary embedding

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#### Model theory
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# Contents
* table of contents
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##Idea

In [[model theory]], [[model|models]] of a [[first-order theory]] $T$ are certain kinds of [[functor|functors]] from the [[syntactic site]] of $T$ to $\mathbf{Set}$. Elementary embeddings are [[natural transformation|natural transformations]] between these functors.

## Definition

In [[model theory]], an **elementary embedding** between [[model|structures]] (over a given [[signature]] $\sigma$) is an [[injection]] that preserves and reflects all of [[first-order logic]] over $\sigma$.  That is, it is an injection $f\colon M\to N$ such that for any first-order formula $\varphi$ written in the language of $\sigma$ and parameters $a_1,\dots,a_n\in M$ (of appropriate [[type]]s), we have
$$ M \vDash \varphi(a_1,\dots,a_n) \;\iff\; N \vDash \varphi(f(a_1),\dots, f(a_n)). $$

In particular, this implies that if either $M$ or $N$ is a [[model]] of some first-order [[theory]], then so is the other.

Note that the condition that $f$ be injective is automatic as long as the logic in question includes equality, since reflecting of the formula $x=y$ implies that $f$ is injective.  If $f$ is (interpreted as) the inclusion of a [[subset]], we say that $M$ is an **elementary substructure** of $N$. We also say here that $N$ is an **elementary extension** of $M$.

More generally, when we consider structures in a [[category]] as in [[categorical logic]], a morphism $f\colon M\to N$ between structures in $C$ is an **elementary embedding** if for any formula $\varphi$, the following square is a [[pullback]]:

$$
 \array{[\varphi]_M & \overset{}{\to} & [\varphi]_N\\
  \downarrow && \downarrow\\
  M A_1 \times\dots \times M A_n & \underset{}{\to} & N A_1 \times \dots \times N A_n}
$$

where $M A_i$ denotes the object of $C$ interpreting the type $A_i$, and $[\varphi]_M$ denotes the corresponding subobject in $C$ interpreting the truth value of the formula $\varphi$.  Note that for an arbitrary morphism of structures, this square need not even commute; one sometimes says that $f$ is an **elementary morphism** if it does.

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[[Urs Schreiber]]: let me try to say this more explicitly, to check if I am following:

The [[theory]] $T$ that we are modelling is exhibited by its syntactic category $Syn(T)$ with finite limits. A model of $T$ in a category $C$ with limits -- equivalently a $T$-structure in $C$ -- is a finite-limit preserving functor $N : Syn(T) \to C$. A morphism $f : M \to N : Syn(T) \to C$ of models is a [[natural transformation]] between such functors. We say that such a natural transformation is an _elementary embedding_ if its naturality squares on certain morphisms of $Syn(T)$ are pullback squares.

[[Mike Shulman]]: Not quite.  First of all, the definition officially happens at the more general level of structures rather than models, but we can of course consider those as models for the empty theory.  And whether we need finite-limit categories and functors, or something else like regular ones, geometric ones, or Heyting ones, depends on what fragment of logic we consider our (possibly empty) theories as living in.  Your rephrasing is correct if we mean finitary first-order theories and therefore [[Heyting categories]] and Heyting functors.  Otherwise, the syntactic category $Syn(T)$ won't have the structure required to construct $[\varphi]$, and the structure wouldn't be preserved by the functors into $C$, so that we wouldn't even have naturality squares to ask to commute (I alluded to this in the last sentence above).

I think I didn't explain this very well, but I have to go now, I'll try to come back to it later and rewrite it to make more sense.

[[Peter Arndt]]: Regarding the statement "this square need not even commute": this suggests that the upper arrow is something that we know to define, but the issue is rather that the restriction of the given morphism to the upper left corner might not factor through the upper right corner. For example given a ring homomorphism $f: R \to S$, we don't necessarily get an induced map $\{x \in R \mid \not \exists y: xy=1\} \to \{x \in S \mid  \not \exists y: xy=1 \}$ because non-units can become units in the codomain ring.

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In practice, it is also useful to separate out the weaker notion of **embedding**.

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###### Definition
Working over a fixed signature $\Sigma$, an injective homomorphism of structures $f: M \to N$ is an *embedding* if for each relation $R \in \Sigma$ of arity $n$, we have $R_M = (f^n)^\ast R_N$, i.e., $R_M$ is obtained by pulling back the inclusion $R_N \hookrightarrow N^n$ along $f^n: M^n \to N^n$.
=--

Notice this is *stronger* than just being an injective homomorphism (where we demand only that $R_M \hookrightarrow (f^n)^\ast R^n$, or that $R_M(a_1, \ldots, a_n) \vdash R_N(f(a_1), \ldots, f(a_n))$), and it is weaker than being an elementary embedding since here we are not demanding a pullback condition over all the formulas of the language, just the atomic ones (including equality, which is handled by injectivity as we noted earlier).

A *substructure* is an embedding $f: M \to N$ where $f$ is a set-theoretic inclusion. (In [[structural set theory]], a substructure would just be an isomorphism class of embeddings.)

## Tarski-Vaught test

A simple criterion for when one structure is an elementary substructure of another is given by the Tarski-Vaught test. Let $L$ be the language (the set of first-order formulas) over some given signature.

+-- {: .num_prop #Tarski-Vaught}
###### Proposition
A substructure $i: M \hookrightarrow N$ is an elementary substructure if, whenever $\varphi$ is an $(n+1)$-ary formula and $b_1, \ldots, b_n \in M$, there exists $a \in N$ such that $\varphi(a, b_1, \ldots, b_n)$ is satisfied in $N$ only if there exists $c \in M$ such that $\varphi(c, b_1, \ldots, b_n)$ is satisfied in $M$.
=--

+-- {: .proof}
###### Proof
By induction on the structure of formulas $\varphi$, using $\neg, \wedge, \exists$ as primitive logical operators. The required pullback condition is satisfied on atomic formulas, by definition of substructure.

If the pullback condition is satisfied for $\varphi$ (of arity $n$ say), then it is trivially satisfied for $\neg \varphi$ because for all $\bar{a} = (a_1, \ldots, a_n) \in M^n$ we have

$$M \models \neg \varphi(\bar{a}) \;\; iff \;\; \not [M \models \varphi(\bar{a})] \;\; iff \;\; \not [N \models \varphi(i^n(\bar{a}))] \;\; iff \;\; N \models \neg\varphi(i^n(\bar{a})).$$

If the pullback condition is satisfied for $\varphi$ and $\psi$, then it is equally trivially satisfied for $\varphi \wedge \psi$:

$$\array{
M \models (\varphi \wedge \psi)(\bar{a}) & iff & M \models \varphi(\bar{a}) \; and \; M \models \psi(\bar{a}) \\
 & iff & N \models \varphi(i^n(\bar{a})) \; and \; N \models \psi(i^n(\bar{a})) \\
 & iff & N \models (\varphi \wedge \psi)(i^n(\bar{a})).
}$$

Using these two steps of the induction, we may say that given a substructure, the pullback condition is satisfied for all quantifier-free formulas in the language.

Finally, suppose the pullback condition is satisfied for $(n+1)$-ary formulas $\varphi(v, \bar{w})$. If $\bar{b} \in (\exists_v \varphi)_M$, then there exists $c \in M$ such that $(c, \bar{b}) \in \varphi_M$, whence there exists $a = i(c)$ such that $(i(c), i^n(\bar{b})) \in \varphi_N$, so $i^n(\bar{b}) \in (\exists_v \varphi)_N$, just using the fact that $i$ is a homomorphism. Conversely: if $i^n(\bar{b}) \in (\exists_v \varphi)_N$, i.e., if there exists $a \in N$ such that $(a, i^n(\bar{b})) \in \varphi_N$, then by hypothesis there exists $c \in M$ such that $(c, \bar{b}) \in \varphi_M$, i.e., $\bar{b} \in (\exists_v \varphi)_M$. This completes the inductive proof.
=--

+-- {: .num_example}
###### Example
In the theory of [[real closed fields]] with signature $(0, 1, +, \cdot, \leq)$, the field of real algebraic numbers is an elementary substructure of the field of real numbers. This follows from the Tarski-Vaught test and the [[Tarski-Seidenberg theorem]] which establishes quantifier elimination over the language generated by the signature (every formula has the same extension as some quantifier-free formula).
=--

Another application is described at [[Löwenheim-Skolem theorem]].

## Properties

+-- {: .num_prop #Elementary embeddings are natural transformations}
###### Proposition

Let $T$ be a [[first-order theory]], and $M, N : \mathsf{Syn}(T) \to \mathbf{Set}$ two [[models]] of $T$.

Let $(\mathsf{Syn}(T))_0$ denote the objects (definable sets of $T$) of $\mathsf{Syn}(T)$.

A $(\mathsf{Syn}(T))_0$-indexed collection of maps $f \overset{df}{=} \left(M(X) \overset{f_X}{\to} N(X) \right)_X$ is a [[natural transformation]] if and only if it preserves and reflects [[first-order logic]].

=--

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

Suppose $f$ is a natural transformation. Then certainly whenever $X$ is a definable set, $x \in M(X) \implies f(x) \in N(X).$

If $f(x) \in N(X)$ but $x$ is _not_ in $M(X)$, then $x \in M(\neg X)$, which implies $f(x) \in N(\neg X)$. Since the functor $N$ preserves disjoint unions, $N(\neg X) \cap N(X)$ is empty, which is impossible. So whenever $f(x) \in N(X)$, $x \in M(X)$.

Now suppose $f$ preserves and reflects first-order logic. If $X \overset{\gamma}{\to} Y$ is a map in $\mathsf{Syn}(T)$ and $y \in M(Y)$, then

$$
M \models \gamma(y) \in M(X) \iff N \models \gamma(f(y)) \in N(X)
$$

and $\gamma(f(y)) = f(\gamma(y))$ since

$$
M \models \Gamma(y,\gamma(y)) \iff N \models \Gamma(f(y), f \gamma(y))
$$

where $\Gamma$ is the graph of the definable function $\gamma$.
=--

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

In the above, we can actually weaken the assumption that we start with a collection of $(\mathsf{Syn}(T))_0$-indexed maps to just starting with a collection of maps $f_C : M(C) \to N(C)$, where $C$ runs along the maximal objects (the [[types]]) in $\mathsf{Syn}(T)$.

Then: the $(\mathsf{Syn}(T))_0$-indexed collection of _pullbacks_ along the canonical embeddings $N(X) \hookrightarrow N(C)$ is a natural transformation $M \to N$ if and only if the maps $f_C$ preserved and reflected first-order logic.

=--

## Elementary embeddings between models of set theory

### In material set theory

Elementary embeddings play an important role in the study of [[large cardinals]] in (material) [[set theory]].

For instance, the existence of a [[measurable cardinal]] is equivalent to the existence of a non-[[surjection|surjective]] elementary embedding $j\colon V\to M$, where $V$ is the universe of sets and $M$ is some [[transitive set|transitive]] model of [[ZF]].  If $\kappa$ is a measurable cardinal with a countably-complete [[ultrafilter]] $\mathcal{U}$, we can form the [[ultrapower]] $V^{\mathcal{U}}$ and then take its [[transitive collapse]] to produce $M$.  (Countable completeness of $\mathcal{U}$ is necessary for $V^{\mathcal{U}}$ to be well-founded and thus have a transitive collapse.)

Conversely, if $j\colon V\to M$ is a nontrivial elementary embedding, it must have a *critical point*, i.e. a least ordinal $\kappa$ such that $j(\kappa)\neq \kappa$.  It follows that $j(\kappa)$ is some [[ordinal]] $\gt \kappa$, so in particular $\kappa\in j(\kappa)$ (using the von Neumann definition of ordinals).  Define $\mathcal{U}\subset P(\kappa)$ by $A\in \mathcal{U}$ iff $\kappa\in j(A)$; then $\mathcal{U}$ is a $\kappa$-complete ultrafilter on $\kappa$.

Stronger large cardinal axioms can be characterized, or defined, as the critical points of elementary embeddings satisfying additional closure axioms on the transitive class $M$.

### In structural set theory

Any elementary embedding of models of ZF induces a [[conservative functor|conservative]] [[logical functor]] between their categories of sets.  In fact, it is much more than that; a conservative logical functor preserves and reflects only first-order logic with [[bounded quantifiers]], while an e.e. preserves and reflects all first-order logic.

The structural meaning of elementary embeddings seems not to be well-explored.

### Inconsistency

The "ultimate" closure property, hence the "strongest" large cardinal axiom, would be having a nontrivial elementary embedding $j\colon V\to V$ (i.e. $M$ is all of $V$).  Sometimes the critical point of such an embedding, if one exists, is called a *Reinhardt cardinal*.  However, having such an e.e. turns out to be inconsistent...sort of.

The technicality is that because any e.e. $V\to V$ is a [[proper class]], the proposition "there does not exist an e.e. $V\to V$" cannot be stated in ZF (one cannot quantify over proper classes).  What we can prove is the following meta-theorem (one instance per formula $\varphi(x,y)$ that might define an e.e.).

+-- {: .un_theorem}
###### Meta-Theorem
For any formula $\varphi(x,y,z)$ and any set $a$, it is not true that defining $j_a(x)=y \iff \varphi(x,y,a)$ makes $j_a$ into an elementary embedding $V\to V$.
=--
+-- {: .proof}
###### Proof
Suppose that $\varphi$ and $a$ exist.  Fix such a $\varphi$.  Fix $\lambda$ as the smallest ordinal such that there exists an $a\in V_\lambda$ such that $\varphi(-,-,a)$ defines an e.e. $V\to V$.  Now the statement "$\lambda$ is the smallest ordinal such that there exists an $a\in V_\lambda$ such that $\varphi(-,-,a)$ defines an e.e. $V\to V$." is definable in the language of ZF (definability of the property "is an e.e. $V\to V$" is tricky, but true).  Therefore, if $b$ is any set such that $j_b$ is an e.e., it preserves the truth of this, so it is also true that $j_b(\lambda)$ is the smallest ordinal such that there exists an $a\in V_{j_b(\lambda)}$ such that $\varphi(-,-,a)$ defines an e.e. $V\to V$.  This clearly implies that $j_b(\lambda)=\lambda$.

Now define $\kappa$ to be the smallest ordinal which is a critical point of an e.e. $V\to V$ of the form $j_a$ for some $a\in V_\lambda$.  Let $b\in V_\lambda$ be such that $j_b$ is an e.e. $V\to V$ and $\kappa$ is the critical point of $j_b$.  The definition of $\kappa$ is again a definable property, so it follows that $j_b(\kappa)$ is the smallest ordinal which is a critical point of an e.e. $V\to V$ of the form $j_a$ for some $a\in V_{j_b(\lambda)} = V_\lambda$.  Therefore, $\kappa= j_b(\kappa)$, a contradiction to $\kappa$ being the critical point of $j_b$.
=--

Now, if we work instead in a theory such as [[NBG]] or [[MK]] which can contain *non-definable* proper classes, in theory there might still be an e.e. $V\to V$ which is not definable.  One can also access such an idea by adding a new symbol "$j$" to ZF and asserting that it is an e.e.  However, it was shown by Kunen in 1971, using a technical combinatorial argument, that the existence of such an e.e. is inconsistent with the [[axiom of choice]].  It is unknown whether it is consistent with [[ZF]].

[[!redirects elementary substructure]]
[[!redirects elementary submodel]]
[[!redirects elementary morphism]]
[[!redirects elementary embeddings]]
[[!redirects elementary substructures]]
[[!redirects elementary submodels]]
[[!redirects elementary morphisms]]
[[!redirects elementary extension]]


elementary equivalence

In [[model theory]] elementary equivalence is a relation among structures of the same [[signature of a structure|signature]]: two such structures are __elementary equivalent__ if they satisfy the same [first order sentences](predicate+logic). This relation has been introduced by Tarski.

* G. E. Sacks, _Saturated model theory_, Benjamin 1972
* wikipedia: [elementary equivalence](http://en.wikipedia.org/wiki/Elementary_equivalence)
[[!redirects elementary equivalent]]


elementary function

__Elementary functions__ are the functions of one [[complex number|complex]] variable obtained from the [[identity function]] $x$, constant functions, the (possibly fractional) [[power function]]s $x^\gamma$, [[trigonometric function]]s, the [[exponential functions]] and [[logarithm|logarithmic functions]] by the algebraic operations (addition, subtraction, multiplication, division) and [[composition]]. Elementary functions form a well-defined class. Some of them are [[multivalued function|multivalued]] and have [[singular point]]s (e.g. the square root $x^{1/2}$ has a [[branch point]] at the origin).

Notice that trigonometric functions (e.g., $\cos x = \frac1{2}(e^{i x} + e^{-i x})$) and inverse trigonometric trigometric functions (e.g., $\arctan (x) = \frac{i}{2}(\log (1 - i x) - \log (1 + i x))$) are elementary. Were we to consider functions of a real variable generated in like manner from polynomials, the exponential function, and the logarithmic function, this would no longer be the case.

Functions of a complex variable which are not elementary are sometimes called transcendental, although usually this term is used for the (strictly larger) class of functions that are not [[algebraic function|algebraic]].

To be included: decidability results of Richardson, etc.

[[!redirects elementary functions]]


elementary function arithmetic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Elementary function arithmetic
* table of contents
{: toc}

## Idea

Elementary function arithmetic, or EFA, is a [[first-order theory]] of [[natural numbers]], one of the weakest fragments of [[arithmetic]] strong enough to do nontrivial mathematics. It is strictly weaker than [[Peano arithmetic]].

Regarding the strength of EFA, [[Harvey Friedman]] has put forth the following "grand conjecture":

+-- {: .standout}
"Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded."
=--

Here "bounded quantifier" refers to a [[quantifier]] of shape $\forall_{m \lt n}$; more formally, if a [[variable]] $n$ does not occur in a formula $\phi$, then $\forall_{m \lt n} \phi(m)$ means $\forall_m (m \lt n) \Rightarrow \phi(m)$.

## Definition

The [[theory|language]] of EFA is one-sorted with

* Two constants $0, 1$

* Three binary function symbols $+, \cdot, \exp$ (where $\exp(x, y)$ is usually written $x^y$).

The [[axioms]] of EFA are

* Those of [Robinson arithmetic](http://ncatlab.org/nlab/show/second+order+arithmetic#firstorder_axioms_10) (where $s$ is translated as $1 + (-)$, i.e., as $+(1, -)$);

* Exponentiation axioms, viz. $x^0 = 1$ and $x^y \cdot x = x^{y+1}$;

* The [induction axiom](http://ncatlab.org/nlab/show/second+order+arithmetic#secondorder_axioms_11) for formulas all of whose quantifiers are bounded.

## Related concepts

* [[Peano arithmetic]]

* [[second-order arithmetic]]

## References

* Wikipedia, _[Elementary function arithmetic](http://en.wikipedia.org/wiki/Elementary_function_arithmetic)_.

"Grand conjectures" by Harvey Friedman may be found here:

* Post from [[Harvey Friedman]] to FOM, April 16, 1999. ([web](http://cs.nyu.edu/pipermail/fom/1999-April/003014.html))

A MathOverflow discussion on Friedman's grand conjecture about EFA may be found here,

* [Charles](http://mathoverflow.net/users/6043), _Status of Harvey Friedman's grand conjecture?_, [http://mathoverflow.net/questions/39452](http://mathoverflow.net/questions/39452) (version: 2011-03-24), ([link](http://mathoverflow.net/questions/39452/status-of-harvey-friedmans-grand-conjecture))

[[!redirects grand conjecture]]
[[!redirects elementary function arithmetic]]
[[!redirects Elementary Function Arithmetic]]
[[!redirects EFA]]


elementary mathematics

## Idea

__Elementary mathematics__ is a conventional term used in the mathematics education denoting the part of mathematics curriculum which does not require advanced, nonstandard or conceptually deep level of background (in particular, no infinitesimal calculus).

Though most good approaches to elementary mathematics include setting up basic foundational issues at the appropriate level (for example the principles of proofs etc.), elementary mathematics should be distinguished from the foundations of mathematics.

#### Terminological remark

In another, related but more narrow sense, in 20-th century education, high school and university mathematics was often divided into "elementary" and "higher" mathematics where elementary the part developed without usage of infinitesimal calculus. Higher mathematics in this obsolete sense denoted calculus and related subjects, but still not any research level, nonstandard or recent mathematics.

## References

Related $n$Lab entries include [[mathematics education]], [[foundations]], [[axioms of plane geometry]]

### Educational theory on elementary mathematics

* wikipedia: [modern elementary mathematics](https://en.wikipedia.org/wiki/Modern_elementary_mathematics)

> Modern elementary mathematics is the theory and practice of teaching elementary mathematics according to contemporary research and thinking about learning. This can include pedagogical ideas, mathematics education research frameworks, and curricular material.

### General references on elementary mathematics

The following is a classics (vol. I in German 1908).

* [[Felix Klein]], _Elementary mathematics from an advanced standpoint_

The following overview of elementary mathematics is written by a Lebesgue's student Lucienne Félix (1901-1994) in French, a well known name in mathematics education (cf. <http://guy-brousseau.com/lucienne-felix>); there are German and Russian translations but no English translation. The introduction says that many of the pedagogical ideas of Lebesgue are incorporated into the text.

* Lucienne Félix, Exposé moderne des mathématiques élémentaires, Dunot Editeur, Paris, 1962, 1966; Translated into Russian by Prosveshenie in 1967, into German by Vieweg Teubner in 1969. Title of Rus. transl.: Элементарная математика в современном изложении, 1967

Soviet mathematicains (A. D. Aleksandrov, complex analyst Markushevich and Hinchin) wrote in 1950s an ambitious 5-volume work "Encyclopaedia of elementary mathematics" (basic calculus included; geometry is volumes 4 and 5, redacted by Yaglom and Boltianskii):

* Александров, Марукушевич, Хинчин, _Энциклопедия элементарной математики_

### References on elementary geometry

Here we try to include only mathematically sound references which brought something new or are very influential, pedagogically original or well written.

The following texts contain useful material for teaching elementary geometry at an intermediate high-school and undergraduate level:

* Robin Hartshorne, _Teaching geometry according to Euclid_ , Notices AMS __47__:4 (2000) 460-465 [pdf](http://www.ams.org/notices/200004/fea-hartshorne.pdf)

* Robin Hartshorne, _Geometry - Euclid and Beyond_ , UTM Springer 2000.

* [[David Hilbert]], Stephan Cohn-Vossen, _Anschauliche Geometrie_ , Springer Berlin 1932. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PID=PPN379425343))

* David Henderson, Daina Taimina, _Experiencing Geometry_ , Prentice-Hall Upper Saddle River 2005. (Supplementary online material: [link](http://www.math.cornell.edu/~henderson/ExpGeom/ExpGeomSuppl.html))

* John Stillwell, _The Four Pillars of Geometry_ , UTM Springer 2005.
* Gustave Choquet, _Geometry in a modern setting_

The following very accurate books were intended to teach geometry to high school teachers of mathematics:

* Jacquelline Lelong-Ferrand, _Les fondements de la géométrie_, Presses Universitaires de France, 1985
* [[Jean Dieudonné]], _Linear algebra and geometry_

Comprehensive modern, more advanced but standard reference for classical geometry (emphasizing the role of transformation groups) starting from basics and going somewhat beyond elementary geometry is

* [[Marcel Berger]], _Geometrie_, 5 vol. Paris: CEDIC, Fernand Nathan 1977; Rus. transl. in 2 vols., Mir 1984; _Geometry. I._ Engl. transl. by M. Cole, S. Levy, Universitext Springer 1987, 1994. xiv+427 pp.

### Basic calculus adapted to physicists and engineers

Infinitesimal calculus is usually not listed as elementary mathematics. In US educational system infinitesimal calculus at rigorous and advanced level is usually
called mathematical analysis while at the beginner's level it is called simply
calculus. The textbooks are very numerous and widely available and will not be listed here; they witnessed many trends and reforms like calculator-intensive courses and so on. As these textbooks are usually rich in accompanying techniques like teacher manuals, problem sets, solution sets etc. their authors are usually professional mathematicians most often those dedicated to teaching, rather than leading research mathematians.

A different project of a beginner calculus textbook written by a leading physicist together with a leading geometer came from former Soviet Union

* Ya. B. Zeldovich, I. M. Yaglom, _Higher math for beginners_, Mir 1987 (translated from Russian original published by Nauka in 1982) [pdf](https://ia801303.us.archive.org/29/items/ZeldovichYaglomHigherMathematics/Zeldovich-Yaglom-Higher-Mathematics.pdf)

Some argue that calculus is more intuitive for engineers if taught in the [[nonstandard analysis]] approach. One of the major such textbooks is free online

* H. Jerome Keistler, _Elementary calculus: an infinitesimal approach_, 2000 [files](http://www.math.wisc.edu/~keisler/calc.html); see also companion booklet
_Foundations of infinitesimal calculus_, [files](http://www.math.wisc.edu/~keisler/foundations.html)

### Foundations of mathematics and categories

Foundations of mathematics is a subject, which is beyond the very basics, NOT belonging to elementary mathematics, nor this entry. However, some readers of this page may find beneficial that we list just a very short choice of standard books on foundations for the first reading (but not that elementary).

* [[Practical Foundations of Mathematics]]
* Elliott Mendelson, _Introduction to mathematical logic_,     Chapman and Hall 1964, 1979, 1987, 1997

The following book is a unique attempt by two prominent mathematicians to give an elementary introduction to [[category theory]] directed at beginning students and general readers which is relevant to school education as well:

* [[F. William Lawvere]], [[Stephen H. Schanuel]], _Conceptual mathematics: a first introduction to categories_ , Cambridge UP 1997.

One should note that the category theory is not traditionally considered a part of elementary mathematics even if some parts can be explained in an elementary way (the same is true for the simplest parts of infinitesimal calculus).

category: education


elementary symmetric polynomial

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

#Contents#
* table of contents
{:toc}

## Definition

The _elementary symmetric polynomial_ on $n$ [[variables]] $\{X_i\}$ of degree $k \leq n$ is the [[polynomial]]

$$
  \sigma_k(X_1, \cdots, X_n)
  \coloneqq
  \sum_{1 \leq i_1 \lneq \cdots \lneq i_k \leq n}
  X_{i_1} \cdots X_{i_k}
  \,.
$$

Equivalently these are the degree-$k$ summands in the polynomial

$$
  (1+X_1)(1+X_2) \cdots (1+X_n)
$$

These polynomials form a [[basis]] for the [[Lambda-ring]] of [[symmetric functions]].

## Examples

* The [[Chern classes]] are, under the [[splitting principle]], elementary symmetric polynomials of the [[first Chern classes]].

## Related concepts

* [[simple root]]

## References

* Wikipedia, _[Elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial)_

[[!redirects elementary symmetric polynomials]]

[[!redirects elementary symmetric function]]
[[!redirects elementary symmetric functions]]


Elements

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

[[Euclid]]'s series of books on [[synthetic geometry|synthetic]] [[Euclidean geometry]], elementary [[number theory]], and other [[mathematics|mathematical]] topics.

* [Book XII](http://aleph0.clarku.edu/~djoyce/elements/bookVII/defVII1.html) discusses [[natural numbers]] as composed of *[[monad (disambiguation)|monads]]*

  ([[prime numbers]] are definted in [Def. 11](monad+disambiguation#ElementsVIIDef11))

* [Book XIII](http://aleph0.clarku.edu/~djoyce/elements/bookXIII/bookXIII.html) discusses the [[Platonic solids]].

## References

* Wikipedia, _[Euclid's Elements](http://en.wikipedia.org/wiki/Euclid%27s_Elements)_

* David Joyce's page on Euclid's Elements ([web](http://aleph0.clarku.edu/~djoyce/java/elements/elements.html))

[[!redirects Euclid's Elements]]

category: reference


Elements of Mathematics

Elements of Mathematics (French: Éléments de mathématique)
is a series of books by [[Nicolas Bourbaki]].

It is named after [[Euclid]]'s _Elements_ (Στοιχεῖα).

Website: <https://www.bourbaki.fr/Ouvrages.html>

Most of these books were quite influential, some of them especially so.

In particular, Bourbaki's presentation of [[multilinear algebra]] in Chapter III of the second book (Algèbre) was the first detailed expository presentation of the [[Grassmann algebra]].

Another influential book was _Groupes et algèbres de Lie_, its presentation of [[root systems]] and related topics in Chapter 4–6 was the first of its kind.

Some criticisms raised against this series:

* Almost complete absence of [[categories]] (a deliberate choice);

* _Théorie des ensembles_ receives a lot of criticism for its somewhat nonstandard treatment of logic, as well as the rather awkward notion of a _structure_, which occupies a similar niche to categories.

* The use of Daniell's approach in _Intégration_, mostly ignoring abstract [[measure spaces]].

* Chapter 10 of _Algèbre_ ([[homological algebra]]) only covers the very classical results, (intentionally) ignoring derived categories.

## Related entries

* [[Nicolas Bourbaki]]

* [[Bourbaki Seminar]]

category: reference


Elements of ∞-Category Theory

This pages compiles material related to the book

* [[Emily Riehl]] and [[Dominic Verity]].

  $\;$

  **Elements of ∞-category theory**

  $\;$

  Cambridge Studies in Advanced Mathematics **194**

  Cambridge University Press, Cambridge, 2022.

  xix+759 pp.

  [doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880)

  ISBN: 978-1-108-83798-9

  [pdf](https://emilyriehl.github.io/files/elements.pdf)

on [[(∞,1)-category theory]] formulated via [[∞-cosmoi]] and the [[homotopy 2-category of (∞,1)-categories]] ([[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]).

Related texts:

* [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics **280** (2015) 549-642 $[$[arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021)$]$

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) $[$[arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf)$]$

* {#Riehl21} [[Emily Riehl]], *The formal theory of ∞-categories*, talk at *[Categories and Companions Symposium June 8–12, 2021](http://web.science.mq.edu.au/groups/coact/seminar/CaCS2021/)* ([video](https://youtu.be/iX5Fxen3K-U))

\linebreak

#Contents#
* table of contents
{:toc}

##    Frontmatter
##    Dedication
##    Contents
##    Preface

##    Part I - Basic ∞-Category Theory
###    1 - ∞-Cosmoi and Their Homotopy 2-Categories
###    2 - Adjunctions, Limits, and Colimits I
###    3 - Comma ∞-Categories
###    4 - Adjunctions, Limits, and Colimits II
###    5 - Fibrations and Yoneda’s Lemma
##    An Interlude On ∞-Cosmology
###    6 - Exotic ∞-Cosmoi
##    Part II - The Calculus Of Modules
###    7 - Two-Sided Fibrations and Modules
###    8 - The Calculus of Modules
###    9 - Formal ∞-Category Theory in a Virtual Equipment
##    Part III - Model Independence
###    10 - Change-of-Model Functors
###    11 - Model Independence
###    12 - Applications of Model Independence
##    Appendix of Abstract Nonsense
###    Appendix A - Basic Concepts of Enriched Category Theory
###    Appendix B - An Introduction to 2-Category Theory
###    Appendix C - Abstract Homotopy Theory
##    Appendix of Concrete Constructions
###    Appendix D - The Combinatorics of (Marked) Simplicial Sets
###    Appendix E - ∞-Cosmoi Found in Nature
###    Appendix F - The Analytic Theory of Quasi-Categories
###    References
###    Glossary of Notation
###    Index

category: reference


Elena Babalic

* [webpage](https://events.theory.nipne.ro/gap/index.php/8-homepage/7-mirela-babalic-homepage)

## Selected writings

On [[M-theory on 8-manifolds]] in terms of [[G-structures]]:

* [[Elena Babalic]], [[Calin Lazaroiu]], _Singular foliations for M-theory compactification_, JHEP 03 (2015) 116 ([arXiv:1411.3497](https://arxiv.org/abs/1411.3497))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Foliated eight-manifolds for M-theory compactification_, JHEP 01 (2015) 140 ([arXiv:1411.3148](https://arxiv.org/abs/1411.3148))

* [[Elena Babalic]], [[Calin Lazaroiu]], _The landscape of $G$-structures in eight-manifold compactifications of M-theory_, JHEP 11 (2015) 007 ([arXiv:1505.02270](https://arxiv.org/abs/1505.02270))

* [[Elena Babalic]], [[Calin Lazaroiu]], _Internal circle uplifts, transversality and stratified $G$-structures_, JHEP 11 (2015) 174 ([arXiv:1505.05238](https://arxiv.org/abs/1505.05238))

category: people


Elena Caviglia

* [personal page](https://elenacaviglia.github.io/)

## Selected writings

On [[quotient stacks]]:

* [[Elena Caviglia]], *Generalized principal bundles and quotient stacks*, Theory and Applications of Categories **39** 20 (2023) 567-597  [[arXiv:2302.07852](https://arxiv.org/abs/2302.07852), [tac:39-20](http://www.tac.mta.ca/tac/volumes/39/20/39-20abs.html)]

On the [[indexed Grothendieck construction]]:

* [[Elena Caviglia]], [[Luca Mesiti]], *Indexed Grothendieck construction* [[arXiv:2307.16076](https://arxiv.org/abs/2307.16076)]

category: people


Elena Konstantinova

* [personal page](http://www.math.nsc.ru/~ekonsta/)

## Selected writings

On [[Cayley graphs]]:

* [[Elena Konstantinova]], *Some problems on Cayley graphs*, Linear Algebra and its Applications Volume 429, Issues 11–12, 1 December 2008, Pages 2754-2769 ([doi:10.1016/j.laa.2008.05.010](https://doi.org/10.1016/j.laa.2008.05.010))

category: people


Elena Martinengo

* [Personal web page] (http://www2.iag.uni-hannover.de/~martinengo/)
* [Mathematics Genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=147835)

category: people


Elena Méndez-Escobar

## Selected writings

On [[M2-brane 3-algebras]] in the [[BLG model]] for coincident [[M2-branes]]:

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Metric Lie 3-algebras in Bagger-Lambert theory_, JHEP 0808 : 045, 2008 ([arXiv:0806.3242](https://arxiv.org/abs/0806.3242))

On the equivalence of [[M2-brane 3-algebras]] to [[Lie algebra representations]] over [[metric Lie algebras]]:

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

On the [[ADE classification]] of [[black brane|black]] [[BPS state|BPS]]-[[M2-brane]] solutions to [[D=11 supergravity]] (via [[spherical space forms]] of the [[7-sphere]]):

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

## Related $n$Lab entries

* [[Freund-Rubin compactification]]

* [[ADE classification]]

category: people


Elham Kashefi

* [Wikipedia entry](https://en.wikipedia.org/wiki/Elham_Kashefi)

* [GoogleScholar page](https://scholar.google.com/citations?user=oXpgxTIAAAAJ&hl=en)

## Selected writings

On [[measurement-based quantum computation]]:

* [[Vincent Danos]], [[Elham Kashefi]], [[Prakash Panangaden]], *The Measurement Calculus*, Journal of the ACM, **54** 2 (2007) [[arXiv:0704.1263](https://arxiv.org/abs/0704.1263), [doi:10.1145/1219092.1219096](https://doi.org/10.1145/1219092.1219096)]

category: people


Eli Cartan

* [Wikipedia entry](http://en.wikipedia.org/wiki/%C3%89lie_Cartan)

category: people


Eli Hawkins

* [webpage](http://maths.york.ac.uk/www/eh555)

## Selected writings

On [[strict deformation quantization]] and specifically of [[strict deformation quantization of the 2-sphere]]:

* {#Hawkins07} [[Eli Hawkins]], Section 4 of: *An Obstruction to Quantization of the Sphere*, Communications in Mathematical Physics **283** (2008) 675–699  [[arXiv:0706.2946](http://arxiv.org/abs/0706.2946), [doi:10.1007/s00220-008-0517-2](https://doi.org/10.1007/s00220-008-0517-2)]

On a form of [[strict deformation quantization]] via higher [[geometric quantization of symplectic groupoids]]:

* {#Hawkins08} [[Eli Hawkins]], *A Groupoid Approach to Quantization*, J. Symplectic Geom. **6** 1  (2008) 61-125 [[arXiv:math/0612363](https://arxiv.org/abs/math/0612363), [euclid](https://projecteuclid.org/journals/journal-of-symplectic-geometry/volume-6/issue-1/A-groupoid-approach-to-quantization/jsg/1215032733.full)]

## Related entries

* [[geometric quantization of symplectic groupoids]]

* [[extended geometric quantization of 2d Chern-Simons theory]]

category: people


Elias Kiritsis

* [webpage](http://hep.physics.uoc.gr/~kiritsis/)

category: people


Eliezer Rabinovici

* [Institute page](https://phys.huji.ac.il/EliezerRabinovici)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Eliezer_Rabinovici)

## Selected writings

On the [[CP^1 sigma-model|$\mathbb{C}P^1$ sigma-model]]:

* [[Shmuel Elitzur]], A. Forge, [[Eliezer Rabinovici]], *On effective theories of topological strings*, Nuclear Physics B **388** 1 (1992) 131-155 [doi:10.1016/0550-3213(92)90548-P]

* [[Shmuel Elitzur]], [[Yaron Oz]], [[Eliezer Rabinovici]], [[Johannes Walcher]], *Open/Closed Topological $\mathbb{C}P^1$ Sigma Model Revisited*,  J. High Energ. Phys. **2012** 101 (2012) [[arXiv:1106.2967](https://arxiv.org/abs/1106.2967)]

On [[NS5-brane]]-[[brane intersections|intersections]] with [[D-branes]]:

* {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], *D-Branes in the Background of NS Fivebranes*, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)]

On [[super-orbifolds]] as [[target spaces]] in [[superstring theory]] ([[orbispace|orbi-]][[superspace]]):

* [[Matthias Gaberdiel]], [[Dan Israel]], [[Eliezer Rabinovici]], _D-branes at multicritical points_, JHEP 0804:086, 2008 [[arXiv:0803.0291](https://arxiv.org/abs/0803.0291)]

On the [[Riemann hypothesis and physics]]:

* {#ACER11} Carlo Angelantonj, Matteo Cardella, [[Shmuel Elitzur]], [[Eliezer Rabinovici]], *Vacuum stability, string density of states and the Riemann zeta function*, JHEP **1102** 024 (2011) [[arXiv:1012.5091](http://arxiv.org/abs/1012.5091)]

category: people


elimination of imaginaries

[[!redirects imaginary element]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

In the [[category of sets]], we have a [[bijection|bijective correspondence]] (up to [[isomorphism]]) between [[surjective functions]] on $X$ and [[equivalence relations]] on $X$, where a surjective function $\pi \colon X \to Y$ is taken to the equivalence relation $E_{\pi}$ gotten by pulling back the [[diagonal]] of $Y$, and an equivalence relation $E$ is taken to the function $f_E$ which [[projection|projects]] $X$ onto the set of $E$-classes. If we replace [[Set]] with the category of [[definable sets]] $\mathbf{Def}(T)$ of a [[first-order theory]] $T$, this correspondence generally fails, because when $E$ is a definable equivalence relation, there may not be a corresponding definable $f_E$. That is to say, [[internal congruences|internal congruences]] in $\mathbf{Def}(T)$ are not generally effective.

## Definition ##

We say that:

1. $T$ has **uniform elimination of imaginaries** if the above correspondence does hold, that is, if internal congruences in $\mathbf{Def}(T)$ are effective.

2. $T$ has **elimination of imaginaries** if for every definable set $X = \{m \in \mathbb{M} \operatorname{ | } \varphi(m,b)\}$, there exists a formula $\psi(x,y)$ and a tuple $c$ with the same sort as $y$ such that $c$ uniquely satisfies $X = \{m \in M \operatorname{ | } \psi(m,c)\}.$

Inside a saturated model $\mathbb{M} \models T$, elimination of imaginaries is equivalent to the following statement: for every definable set $X$, there exists a tuple $c$ such that for all $\sigma \in \operatorname{Aut}(\mathbb{M})$, $\sigma(X) = X$ setwise if and only if $\sigma(c) = c$ pointwise. This is sometimes called the **coding of definable sets**.

Poizat noticed that $T$ having elimination of imaginaries allows one to develop a classical Galois theory classifying definably-closed extensions of small parameter sets in terms of the closed subgroups of a profinite automorphism group in any sufficiently saturated model of $T$.

The _imaginary elements_ of a theory $T$ are precisely $E$-classes, where $E$ is a $0$-definable equivalence relation on $T$.

## The $(-)^{\operatorname{eq}}$ construction ##
In the process of inventing [[stability theory]], i.e. classifying theories according to the number of isomorphism classes of their models in each cardinality, Shelah found he needed to eliminate imaginaries in a universal way, so he found a way to conservatively interpret each theory $T$ in a theory $T^{\operatorname{eq}}$ which eliminated all the imaginaries of $T$. The process just involves throwing in for each congruence $E$ on $X$ a new sort ([[type]] in type theory) for $X/E$ and a projection map $X \twoheadrightarrow X/E$.

$T^{\operatorname{eq}}$ satisfies the following universal property: any theory which interprets $T$ and eliminates imaginaries must uniquely interpret $T^{\operatorname{eq}}$ as theories under $T$ (in the $2$-category of theories and interpretations.)

Makkai realized not long after that the $(-)^{\operatorname{eq}}$ construction corresponded to the [[pretopos completion]] of the [[syntactic category]] of $T$.

(Indeed, pretoposes are just toposes that might be missing some power objects. According to the next section, inside a Boolean pretopos these power objects will still be there, just generally only formally.)

## Relation to ind-representable power objects ##
If a first-order theory eliminates imaginaries, it interprets infinitely many constants. This is because it interprets at least two constants: a code for the diagonal relation and its complement. Taking binary sequences of these two constants in higher and higher Cartesian products of the model with itself produces infinitely many constants.

In particular, when $T$ eliminates imaginaries, its [[syntactic category]] has finite [[coproducts]]. In particular, $\mathbf{2}$ exists, and all definable subsets have definable classifying maps.

The following characterization is due to Moshe Kamensky, and can be found [here](https://arxiv.org/abs/1012.3185).

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

Let \(T\) be a first-order theory which interprets two constants. Then $T$ eliminates imaginaries if and only if for all $X$ and $Y$ in $\mathbf{Def}(T)$, the presheaf $Z \mapsto \operatorname{Hom}_{\mathbf{Def}(T)}(Z \times X, Y)$ is ind-representable.

=--

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

Suppose first that the presheaves $Y^X \overset{\operatorname{df}}{=}Z \mapsto \operatorname{Hom}_{\mathbf{Def}(T)}(Z \times X,Y)$ are ind-representable. Let $E \overset{s}{\underset{t}{\rightrightarrows}} X$ be a definable equivalence relation on $X$. Let $\phi_E : X \times X \to 2$ classify $E \overset{(s,t)}{\hookrightarrow} X \times X$. Let $\mathbf{y} : \mathbf{Def}(T) \hookrightarrow \widehat{\mathbf{Def}(T)}$ be the Yoneda embedding. Then $\mathbf{y} \phi_E$ has a product-exponential transpose $\overline{\mathbf{y} \phi_E} : \mathbf{y}X \to 2^X$. The Yoneda lemma says that $\operatorname{Hom}_{\widehat{\mathbf{Def}(T)}}(\mathbf{y}X, 2^X) \simeq 2^X(X)$, and by assumption $2^X(X) \simeq \underset{\longrightarrow}{\lim}\operatorname{Hom}(X,J(i))$ for some $J : I \to \mathbf{Def}(T)$ a filtered diagram of definable sets. Since this is filtered, the colimit as a set can be computed as

$$\bigsqcup_{i \in I} J(i) / \sim,$$

where the equivalence relation $\sim$ identifies elements that are eventually sent to the same element by transition maps in the diagram. Hence, we can identify $\overline{\mathbf{y}\phi_E}$ with its equivalence class of maps $\{\phi_i : X \to J(i)\}_{i \in I}$. Now, $\mathbf{y}E$ is the kernel pair of $\overline{\mathbf{y} \phi_E}$ in $\widehat{\mathbf{Def}(T)}$, so $E = \bigvee_{i \in I} \ker(\phi_i)$. By compactness, we can replace $I$ with a finite subset $I_0 \subseteq I$. Since $I$ is filtered, there exists a weak coproduct $j$ to $I_0$. Since the $\{\phi_i\}_{i \in I}$ turn $X$ into a cone to $J$, for each $i \in I_0$ there is a transition map $h_{ij} : J(i) \to J(j)$, so $\ker(\phi_j) = \ker(h_{ij} \circ \phi_i) \supseteq \ker(\phi_i)$. Replacing each $\ker(\phi_i)$ with $\ker(\phi_j)$, we see $E = \ker(\phi_j)$, and $\phi_j$ is definable.

On the other hand, suppose we have elimination of imaginaries. Fix $X$ and $Y$. For each $Z$ and $f : Z \times X \to Y$, we may form the equivalence relation $E_f$ on $Z$ which says $z_1 \sim_{E_f} z_2$ if and only if they define the same fiber of $\Gamma(f)$. The projection map induces a map

$$(-/E_f) \times X : Z \times X \to Z/E_f \times X$$

which is also a map $f \to f/E_f$ of objects over $Y$.

Now, the [[category of elements]] $\operatorname{Pt}(1, Y^X)$ of the presheaf $\operatorname{Hom}(- \times X, Y)$ has objects maps $f : Z \times X \to Y$ for $Z$ ranging over $\mathbf{Def}(T)$, and morphisms from $f_1 : Z_1 \times X \to Y$ to $f_2 : Z_2 \times X \to Y$ the maps $g \times X : Z_1 \times X \to Z_2 \times X$ for $g : Z_1 \to Z_2$ such that $f_1 = f_2 \circ g$. For each object $f \in \operatorname{Pt}(1, Y^X)$, $-/ E_f \times X$ as above coequalizes all pairs of maps in $\operatorname{Pt}(1, Y^X)$ into $f$. Furthermore, the natural projection $p$ from $\operatorname{Pt}(1, Y^X) \to \mathbf{Def}(T)$ by sending $f : Z \times X \to Y \mapsto Z$ creates colimits, in particular finite coproducts, which by assumption exist in $\mathbf{Def}(T)$. So $\operatorname{Pt}(1, Y^X)$ is filtered, and since for any presheaf $P$, we have that

$$P \simeq \underset{\longrightarrow}{\lim}\left(\operatorname{Pt}(1,P) \overset{p}{\to} \mathbf{Def}(T) \overset{\mathbf{y}}{\hookrightarrow} \widehat{\mathbf{Def}(T)}\right),$$

so $Y^X$ is ind-representable.
=--

+-- {: .num_remark}
###### Remark
In the above proof, complementation is required to characterize compactness as "every covering of a definable set by an infinite family of definable sets is finitely supported." The same proof works if we replace inner homs with power objects.

=--

## Imaginaries and automorphisms
Let $M \models T$ be a model. If a formula $\varphi(x_1, x_2)$ is a $0$-definable equivalence relation $E$ (i.e. an [[internal congruence]] in $\mathbf{Def}(T)$) on a definable set $X$, any automorphism $\sigma$ of $M$ must preserve and reflect $E$-equivalence, so that $\sigma$ extends along $X \twoheadrightarrow X/E$ to a permutation on $X/E$. So automorphisms of models "already see" imaginaries. This can be stated rigorously as: the expansion of a monster model $\mathbb{M}$ of $T$ to a monster model $\mathbb{M}^{\operatorname{eq}}$ of $T^{\operatorname{eq}}$ will have the same automorphism group as $\mathbb{M}$.

To make things concrete, let's consider the structure $(\mathbb{Z}, +)$. Since there is an automorphism which switches signs, the only constant is $0$. So the theory of this structure cannot eliminate imaginaries, since such theories must interpret infinitely many distinct constants.

However, we can tell that this structure has many imaginaries by its rigidity: the above automorphism is the only nontrivial one. This is because any such automorphism of $(\mathbb{Z},+)$ must also act on the internal congruences $E_n$ which say

$$x \sim_{E_n} y \iff x \equiv y (\operatorname{mod} n) \iff (\exists \ell)\left[x - y = \ell + \dots \text{ (n times) } \dots + \ell\right],$$

that is, must extend along the abelian group epimorphisms $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/n\mathbb{Z}$ for each $n$, so that $k \mapsto -k$ is the only one that works.

(It's essentially for this reason that the automorphism group of the [[forgetful functor]] $\mathbf{Grp} \to \mathbf{Set}$ is also $\mathbb{Z}/2 \mathbb{Z}$.)

## Related entries

 - [[pretopos completion]]

 - [[Barr-exact category]]

 - [[hyperimaginary element]]

 - [[Galois theory]]

## References

* Bruno Poizat, _Une théorie de Galois imaginaire_, J. Symbolic Logic __48__ (1984), no.4, 1151-1170, [MR85e:03083](http://www.ams.org/mathscinet-getitem?mr=727805), [doi](http://dx.doi.org/10.2307/2273680)
* wikipedia [imaginary element](http://en.wikipedia.org/wiki/Imaginary_element)
* Anand Pillay, _Some remarks on definable equivalence relations in O-minimal structures_, J. Symbolic Logic __51__ (1986), 709-714, [MR87h:03046](http://www.ams.org/mathscinet-getitem?mr=853850), [doi](http://dx.doi.org/10.2307/2274024)
* Jan Holly, _Definable operations on sets and elimination of imaginaries_, Proc. Amer. Math. Soc. __117__ (1993), no. 4, 1149–1157, [MR93e:03052](http://www.ams.org/mathscinet-getitem?mr=1116261), [doi](http://dx.doi.org/10.2307/2159546), [pdf](http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116261-6/S0002-9939-1993-1116261-6.pdf)
* [[Ehud Hrushovski]], _Groupoids, imaginaries and internal covers_, [arxiv/math.LO/0603413](http://arxiv.org/abs/math/0603413); _On finite imaginaries_, [arxiv/0902.0842](http://arxiv.org/abs/0902.0842)
* D. Haskell, E. Hrushovski, H.D.Macpherson, _Definable sets in algebraically closed valued fields: elimination of imaginaries_, J. reine und angewandte Mathematik __597__ (2006)
* [[Saharon Shelah]], _Classification theory and the number of non-isomorphic models_, Studies in Logic and the Foundations of Mathematics __92__, North Holland, Amsterdam 1978
[[!redirects elimination of imaginaries]]
* Moshe Kamensky, [_A categorical approach to internality_](http://arxiv.org/abs/1012.3185)

[[!redirects imaginary element]]
[[!redirects imaginaries]]


elimination of quantifiers

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

#Contents#
* table of contents
{:toc}

## Idea

One thinks of [[existential quantifier | existential quantification]] as _[[projection]]_. A theory eliminates quantifiers when "the projection of a primitive (quantifier-free) definable set is still primitive (quantifier-free)."

For example, suppose we're working in the theory of a [[vector space]] over a fixed field $\mathbb{F}$. Let $A$ be a linear transformation $\mathbb{F}^{n+1} \to \mathbb{F}^m$. Let $\vec{b}$ be an $m$-tuple, $x$ a variable, and $\vec{y}$ an $n$-tuple of variables. Then the solution set for

$$
\exists x A(x, \vec{y}) = \vec{b}
$$

is the projection of the solution set for

$$
A(x, \vec{y}) = \vec{b}
$$

to the $\vec{y}$-hyperplane. Since the projection of a (translated) hyperplane is again a (translated) hyperplane, there exists some $B : \mathbb{F}^n \to \mathbb{F}^m$ such that $B(\vec{y}) = \vec{b} \iff (\exists x) \left(A(x, \vec{y}) = \vec{b}\right).$

This is an instance of the fact that the theories of [[vector space | vector spaces]] over division rings admits quantifier elimination. It's nice when this happens, and when it doesn't, one asks for minimal ways to expand the language to make it happen. That [[ACF]] eliminates quantifiers is a special case of [[Chevalley's theorem on constructible sets]] ("the image of a constructible set is constructible").

Often, in order to prove quantifier-elimination, one develops an _effective_ procedure for reducing an arbitrary sentence to a quantifier-free one, which shows that the theory is [[decidability|decidable]]; this is how Tarski proved that the theory [[RCF]] of real closed fields is decidable.

## Definition
An $\mathcal{L}$-theory $T$ **eliminates quantifiers** if every $\mathcal{L}$-formula $\varphi(x)$ is equivalent to a quantifier-free $\mathcal{L}$-formula $\psi(x)$, i.e.

$$
T \models \varphi(x) \leftrightarrow \psi(x).
$$

## Remarks

* Any theory is [[interpretation|bi-interpretable]] with an expanded theory where every definable set is named with a predicate, called its [[coherent logic|Morleyization]], which eliminates quantifiers.

* Therefore, whether or not a structure has quantifier elimination depends on the language being used to describe the structure.

* Quantifier elimination implies [[model complete theory | model completeness]]: all submodels are elementary submodels and all models are [[existentially closed theory | existentially closed]].

* It is an informative exercise to try to prove QE syntactically (a "tractable" case like [[DLO]] suffices) to appreciate the usefulness of the semantic/model-theoretic characterization of QE below.

* The characterizations below (and in general important tools in model theory like the [[compactness theorem]], the [[Lowenheim-Skolem theorem]], etc.) only work for _infinite_ models.

* Since quantifier elimination is equivalent to [[substructure complete theory#substructureconverse|substructure completeness]] and [[elementary equivalence|elementarily equivalent]] finite structures are isomorphic, (the theory of a) finite [[first-order structure]] eliminates quantifiers if and only if it is [[Fraisse limit|ultrahomogeneous]].

## Semantic characterizations of QE

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

A first-order theory $T$ eliminates quantifiers if and only if it is substructure-complete: $T$ plus the [[quantifier-free diagram]] of any substructure of any model of $T$ form a complete theory.

=--

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

See [[substructure complete theory#QEiffsubstructurecomplete|substructure completeness]].

=--

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

Let $T$ be a theory. The following are equivalent:

1. $T$ eliminates quantifiers.

2. For every $M$ and $N$ models of $T$ such that $N$ is $|M|^+$-[[type (in model theory)|saturated]] and every proper substructure $A \subseteq M$ such that there is an embedding $i \colon A \to N$, there exists a proper intermediate substructure $A'$, $A \subsetneq A' \subseteq M$ and an extension $i' \colon A' \to N$ of $i$ to $A'$.

=--

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

(1. $\implies$ 2.) Suppose $T$ eliminates quantifiers. Since this is equivalent to [[substructure-completeness]], choose $x \in M \backslash A$, and let $B \overset{df}{=} \langle A \cup \{x\} \rangle$ be the substructure of $M$ generated by $A \cup \{x\}.$ Then, since quantifier-free statements are transferred by embeddings, the [[type (in model theory)]] of $x$ over $A$ in $M$ is finitely realized in $N$ and therefore realized via $|M|^+$-saturation by some $x'$.

A standard [[back-and-forth argument]] starting at $x'$ lets us build our required extension.

(2. $\implies$ 1.) By possibly transfinitely iterating 2., we obtain an embedding $M \to N.$ As we can always embed a $T$-model inside a larger, arbitrarily saturated $T$-model, we therefore have [[model completeness|amalgamation]].

This gives [[model completeness]]: since model completeness is equivalent to every $T$-submodel being an [[elementary embedding|elementary submodel]], it suffices by the Tarski-Vaught test (and an induction on complexity of formulas) to test that whenever $m$ is a tuple from $M$, $\varphi(x,y)$ is a quantifier-free formula, and $N \models \exists x \varphi(x, m)$, then $M \models \exists x \varphi(x,m)$ (i.e. that $M$ is [[existentially closed]].)

Embed $M$ into a sufficiently saturated elementary extension $^*M$ (by taking a large enough [[ultrapower]], if you like). Then by (2.), the elementary embedding $M \preceq ^*M$ extends along the embedding $M \hookrightarrow N$ to an embedding $N \hookrightarrow ^*M$. If $N \models \exists x \varphi(x,m)$, then there is a witness $n \in N$ such that $N \models \varphi(n,m)$. Since $\varphi$ was quantifier-free, this transfers along the embedding $N \hookrightarrow ^*M$, so that $^*M \models \varphi(n,m)$, and therefore $^*M \models \exists x \varphi(x,m).$ This then transfers down the elementary embedding $M \preceq ^*M$, so $M \models \exists x \varphi(x,m).$ Since $M$ and $N$ were arbitrary, $T$ is model complete.

Since [[model completeness#substructureconverse|model completeness + amalgamation implies substructure completeness]] and [[substructure complete theory#QEiffsubstructurecomplete|substructure completeness is equivalent to quantifier elimination]], the theorem is proved.

=--

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

  [[ACF]] admits quantifier elimination. Furthermore, the theories $\mathsf{ACF}_p$ (where $p$ is specification of the characteristic) are complete.

=--

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

We put ourselves into the situation of the above theorem. Let $E$ and $F$ be algebraically closed fields such that $F$ is $|E|^+$-saturated. Let $R$ be a subring of $E$ with $i : R \to F$ an embedding. As $E$ is a field, $R$ is an integral domain, and thus $E$ contains the fraction field $\operatorname{Frac}(R)$ of $R$. The embedding $i$ extends uniquely to $\operatorname{Frac}(R)$. By a back-and-forth argument, $i$ extends uniquely to $\operatorname{Frac}(R)^{\operatorname{alg}}$. Therefore, we can assume that $R$ is algebraically closed. Let $a \in E \backslash R$. Then $a$ is transcendental over $R$, and by saturation its [[type (in model theory) | type]] is realized by some $b \in F$ which is transcendental over the image $i(R)$ of $R$ under $i$. Then $a \mapsto b$ induces an embedding $R[a] \overset{i'}{\to} i(R)[b]$, which extends $i$. Thus the hypotheses of the theorem are satisfied and [[ACF]] admits QE.

That $\mathsf{ACF}$ becomes complete after specifying a characteristic follows from the fact that in any theory with quantifier elimination, naming a substructure (thus passing to the theory of all models which contain a copy of that substructure) makes the theory complete, and every field of characteristic $p$ contains the prime field of characteristic $p$.

=--

## QE extends to co-slice categories
Let $T$ eliminate quantifiers. Let $K$ be a substructure (not necessarily a submodel) of some model $A$ of $T$. Then the theory $T/K$ of models of $T$ interpreting a copy of $K$ also has quantifier elimination.

In particular, if we specialize to [[ACF]], this tells us that not only do all algebraically closed fields admit quantifier elimination in the language of rings, but so do all algebraically closed fields containing a fixed ring $R$ (i.e. the geometric points of $\operatorname{Spec}(R)$), in the language of rings with constants for the elements of $R$.

## Related concepts
* [[quantification]]

* [[quantifier-free diagram]]

* [[decidability]]

## References

* wikipedia [quantifier elimination](http://en.wikipedia.org/wiki/Quantifier_elimination)
* Wilfrid Hodges, _Model Theory_, Cambridge University Press 1993

category: logic, model theory
[[!redirects quantifier elimination]]
[[!redirects quantifier-elimination]]


Elisabeth Stenholm

Elisabeth Stenholm

* [personal page](https://elisabeth.bonnevier.one/)

* [github page](https://github.com/elisabethstenholm/)

* [github (old)](https://git.app.uib.no/Elisabeth.Bonnevier)

## Selected writings

On [[constructive set theory]] in [[homotopy type theory]]:

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* [[arXiv:2001.06696](https://arxiv.org/abs/2001.06696), talk video:[YT](https://www.youtube.com/watch?v=ALj5hOmwXps)]

category: people

[[!redirects Elisabeth Bonnevier]]


Ellen Maycock Parker

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Ellen_Maycock)

## Selected writings

The $\mathbb{Z}/2$-[[graded PU(H)|graded projective unitary group]], (reflecting the twists of [[twisted K-theory]] not just in degree 3 but also in degree 1):

* [[Ellen Maycock Parker]], *The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras*, Transactions of the American Mathematical Society **308** 1 (1988) ([jstor:2000953](https://www.jstor.org/stable/2000953))

category: people

[[!redirects Ellen Maycock]]
[[!redirects Ellen Parker]]


Elliott Lieb

Professor of Mathematics and Physics at University of Princeton.

* [webpage](https://web.math.princeton.edu/~lieb/)

##Related concepts

* [[stability of matter]]

category: people


ellipse

## Related concepts

* [[conic section]]

  * [[parabola]], [[hyperbola]]

* [[circle]]

[[!redirects ellipses]]


elliptic chain complex

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{:toc}

## Idea

The notion of _elliptic chain complex_ is the generalization of the notion of [[elliptic operator]] from single [[linear maps]] to [[chain complexes]] of linear maps.

## Definition

For $X$ a [[smooth manifold]] and $\{E_k\}_{k \in \mathbb{Z}}$ a collection of [[vector bundles]] over $X$, a [[chain complex]] of [[differential operators]] between the spaces of [[sections]] of these bundles

$$
  \cdots \to \Gamma(E_{k+1}) \stackrel{P_k}{\to} \Gamma(E_k) \to \cdots
$$

is called an **elliptic chain complex** if the corresponding sequence of [[symbol of a differential operator|symbols]]

$$
  \cdots \to \pi^* E_{k+1} \stackrel{\sigma(P_k)}{\to} \pi^* E_k \to \cdots
$$

(where $\pi \colon T^* X \to X$ is the [[cotangent bundle]]) is an [[exact sequence]].

For instance ([Pati, def. 9.4.1](#Pati)).

For a single differential operator $P$ this says that $0 \to \pi^* E_1 \stackrel{\sigma(P)}{\to} \pi^* E_0 \to 0$ is exact, which means that $\sigma(P)$ is an [[isomorphism]], hence that $P$ is an [[elliptic operator]].

## Properties

### Atiyah-Bott lemma
 {#AtiyahBottLemma}

If $(\mathcal{E}, d)$ is an elliptic complex of smooth [[sections]] $\mathcal{E} = \Gamma_X(E)$ of a [[vector bundle]] $E \to X$ overa [[compact topological space|compact]] [[closed manifold]] $X$, then the inclusion

$$
  (\mathcal{E},d) \hookrightarrow (\overline{\mathcal{E}}, d)
$$

into the complex of [[distribution|distributional]] [[sections]] is a [[quasi-isomorphism]], in fact a [[homotopy equivalence]].

This is due to ([Atiyah-Bott](#AtiyahBott)). A localized refinement (suitable for [[factorization algebras of local observables]]) appears as [Gwilliam, lemma 5.2.13](#Gwilliam).

## Examples

The classical examples of elliptic complexes are discussed also in ([Gilkey section 3](#Gilkey)).

### de Rham complex

Let $X$ be a compact smooth manifold. Then the [[de Rham complex]] is an elliptic complex. The corresponding [[index of an elliptic complex]] is the [[Euler characteristic]]

$$
  Ind(\Omega^\bullet(X),d)
  =
  \chi(X)
  =
  \sum_{p = 0}^{dim X} (-1)^p  dim H_{dR}^p(X, \mathbb{C})
$$

### The Yang-Mills complex

(...)

### The Dolbeault complex

(...) [[Dolbeault complex]]

The [[index of an elliptic complex]] of the Dolbeault complex is the [[arithmetic genus]]

### Spin complex

(...) index is [[A-hat genus]] (...)

## Related concepts

* [[index of an elliptic complex]]

## References

* V. Pati, _Elliptic complexes and index theory_ ([pdf](http://www.isibang.ac.in/~statmath/resource/elliptic.pdf))
 {#Pati}

* [[Peter Gilkey]], _The Atiyah-Singer Index Theorem -- Chapter 5_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/gilkey3.pdf))
 {#Gilkey}

* [[Michael Atiyah]], [[Raoul Bott]], _A Lefschetz fixed point formula for elliptic complexes. I_, Ann. of Math. (2) 86 (1967),
374–407. MR 0212836 (35 #3701)
 {#AtiyahBott}

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))
 {#Gwilliam}

[[!redirects elliptic chain complexes]]
[[!redirects elliptic complex]]
[[!redirects elliptic complexes]]

[[!redirects Atiyah-Bott lemma]]


elliptic Chern character

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{:toc}

## Idea

A [[higher chromatic Chern character]] from [[elliptic cohomology]]/[[tmf]] with values in [[Laurent series]]  with [[coefficients]] in [[K-theory]] ([[Tate K-theory]]).

## Related concepts

* [[Witten genus]]

* [[Tate K-theory]]

## References

In the context of the [[Witten genus]] the elliptic Chern character was introduced in

* [[Haynes Miller]], _The elliptic character and the Witten genus_, in: [[Mark Mahowald]], [[Stewart Priddy]] (eds.), _Algebraic topology_ (Evanston, IL, 1988), Contemp. Math. 96, Amer. Math. Soc., Providence, RI (1989) 281–289 ([pdf](http://dedekind.mit.edu/~hrm/papers/ell-char.pdf), [doi:10.1090/conm/096](http://dx.doi.org/10.1090/conm/096))

Discussion in the context of [[equivariant elliptic cohomology]] is in

* {#Ganter12} [[Nora Ganter]], section 3.2 of _The elliptic Weyl character formula_ ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))

based on

* [[Ioanid Rosu]], [[Allen Knutson]], _Equivariant K-theory and Equivariant Cohomology_, Mathematische Zeitschrift 243 (2003), 423-448 ([arXiv:math/9912088](http://arxiv.org/abs/math/9912088))

[[!redirects elliptic Chern characters]]


elliptic cohomology

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#Contents#
* table of contents
{:toc}

## Idea

An _elliptic cohomology theory_ is a type of _[[generalized (Eilenberg-Steenrod) cohomology]]_ theory associated with the datum of an [[elliptic curve]].

Even ([[weakly periodic cohomology theory|weakly]]) [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology]] theories $A$ are characterized by the [[formal group]] whose ring of functions $A(\mathbb{C}P^\infty)$ is the [[cohomology ring]] of $A$ evaluated on the [[complex projective space]] $\mathbb{C}P^\infty$ and whose group product is induced from the canonical morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that describes the tensor product of complex [[line bundle]]s under the identification $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$.

There are precisely three types of 1-dimensional such [[formal group laws]]:

* the simple additive group structure -- this corresponds to  integral [[ordinary cohomology]] given by the [[Eilenberg-MacLane spectrum]];

* the multiplicative group that corresponds to complex [[topological K-theory]]

* the formal group law on an [[elliptic curve]].

An **elliptic cohomology** theory is an even [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology]] theory whose corresponding formal group is an elliptic curve, hence which is [[Brown representability theorem|represented]] by an [[elliptic spectrum]].

(e.g. [Lurie, def. 1.2](#Lurie)).

The [[Goerss-Hopkins-Miller-Lurie theorem]] shows that the assignment of [[generalized (Eilenberg-Steenrod) cohomology]] theories to [[elliptic curves]] lifts to an assignment of representing [[spectrum|spectra]] in a structure-preserving way.

The [[homotopy limit]] of this assignment functor, i.e. the "gluing" of all spectra representing all elliptic cohomology theories is the [[spectrum]] that represents the cohomology theory called [[tmf]].

## Properties

### Genera, the elliptic genus and relation to string theory

> A properly developed theory of [[elliptic cohomology]] is likely to shed some light on what [[string theory]] really means. ([Witten 87, very last sentence](#Witten87))

The following is **rough material** originating from notes taken live (and long ago), to be polished. See also at _[[elliptic genus]]_ and _[[Witten genus]]_

Some topological invariants of [[manifolds]] that are of interest:

we restricted attention to [[closed manifold|closed]] connected  [[smooth manifolds]] $X$

* the [[Euler characteristic]] $e(X) \in \mathbb{Z}$

  * takes all values in $\mathbb{Z}$

  * is the obstruction to the existence of a nowhere vanishing [[vector field]] on $X$:

    $$
      (e(X)= 0)
      \Leftrightarrow
      (\exists  v \in \Gamma(T X) : \forall x \in X : v(x) \neq 0)
    $$

* [[signature]] $sgn(X)$

  this is the obstruction to $X$ being cobordant to a fiber bundle over the circle:

  $X$ is [[bordant manifold|bordant]] to a [[fiber bundle]] over $S^1$ precisely if $sgn(X) = 0$

* when $X$ has a [[spin structure]]

  the index of the [[Dirac operator]] $D$:

  $$
    ind D_X \in \mathbb{Z}
  $$

  $$
    \alpha(D) \in
    \left\{
      \array{
           \mathbb{Z} & dim X = 0 mod 4
           \\
           \mathbb{Z}_2 & dim X = 1, 2 mod 8
           \\
           0 & otherwise
      }
    \right.
  $$

  **theorem** (Gromov-Lawson / [[Stephan Stolz|Stolz]]) let $dim X \geq 5$ and

  then $X$ admits a [[Riemannian metric]] of positive [[scalar curvature]] precisely when $\alpha(X) = 0$

These invariants share the following properties:

* they are additive under [[disjoint union]] of [[manifold]]s

* they are multiplicative under [[cartesian product]] of manifolds

* $e(X) mod 2, sgn(X), ind(D_X)$ all vanish when $X$ is a boundary, $\exists W : X = \partial W$, which means that $X$ is [[cobordant manifold|cobordant]] to the empty manifold $\emptyset$.

  in other words, these invariants are [[genus|genera]], namely [[ring]] homomorphisms

  $$
    \Omega \to R
  $$

  form the [[cobordism ring]] $\Omega$ to some commutative [[ring]] $R$

* good [[genus|genera]] are those which reflect geometric properties of $X$.

* now for $X$ a [[topological space]] consider the [[cobordism ring]] _over $X$_:

  $$
    \Omega(X) := \{(M,f)| f : M \stackrel{cont}{\to X}\}/_{bordism}
  $$

  where addition and multiplication are again given by disjoint union and cartesian product.

  this assignment of rings to topological spaces is a [[generalized homology]] theory: [[cobordism homology theory]]

  **question** given a [[genus]] $\Omega \to R$, can we find a [[homology theory]] $R(-)$ with $R  = R(pt)$ its [[homology ring]] over the point and such that it all fits into a natural diagram

  $$
    \array{
     \Omega &\to& R
     \\
     \uparrow && \uparrow
     \\
     \Omega(X) &\stackrel{\rho}{\to}& R(X)
    }
  $$

  This would be a _parameterized extension_ $\rho = R(-)$ of $R$ .

  Now let $X$ be a closed manifold.

  consider $u_X : X \to K(\pi_1(X),1)$ (on the right an [[Eilenberg-MacLane space]]) which is the classifying map for the [[universal cover]]

  $$
    u_* \pi_1(X) \stackrel{\simeq_{canon}}{\to}
    \pi_1(K(\pi_1(X), 1))
  $$

  then consider

  $$
    \rho_X[X, u_X] \in R(K(\pi_1(X),1))
  $$

  **theorem (Julia Weber)**

  take the [[Euler characteristic]]  mod 2, $Eu(X)$

  $$
    \array{
      \Omega^0 &\stackrel{Eu(M)\cdot t^{dim M}}{\to}& \mathbb{Z}_2[t]
      \\
      \uparrow && \uparrow
      \\
      \Omega^0(X) &\to& Eu(X) &
      \simeq H_\bullet(X; \mathbb{Z}_2[t])
    }
  $$

  for $X$ smooth we have then:

  $$
    Eu_X[X, id] = Poincare\; dual\;  of\; total\; Stiefel-Whitney\; class
  $$

  **theorem** (Minalta)

  something analogous for [[signature genus]]

  $$
    \array{
      \Omega_\bullet^{SO}
      &\to&
      Sig_\bullet(X)
    }
  $$

  $sign_X[X,u] \in sig_\bullet(K) \otimes \mathbb{Q}$

  this is the Novikov higher signature

  now the same for the $\alpha$-genus

  $$
    \array{
      \Omega_{X}^{Spin} &\stackrel{\alpha}{\to}&
      KO_\bullet(pt)
      \\
      \uparrow && \uparrow
      \\
      \Omega_\bullet^{Spin}
      &\to&
      KO_\bullet(X)
    }
  $$

now towards elliptic genera: recall the notion of [[string structure]] of a [[manifold]] $X$: a lift of the structure map $X \to \mathcal{B}O(n)$ through the 4th connected universal cover $\mathcal{B}String(n) := \mathcal{B}O(n)\langle 4\rangle \to \mathcal{B} O$:

so consider [[string structure|String manifold]]s and the bordism ring $\Omega_\bullet^{String}$ of String manifold, let $M_\bullet$ be the ring of integral [[modular form]]s, then there is a [[genus]] -- the [[Witten genus]] $W$--

$$
  \array{
    \Omega_\bullet^{String} &\stackrel{W}{\to}& M_\bullet
    \\
    \uparrow && \uparrow
    \\
    \Omega_\bullet^{String}(X)
    &\to&
    M_\bullet(X)
    \\
    &\searrow&
    \\
    && tmf_\bullet(X)
  }
$$

where $\Omega_\bullet^{String}(pt) \to tmf_\bullet(pt)$ is surjective

**conjecture ([[Stolz conjecture]])**

If a String manifold $Y$ has a positive [[Ricci curvature]] [[Riemannian metric|metric]], then the [[Witten genus]] vanishes.

The attempted "Proof" of this is the motivation for the [[Stephan Stolz|Stolz]]-[[Peter Teichner|Teichner]]-program for [[geometric models for elliptic cohomology]]:

**"Proof"** If $Y$ is String, then the [[loop space]] $L Y$ is has [[spin structure]], so if $Y$ has positive [[Ricci curvature]] the $L Y$ has positive [[scalar curvature]] which implies by the above that $ind^{S^1} D_{L Y} = 0$ which by the index formula is the [[Witten genus]].

### Equivariant elliptic cohomology and loop group representations

The analog of the [[orbit method]] with [[equivariant K-theory]]
replaced by [[equivariant elliptic cohomology]] yields (aspects of) the [[representation theory]] of [[loop groups]]. ([Ganter 12](#Ganter12))

### Chromatic filtration

[[!include chromatic tower examples - table]]

## Related concepts

* [[elliptic spectrum]]

* [[elliptic genus]], [[equivariant elliptic genus]]

* [[SO orientation of elliptic cohomology]], [[spin orientation of elliptic cohomology]]

* [[elliptic Chern character]]

* [[equivariant elliptic cohomology]]

* [[sigma-orientation]]

* [[tmf]]

[[!include moduli of higher lines -- table]]

## References

[[!include elliptic cohomology -- references]]

[[!redirects elliptic cohomologies]]

[[!redirects elliptic cohomology theory]]
[[!redirects elliptic cohomology theories]]

[[!redirects elliptic generalized cohomology theory]]
[[!redirects elliptic generalized cohomology theories]]


elliptic cohomology -- contents

**[[elliptic cohomology]], [[tmf]], [[string theory]]**

[[complex oriented cohomology theory|complex oriented]] [[cohomology]] of [[chromatic level]] 2

* [[elliptic curve]]

  * [[supersingular elliptic curve]]

  * [[derived elliptic curve]]

* [[moduli stack of elliptic curves]]

* [[modular form]], [[Jacobi form]]

  * [[Eisenstein series]], [[j-invariant]], [[Weierstrass sigma-function]], [[Dedekind eta function]]

  * [[elliptic genus]], [[Witten genus]]

* [[topological modular form]]

  * [[string orientation of tmf]]