tiny object

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#### Compact objects
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[[!include compact object - contents]]
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## Definition

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

Let $E$ be a [[locally small category]] with all small [[colimits]]. An object $e$ of $E$ is called **tiny**, **small-[[projective object]]** ([Kelly, §5.5](#Kelly)) if the [[hom-functor]] $E(e, -) : E \to Set$ preserves small colimits. It is called **absolutely presentable** if the functor preserves all colimits.

More generally, for $V$ a [[cosmos]] and $E$ a $V$-[[enriched category]], $e \in E$ is called tiny if $E(e,-) : E \to V$ preserves all small colimits.
=--

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

Since being an [[epimorphism]] is a "colimit-property" (a morphism is epic iff its [[pushout]] with itself consists of identities), if $e$ is tiny then $E(e,-)$ preserves epimorphisms, which is to say that $e$ is [[projective object|projective]] (with respect to epimorphisms).  This is presumably the origin of the term "small-projective", i.e. the corepresentable functor preserves small colimits instead of just a certain type of finite one.

=--

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

If $E$ is [[cartesian closed category|cartesian closed]] and the
[[inner hom]] $(-)^e$ has a [[right adjoint]] (and hence preserves all colimits), $e$ is called (internally) **atomic** or [[infinitesimal object|infinitesimal]].

=--

(See for instance [Lawvere 97](#Lawvere97).)

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

The [[right adjoint]] in def. \ref{Atomic} is sometimes called an "[[amazing right adjoint]]", particularly in the context of [[synthetic differential geometry]].

=--

+-- {: .num_remark}
###### Remark
Various terminological discrepancies in the literature hinge on the distinction between internal notions and external notions. Thus, if $E$ is a [[cartesian closed category]] with small colimits, we may say $e \in Ob(E)$ is *internally* tiny if the functor $(-)^e: E \to E$ preserves small colimits. Relatedly, the word "atomic" has been used in both an external sense where $E(e, -): E \to Set$ has a right adjoint, as in Bunge's thesis, and in an internal sense, as when Lawvere refers to $e$ as an a.t.o.m. ("amazingly tiny object model") if $(-)^e: E \to E$ has a right adjoint. But under certain hypotheses, the two notions coincide; see for instance Proposition \ref{coincide}.
=--

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

If $E$ is a [[sheaf topos]], then (externally) tiny objects and externally atomic objects coincide.

=--

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

Clearly any externally atomic object is tiny. For the converse, use a dual form of the special [[adjoint functor theorem]] (SAFT): $E$ is [[locally small category|locally small]], [[cocomplete category|cocomplete]], and [[well-powered category|co-well-powered]] (because for any object $X$, equivalence classes of [[epimorphisms]] with domain $X$ are in natural bijection with [[internal equivalence relations]] on $X$, and there is a small set of these because they are contained in a set isomorphic to $\hom(X \times X, \Omega)$), and finally $E$ has a [[generating set]] (namely, the set of associated [[sheaves]] of [[representables]] coming from a small [[site]] presentation for $E$). Under these conditions, the SAFT guarantees that any [[cocontinuous functor]] $E \to C$ has a right adjoint, provided that $C$ is locally small; then apply this to the case $C = Set$.

=--

Clearly the statement and the proof of Proposition \ref{InSheafToposTinyImpliesAtomic} carry over when "external" is replaced by "internal" throughout.

## Properties

### General

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

Any [[retract]] of a tiny object is tiny, since [[split idempotent|splitting of idempotents]] is an [[absolute colimit]] (see also [Kelly, prop. 5.25](#Kelly)).

=--

### In categories of modules over rings
 {#InCategoriesOfModulesOverRings}

The notion of tiny object is clearly highly dependent on the base of [[enriched category|enrichment]]. For example, for a [[ring]] $R$, the tiny objects in the category of left $R$-[[category of modules|modules]] $Ab^R$, considered as an [[Ab]]-[[enriched category]], are the [[finitely generated module|finitely generated]] [[projective modules]]. Certainly f.g. projective modules are tiny because $R$ is tiny (the [[forgetful functor]] $\hom(R, -): Ab^R \to Ab$ preserves $Ab$-[[colimits]]) and the closure of $R$ under finite direct sums and retracts, which are absolute $Ab$-[[colimits]], comprise finitely generated projective modules. See also _[[Cauchy completion]]_.

On the other hand, when the category $Ab^R$ is considered as a [[Set]]-[[enriched category]], there are _no_ tiny objects. In fact this is true for any Set-enriched category with a [[zero object]]: Let $X$ be a tiny object. The morphism $X \to 0$ induces a map $Hom(X,X) \to Hom(X,0)$. This map has empty codomain (since $Hom(X,-)$ preserves the zero object, as an empty colimit). Thus $Hom(X,X) = \emptyset$ in contradiction to $id_X \in Hom(X,X)$.

### In presheaf categories

+-- {: .num_example}
###### Example

In a [[presheaf category]] every [[representable functor|representable]] is a tiny object:

since colimits of presheaves are computed objectwise (see [[limits and colimits by example]]) and using the [[Yoneda lemma]] we have for $U$ a [[representable]] functor and $F : J \to PSh$ a diagram that

$$
  Hom(U, \lim_\to F) \simeq (\lim_\to F)(U) \simeq \lim_\to F(U)
$$

where now the last [[colimit]] is in [[Set]].

=--

Thus, in a presheaf category, any [[retract]] of a representable functor is tiny. In fact the converse also holds:

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

The tiny objects in a [[presheaf category]] are precisely the [[retracts]] of [[representable functor]]s.

=--

This is for instance ([BorceuxDejean, prop 2](#BorceuxDejean)). For instance, the only tiny object in [[G-set]] is $G$ itself with its regular action.

Thus, if the domain category is [[Cauchy complete category|Cauchy complete]] (has [[split idempotent]]s), then every tiny presheaf is representable; and more generally the Cauchy completion or [[Karoubi envelope]] of a category can be defined to consist of the tiny presheaves on it. See [[Cauchy complete category]] for more on this.

+-- {: .num_prop #coincide}
###### Proposition
For presheaves on a category $C$ with finite products, the notions of externally tiny object and internally tiny object coincide.
=--

+-- {: .proof}
###### Proof
Without loss of generality, we may assume $C$ is Cauchy complete (note that the Cauchy completion of a category with finite products again has finite products), so that tiny presheaves coincide with representable functors $C(-, c)$.

Let $E$ denote the presheaf category. Given that the empty product $1$ is tiny, if $e \in Ob(E)$ is internally tiny, then the composite

$$E(e, -): E \to Set = \left(E \stackrel{(-)^e}{\to} E \stackrel{E(1,-)}{\to} Set \right)$$

is cocontinuous, hence $e$ is externally tiny.

In the other direction, recall how exponentials $G^F$ in $E = PSh(C)$ are constructed: we have the formula

$$G^F(c) = E(C(-, c) \times F, G).$$

In particular, if $F$ is externally tiny, hence a representable $C(-, c')$, we have

$$G^F(c) = E(C(-, c) \times C(-, c'), G) \cong E(C(-, c \times c'), G) \cong G(c \times c')$$

where the last isomorphism is by the [[Yoneda lemma]]. Since colimits in $PSh(C)$ are computed pointwise, whereby evaluation functors $ev_c: PSh(C) \to Set$ preserve colimits, we see that $(-)^{C(-, c')}: G \mapsto G(- \times c')$ preserves colimits, so that $F = C(-, c')$ is internally tiny. The amazing right adjoint $R$ in this case takes a presheaf $H$ to the presheaf $R H$ that takes an object $d$ to the set $(R H)(d) = E(C(- \times c', d), H)$.
=--

(Compare the result [here](https://ncatlab.org/nlab/show/internally+projective+object#enough).)

In the context of [[topos theory]] we say, for $C$ [[small category]], that an [[adjoint triple]] of [[functors]]

$$
  Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  [C,Set]
$$

is an [[essential geometric morphism]] of [[topos]]es $f : Set \to [C,Set]$; or an **[[point of a topos|essential point]]** of $[C,Set]$.

By the [[adjoint functor theorem]] this is equivalently simply a single functor $f^* : [C, Set] \to Set$ that preserves all small [[limits]] and [[colimits]]. Write

$$
  Topos_{ess}(Set,[C,Set])
   \simeq
  LRFunc([C,Set], Set)
   \subset
  Func([C,Set], Set)
$$

for the [[full subcategory]] of the [[functor category]] on functors that have a [[left adjoint]] and a [[right adjoint]].

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

For $C$ a [[small category]] there is an [[equivalence of categories]]

$$
  \overline{C} := TinyObjects([C,Set]) \simeq Topos_{ess}(Set, [C,Set])^{op}
$$

of the tiny objects of $[C,Set]$ with the category of essential points of $[C,Set]$.

=--

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

We first exhibit a [[full subcategory|full inclusion]] $Topos_{ess}(Set,[C,Set])^{op} \hookrightarrow \overline{C}$.

So let $Set \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [C,Set]$ be an [[essential geometric morphism]]. Then because $f_!$ is [[left adjoint]] and thus preserves all small [[colimits]] and because every [[set]] $S \in Set$ is the colimit over itself of the singleton set we have that

$$
  f_! S \simeq \coprod_{s \in S} f_!(*)
$$

is fixed by a choice of [[copresheaf]]

$$
  F := f_!(*) \in [C, Set]
  \,.
$$

The $(f_! \dashv f^*)$-[[adjunction]] [[isomorphism]] then implies that for all $H \in [C,Set]$ we have

$$
  f^* H \simeq Set(*, f^* H) \simeq [C,Set](f_! *, H)
  \simeq [C,Set](F,H)
  \,.
$$

naturally in $H$, and hence that

$$
  f^*(-) \simeq [C,Set](F,-) : Set \to [C,Set]
  \,.
$$

By assumption this has a further right adjoint $f_!$ and hence preserves all [[colimits]]. By the discussion at [[tiny object]] it follows that $F \in [C,Set]$ is a tiny object. By prop. \ref{CauchyComplIsFullSubcatOnTinyObjects} this means that $F$ belongs to $\overline{C} \subset [C,Set]$.

A morphism $f \Rightarrow g$ between [[geometric morphisms]] $f,g : Set \to [C,Set]$ is a [[geometric transformation]], which is a [[natural transformation]] $f^* \Rightarrow g^*$, hence by the above a natural transformation $[C,Set](F,-) \Rightarrow [C,Set](G,-)$. By the [[Yoneda lemma]] these are in bijection with morphisms $G \to H$ in $[C,Set]$. This gives the full inclusion $Topos_{ess}(Set,[C,Set])^{op} \subset \overline{C}$.

The converse inclusion is now immediate by the same arguments: since the objects in $\overline{C}$ are precisely the [[tiny object]]s $F \in [C,Set]$ each of them corresponds to a functor $[C,Set](F,-) : [C,Set] \to Set$ that has a [[right adjoint]]. Since this generally also has a left adjoint, it is the [[inverse image]] of an essential geometric morphism $f : Set \to [C,Set]$.

=--

### In a local topos
 {#AtomsInALocalTopos}

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

The [[terminal object]] in any [[local topos]] is atomic.

In particular for $\mathbf{H}$ a [[topos]] and $X \in \mathbf{H}$ an object, the [[slice topos]] $\mathbf{H}_{/X}$ is [[local topos|local]] precisely if $X$ is atomic.

=--

This is discuss at [local geometric morphism -- Local over-toposes](local+geometric+morphism#LocalOverTopoes).

### In a cohesive topos
 {#AtomsInACohesiveTopos}

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]]. Write $(\int \dashv \flat \dashv \sharp)$ for its [[adjoint triple]] of [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]].
Consider the following basic notion from _[[cohesive (∞,1)-topos -- structures]]_.

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

An [[object]] $X \in \mathbf{H}$ is called _geometrically contractible_ if its [[shape modality|shape]] is [[contractible]], in that $\int X \simeq \ast$.

=--

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

Over the [[base (∞,1)-topos]] [[∞Grpd]],
every atom in a [[cohesive (∞,1)-topos]] is geometrically contractible.

=--

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

By [[reflective sub-(∞,1)-category|reflection]] of the [[discrete objects]] it will be sufficient to show that for all [[discrete objects]] $S \in \infty Grpd \hookrightarrow \mathbf{H}$ we have an [[equivalence]]

$$
  \left[\int X , S\right] \simeq S
  \,.
$$

Now notice that, by the discussion at _[∞-tensoring](limit+in+a+quasi-category#Tensoring)_, every [[discrete object]] here is the [[homotopy colimit]] indexed by itself of the [[(∞,1)-functor]] constant on the [[terminal object]]:

$$
  S \simeq \underset{\rightarrow}{\lim}_S \ast
  \,.
$$

Using this we have

$$
  \begin{aligned}
    \left[\int X, S\right]
    &\simeq
    \left[
       X, \flat S
    \right]
    \\
    & \simeq
    \left[
       X, \flat \underset{\rightarrow}{\lim}_S \ast
    \right]
    \\
    & \simeq
    \left[
       X, \underset{\rightarrow}{\lim}_S \flat  \ast
    \right]
    \\
    & \simeq
    \underset{\rightarrow}{\lim}_S
    \left[
       X, \flat  \ast
    \right]
    \\
    & \simeq
    \underset{\rightarrow}{\lim}_S
    \left[
       X, \ast
    \right]
    \\
    & \simeq
    \underset{\rightarrow}{\lim}_S
    \ast
    \\
    & \simeq
    S
  \end{aligned}
  \,.
$$

where we applied, in order of appearance: the $(\int \dashv \flat)$-[[adjunction]], the $\infty$-[[tensoring]], the fact that $\flat$ is also [[left adjoint]] (hence the existence of the [[sharp modality]]), the assumption that $X$ is atomic, then again the fact that $\flat$ is right adjoint, that $\ast$ is the terminal object and finally again the $\infty$-tensoring.

=--

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

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] over [[∞Grpd]] and let $X \in \mathbf{H}$ be an atomic object. Then the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ sits by an [[adjoint quadruple]] over [[∞Grpd]] whose leftmost adjoint preserves the terminal object.

=--

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

By the discussion at [[étale geometric morphism]], the [[slice (∞,1)-topos]] comes with an [[adjoint triple]] of the form

$$
  \mathbf{H}_{/X}
   \stackrel{\overset{\sum_X}{\longrightarrow}}{\stackrel{\overset{(-)\times X}{\leftarrow}}{\stackrel{\overset{\prod_X}{\longrightarrow}}{\underset{}{}}}}
  \mathbf{H}
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{CoDisc}{\leftarrow}}}}
  \infty Grpd
  \,.
$$

The bottom composite $\Gamma\circ \prod_X$
has an extra right adjoint by prop \ref{SliceOverAtomicObject}. The extra left adjoint $\Pi \circ \sum_X$ preserves the terminal object by prop. \ref{InCohesionAtomicObjectIsGeometricallyContractible}.

=--

## Related concepts

* [[atom]] (in a [[poset]])

* [[atomic category]]

## References

The term _small projective object_ is used in  section 5.5. of

* [[Max Kelly]], _Basic Concepts of Enriched Category Theory_ ([pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))
{#Kelly}

Tiny objects in presheaf categories ([[Cauchy complete categories|Cauchy completion]]) are discussed in

* [[Francis Borceux]] and D. Dejean, _Cauchy completion in category theory_  Cahiers Topologie Géom. Différentielle Catégoriques, 27:133–146, (1986) ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0))
  {#BorceuxDejean}

* David Yetter, _On right adjoints to exponential functors_ [link](https://www.sciencedirect.com/science/article/pii/0022404987900776)

The term "atomic object" or rather "a.t.o.m" is suggested in

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

[[!redirects small-projective object]]
[[!redirects small-projective objects]]
[[!redirects small-projective]]

[[!redirects tiny objects]]
[[!redirects tiny]]

[[!redirects atomic object]]
[[!redirects atomic objects]]

[[!redirects absolutely presentable]]


Titus Neupert

* [institute page](https://www.physik.uzh.ch/en/groups/neupert/team/neupert.html)

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

## Selected writings

On higher order [[topological insulators]] (with protected corner-modes beyond the edge-modes):

* [[Frank Schindler]], Ashley M. Cook, Maia G. Vergniory, Zhijun Wang, Stuart S. P. Parkin, [[B. Andrei Bernevig]], [[Titus Neupert]], *Higher-Order Topological Insulators*, Science Advances 01  **4** 6 (2018) eaat0346 [[arXiv:1708.03636](https://arxiv.org/abs/1708.03636), [doi:10.1126/sciadv.aat0346](https://doi.org/10.1126/sciadv.aat0346)]

On [[topological crystalline insulators]] with focus on the [[Wilson lines]] of the [[Berry connection]]:

* [[Titus Neupert]], [[Frank Schindler]], *Lecture Notes on Topological Crystalline Insulators*, in: *Topological Matter* Springer Series in Solid-State Sciences **190**, Springer (2018) [[arXiv:1810.03484](https://arxiv.org/abs/1810.03484), [doi:10.1007/978-3-319-76388-0_2](https://doi.org/10.1007/978-3-319-76388-0_2)]

category: people


Tiziano Gaudio

* [GoogleScholar page](https://scholar.google.co.uk/citations?user=EA8QTnMAAAAJ&hl=en)

## Selected writings

Relating [[super vertex operator algebras]] (and particularly the [[monster vertex operator algebra]]) to [[conformal nets]]:

* [[Sebastiano Carpi]], [[Tiziano Gaudio]], [[Robin Hillier]], *From vertex operator superalgebras to graded-local conformal nets and back* [[arXiv:2304.14263](https://arxiv.org/abs/2304.14263)]

category: people


Tjark Tjin

* [InSpire page](https://inspirehep.net/authors/986037)

## Selected writings

On [[quantum groups]]:

* {#Tjin92} [[Tjark Tjin]], *An introduction to quantized Lie groups and algebras*, Int. J. Mod. Phys. A **7** (1992) 6175-6213 [[arXiv:hep-th/9111043](https://arxiv.org/abs/hep-th/9111043), [doi:10.1142/S0217751X92002805](https://doi.org/10.1142/S0217751X92002805)]

## Related entries

* [[universal enveloping algebra]]

category: people


tmf

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###Context###
#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
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## Idea

The [[generalized (Eilenberg-Steenrod) cohomology]] theory/[[spectrum]] called $tmf$ -- for _[[topological modular forms]]_ -- is in a precise sense the union of all [[elliptic cohomology theories]]/[[elliptic spectra]] ([Hopkins 94](#Hopkins94)).

More precisely, $tmf$  is the [[homotopy limit]] in [[E-∞ rings]] of the [[elliptic spectra]] of all [[elliptic cohomology]] theories, parameterized over the [[moduli stack of elliptic curves]] $\mathcal{M}_{ell}$. That such a parameterization exists, coherently, in the first place is due to the [[Goerss-Hopkins-Miller theorem]]. In the language of [[derived algebraic geometry]] this refines the [[commutative ring]]-valued [[structure sheaf]] $\mathcal{O}$ of the [[moduli stack of elliptic curves]] to an [[E-∞ ring]]-valued sheaf $\mathcal{O}^{top}$, making $(\mathcal{M}_{ell}, \mathcal{O}^{top})$ a [[spectral Deligne-Mumford stack]], and $tmf$ is the [[E-∞ ring]] of [[global sections]] of that structure sheaf ([[A Survey of Elliptic Cohomology|Lurie]]).

The construction of $tmf$ has motivation from [[physics]] ([[string theory]]) and from [[chromatic homotopy theory]]:

1. **from [[string theory]].** Associating to a [[space]], roughly, the [[partition function]] of the [[spinning string]]/[[superstring]] [[sigma-model]] with that space as [[target space|target]] [[spacetime]] defines a [[genus]] known as the _[[Witten genus]]_, with [[coefficients]] in ordinary [[modular forms]]. Now, the interesting genera typically appear as the values on [[homotopy groups]] (the [[decategorification]]) of [[orientation in generalized cohomology|orientations]] of [[multiplicative cohomology theories]]; for instance the [[A-hat genus]], which is the [[partition function]] of the [[spinning particle]]/[[superparticle]] is a shadow of the [[Atiyah-Bott-Shapiro orientation|Atiyah-Bott-Shapiro]] [[Spin structure]]-orientation of the [[KO]] spectrum. Therefore an obvious question is which spectrum lifts this classical statement from point [[particles]] to [[strings]]. The spectrum $tmf$ solves this: there is a [[String structure]] [[string orientation of tmf|orientation of tmf]] such that on homotopy groups it reduces to the [[Witten genus]] of the [[superstring]] ([Ando-Hopkins-Rezk 10](#AndoHopkinsRezk10)).

   Mathematically this means for instance that $tmf$-cohomology classes help to detect elements in the [[String structure|string]] [[cobordism ring]]. Physically it means that the small aspect of [[string theory]] which is captured by the [[Witten genus]] is realized more deeply as part of fundamental mathematics ([[chromatic homotopy theory|chromatic]] [[stable homotopy theory]], see the next point) and specifically of [[elliptic cohomology]]. Since the full mathematical structure of [[string theory]] is still under investigation, this might point the way:

   > 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))

1. **from [[chromatic homotopy theory]].** The [[symmetric monoidal (∞,1)-category|symmetric monoidal]] [[stable (∞,1)-category|stable]] [[(∞,1)-category of spectra]] ([[finite spectra]]) has its [[prime spectrum of a symmetric monoidal stable (∞,1)-category|prime spectrum]] parameterized by [[prime numbers]] $p$ and [[Morava K-theory]] spectra $K(n)$ at these primes, for natural numbers $n$. The level $n$ here is called the _[[chromatic level]]_. In some sense the part of this prime spectrum at chromatic level 0 is [[ordinary cohomology]] and that at level 1 is [[topological K-theory]]. Therefore an obvious question is what the part at level 2 would be, and in some sense the answer is $tmf$. (This point of view has been particularly amplified in the review ([Mazel-Gee 13](#MazelGee13)) of the writeup of the construction in ([Behrens 13](#Behrens13)), which in turn is based on unpublished results based on ([Hopkins 02](#Hopkins02))). For purposes of [[stable homotopy theory]] this means for instance that $tmf$ provides new tools for computing more [[homotopy groups of spheres]] via an [[Adams-Novikov spectral sequence]].

## Definition

Write

* $\mathcal{M}_{cub}$ for the [[moduli stack of curves]] for [[cubic curves]];

* $\mathcal{M}_{ell}$ for the [[moduli stack of elliptic curves]];

* $\mathcal{M}_{\overline{ell}}$ for its [[Deligne-Mumford compactification]] obtained by adding the [[nodal cubic curve]].

(Here $\mathcal{M}_{cub}$ is obatined by furthermor adding also the [[cuspidal cubic curve]], hence we have canonical maps $\mathcal{M}_{ell}\to \mathcal{M}_{\overline{ell}}\to \mathcal{M}_{cusp} \to \mathcal{M}_{FG}$).

The [[Goerss-Hopkins-Miller theorem]] equips these three moduli stacks with [[E-∞ ring]]-valued [[structure sheaves]] $\mathcal{O}^{top}$ (and by [[A Survey of Elliptic Cohomology|Lurie (Survey)]] that makes them into [[spectral Deligne-Mumford stacks]] which are moduli spaces for [[derived elliptic curves]] etc.)

The _$tmf$-spectrum_ is defined to be the $E_\infty$-ring of [[global sections]] of $\mathcal{O}^{top}$ (in the sense of [[derived algebraic geometry]], hence the [[homotopy limit]] of $\mathcal{O}^{top}$ over the [[etale site]] of $\mathcal{M}$). More precisely one sets

* $TMF \coloneqq \Gamma(\mathcal{M}_{ell}, \mathcal{O}^{top})$;

* $Tmf \coloneqq \Gamma(\mathcal{M}_{\overline{ell}}, \mathcal{O}^{top})$;

* $tmf \coloneqq$ the [[connective cover]] of $Tmf$ (also $\simeq \Gamma(\mathcal{M}_{\overline{cub}}, \mathcal{O}^{top})$ ([Hill-Lawson 13, p. 2](#HillLawson13}) (?)).

##Constructions

### Decomposition via Arithmetic fracture squares
 {#DecomopositionViaArithmeticSquares}

We survey here some aspects of the explicit construction in ([Behrens 13](#Behrens13)), a review is also in ([Mazel-Gee 13](#MazelGee13)),

The basic strategy here is to use [[arithmetic squares]] in order to decompose the problem into smaller more manageable pieces.

Write $\overline{\mathcal{M}_{ell}}$ for the compactified [[moduli stack of elliptic curves]]. In there one finds the pieces

$$
  \array{
    \overline{\mathcal{M}_{ell}}
    &\stackrel{\iota_{p}}{\leftarrow}&
    (\overline{\mathcal{M}_{ell}})_p
    \\
    {}^{\mathllap{\iota_{\mathbb{Q}}}}\uparrow
    \\
    (\overline{\mathcal{M}_{ell}})_{\mathbb{Q}}
  }
$$

given by [[rationalization]]

$$
  (\overline{\mathcal{M}_{ell}})_{\mathbb{Q}}
  =
  \overline{\mathcal{M}_{ell}}
  \underset{Spec(\mathbb{Z})}{\times}
  Spec(\mathbb{Q})
$$

(hence this is the moduli of [elliptic curves over the rational numbers](elliptic+curve#OverTheRationalNumbers))
and by [[p-completion]]

$$
  (\overline{\mathcal{M}_{ell}})_p
  =
  (\overline{\mathcal{M}_{ell}})
  \underset{Spec(\mathbb{Z})}{\times}
  Spf(\mathbb{Z}_p)
$$

for any [[prime number]] $p$, where $\mathbb{Z}_p$ denotes the [[p-adic integers]] and $Spf(-)$ the [[formal spectrum]]. (Hence this is the moduli  of [elliptic curves over p-adic integers](elliptic+curve#OverpAdics)).

This induces the [[arithmetic square]] decomposition which realizes $\mathcal{O}^{top}$ as the [[homotopy fiber product]] in

$$
  \array{
    \mathcal{O}^{top} &\to& \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p
    \\
    \downarrow && \downarrow^{\mathrlap{L_{\mathbb{Q}}}}
    \\
    (\iota_{\mathbb{Q}})_\ast \mathcal{O}^{top}_{\mathbb{Q}}
    &\stackrel{\alpha_{arith}}{\to}&
    \left(
     \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p
    \right)_{\mathbb{Q}}
  }
$$

Here $\mathcal{O}^{top}_{\mathbb{Q}}$ can be obtained directly, and to obtain $\mathcal{O}^{top}_p$ one uses in turn another [[fracture square]], now decomposing via [[K(n)-localization]] into $K(1)$-local and $K(2)$-local pieces.

(...)

### Stacks from spectra

There is a way to "construct" the tmf-spectrum as the [[E-∞ ring]] of [[global section]]s of a [[structured (∞,1)-topos]] whose underlying space is essentially the [[moduli stack]] of [[elliptic curve]]s. We sketch some main ideas of this construction.

#### The context -- derived geometry over formal duals of $E_\infty$-rings

The discussion happens in the context of [[derived geometry]] in the [[(∞,1)-topos]] $\mathbf{H}$ over a [[small (∞,1)-category|small]] version of the [[(∞,1)-site]] of formal duals of [[E-∞ ring]]s ([[ring spectra]]). This is equipped with some [[subcanonical coverage]]. For $R \in E_\infty Ring$ we write $Spec R$ for its image under the [[(∞,1)-Yoneda embedding]] $(E_\infty Ring)^{op} \hookrightarrow \mathbf{H}$.

\begin{remark}
The [[terminal object in an (∞,1)-category|terminal object]] in $\mathbf{H}$ is the formal dual of the [[sphere spectrum]]

$$
  * \simeq Spec(\mathbb{S})
  \,.
$$

Because the sphere spectrum is the [[initial object]] in $E_\infty Ring$.
\end{remark}

#### Coverings by the Thom spectrum

The crucial input for the entire construction is the following statement.

The idea is that the formal dual of the [[complex cobordism cohomology|complex cobordism]] [[Thom spectrum]] $M U$ is in a suitable sense a covering

$$
  Spec M U \to  Spec \mathbb{S}
$$

of the [[terminal object]] in $\mathbf{H}$. (See at _[Adams spectral sequence -- As derived descent](Adams%20spectral%20sequence#DefinitionInHigherAlgebra)_)

This means that $Spec M U$ plays the role of a [[cover]] of the point. This allows to do some computations with ring spectra _locally on the cover $Spec M U$_ . Since $M U^*$ is the [[Lazard ring]], this explains why [[formal group law]]s show up all over the place.

To see this, first notice that the problem of realizing $R = tmf$ or any other ring spectrum as the ring of global sections on something has a _tautological solution_ : almost by definition (see [[generalized scheme]]) there is an $E_\infty$-ring valued [[structure sheaf]] $\mathcal{O}Spec(R)$ on $Spec R$ and its global sections is $R$. So we have in particular

$$
  tmf \simeq \mathcal{O}(Spec(tmf))
  \,.
$$

In order to get a less tautological and more insightful characterization, the strategy is now to pass on the right to the $Spec M U$-cover by forming the [[(∞,1)-pullback]]

$$
  \array{
    Spec(tmf) \times Spec(M U) &\to& Spec(tmf)
    \\
    \downarrow && \downarrow
    \\
    Spec(M U) &\to& * \simeq Spec(\mathbb{S})
  }
  \,.
$$

The resulting [[Cech nerve]] is a [[groupoid object in an (∞,1)-category]] given by

$$
  \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec(tmf) \times Spec(MU) \times Spec(MU) \stackrel{\to}{\to} Spec(tmf) \times Spec(MU)
$$

which by formal duality is

$$
  \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (tmf \wedge MU \wedge MU) \stackrel{\to}{\to} Spec ( tmf \wedge MU)

$$

where the [[smash product]] $\wedge$ of ring spectra over the [[sphere spectrum]] $\mathbb{S}$ is the [[tensor product]] operation on function algebras formally dual to forming products of spaces.

As a [[groupoid object in an (∞,1)-category|groupoid object]] this is still equivalent to just $Spec(tmf)$.

#### Decategorification: the ordinary moduli stack of elliptic curves

To simplify this we take a drastic step and apply a lot of [[decategorification]]: by applying the [[homotopy group]] [[(∞,1)-functor]] to all the $E_\infty$-rings involved these are sent to graded ordinary [[ring]]s $\pi_*(tmf)$, $\pi_*(M U)$ etc. The result is an ordinary [[simplicial object|simplicial]] [[scheme]]

$$
  \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(tmf \wedge M U \wedge  M U)) \stackrel{\to}{\to} Spec ( \pi_*(tmf \wedge M U))
  \,,
$$

which remembers the fact that its structure rings are graded by being equipped with an [[action]] of the multiplicative group $\mathbb{G} = \mathbb{A}^\times$ (see [[line object]]).

This general Ansatz is discussed in ([Hopkins](#Hopkins)).

This simplicial scheme, which is degreewise the formal dual of a graded ring of [[generalized (Eilenberg-Steenrod) cohomology|generalized homology]]-groups one can show is in fact a [[groupoid]], hence a [[stack]]: effectively the [[moduli stack of elliptic curves]]. $\mathcal{M}_{ell}$. See ([Henriques](#HenriquesModuli)).

In fact if in this construction one replaced $Spec tmf$ by the point, one obtains the simplicial scheme

$$
  \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(M U \wedge M U)) \stackrel{\to}{\to} Spec ( \pi_*(M U))
$$

which one finds is the [[moduli stack  of formal group laws]] $\mathcal{M}_{fg}$.

#### Explicit computation of homotopy groups by a spectral sequence

Now, a priori these underived stacks remember little about the original [[derived scheme]]s $Spec tmf$ etc. They may not even carry any $E_\infty$-ring valued structure sheaf anymore (though some of them do).

If they do carry an $E_\infty$-ring valued structure sheaf $\mathcal{O}$, one can compute the homotopy groups of its global sections by a [[spectral sequence]]

$$
  H^p(\mathcal{M}_{ell}, \pi_q(\mathcal{O}))
  \Rightarrow
  \pi_{p+q} \mathcal{O}(\mathcal{M}_{ell})
  \,.
$$

But it turns out that even if the derived structure sheaf does not exist, this spectral sequence may still converge and may still compute the homotopy groups of the ring spectrum that one started with. This gives one way to compute the homotopy groups of $tmf$.

For the case of $tmf$ one finds that the homotopy sheaves $\pi_q(\mathcal{O}(\mathcal{M}_{ell}))$ are simple: they vanish in odd degree and are [[tensor power]]s $\omega^{\otimes k}$ of the canonical line bundle $\omega$ in even degree $2 k$, where the fiber of $\omega$ over an [[elliptic curve]] is the [[tangent space]] of that curve at its identity element. A [[section]] of $\omega^{\otimes k}$ is a [[modular form]] of weight $k$. So the whole problem of computing the homotopy groups of $tmf$ boils down to computing the [[abelian sheaf cohomology]] of the moduli stack of elliptic curves with coefficients in these abelian groups of modular forms --- and then examining the resulting spectral sequence.

This can be done quite explicitly in terms of a long but fairly elementary computation  in ordinary algebra. A detailed discussion of this computation is in ([Henriques](#Henriques))

### With Level structure
 {#WithLevelStructure}

The [[moduli stack of elliptic curves]] has covers by that of [[elliptic curves with level structure]] $\Gamma$. Under some conditions these covers inherit derived structure sheaves $\mathcal{O}^{top}$ and hence induce spectra of "topological forms with level structure", $tmf(\Gamma)$ ([Mahowald-Rezk 09](#MahowaldRezk09)). For more on this see at _[[modular equivariant elliptic cohomology]]_.

For instance $tmf_0(2)$ (for the [[congruence subgroup]] which preserves an NS-R [[spin structure]] on elliptic curves over the complex numbers) is the [[elliptic spectrum]] $Ell$ of ([Landweber-Ravenel-Stong 93](#spin+orientation+of+Ochanine+elliptic+cohomology#LandweberRavenelStong93)), see at _[[tmf0(2)]]_.

Discussion of level structure also governs the relation of tmf to K-theory, see at _[Maps to K-theory and Tate K-theory](#MapToTateKTheory)_.

## Properties

### Homotopy groups
 {#HomotopyGroups}

The first few [[homotopy groups]] of $tmf$ are ([Hopkins 02, section 4.3](#Hopkins02))

| $k$ |  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
|-----|----|---|---|---|---|---|---|---|---|----|----|----|----|----|----|
| $\pi_k(tmf)$ | $\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/24\mathbb{Z}$ | 0 | 0 | $\mathbb{Z}/2\mathbb{Z}$ | 0 | $\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$ | $(\mathbb{Z}/2\mathbb{Z})^2$ | $\mathbb{Z}/6\mathbb{Z}$ | 0 | $\mathbb{Z}$ | $\mathbb{Z}/3\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}$ | $\mathbb{Z}/2\mathbb{Z}$ |

### Boardman homomorphism
 {#BoardmanHomomorphism}

Write $\mathbb{S}$ for the [[sphere spectrum]] and [[tmf]] for the [[connective spectrum]] of [[topological modular forms]].
Since [[tmf]] is an [[E-∞ ring|E-∞]][[ring spectrum]], there is an essentially unique homomorphism of [[E-∞ ring|E-∞]][[ring spectra]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
  \,.
$$

Regarded as a morphism of [[generalized homology]]-theories, this is also called the [[Hurewicz homomorphism]], or rather the [[Boardman homomorphism]] for $tmf$

+-- {: .num_prop #BoardmanHomomorphismInTmfIs6Connected}
###### Proposition
**(Boardman homomorphism in $tmf$ is 6-connected)**

The [[Boardman homomorphism in tmf]]

$$
  \mathbb{S}
  \overset{e_{tmf}}{\longrightarrow}
  tmf
$$

induces an [[isomorphism]] on [[stable homotopy groups]]
(hence from the [[stable homotopy groups of spheres]]), up to degree 6:

$$
  \pi_{\bullet \leq 6}(\mathbb{S})
  \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow}
  \pi_{\bullet\leq 6}(tmf)
  \,.
$$

=--

([Hopkins 02, Prop. 4.6](#Hopkins02), [DFHH 14, Ch. 13](#DFHH14))

### Inclusion of circle 2-bundles
 {#InclusionOfCircle2Bundles}

Write $B^2 U(1) \simeq K(\mathbb{Z},3)$ for the [[abelian ∞-group]] whose underlying [[homotopy type]] is the [[classifying space]] for [[circle 2-bundle]]. Write $\mathbb{S}[B^2 U(1)]$ for its [[∞-group ∞-ring]].

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

There is a canonical homomorphism of [[E-∞ rings]]

$$
  \mathbb{S}[B^2 U(1)] \to tmf
  \,.
$$

=--

See ([Ando-Blumberg-Gepner 10, section 8](#ABG10)).

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

This means that every [[circle 2-bundle]] ($U(1)$-[[bundle gerbe]]) given by a modulating map $\chi \colon X \to B^2 U(1)$ determines a class represented by

$$
  X \stackrel{\chi}{\to} B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf
$$

in the $tmf$-[[generalized cohomology]] of its base space $X$.

=--

### Maps to K-theory and to Tate K-theory
 {#MapToTateKTheory}

The inclusion of the compactification point (representing the [[nodal curve]] but being itself the [[cusp]] of $\mathcal{M}_{\overline{ell}}$) into the [[Deligne-Mumford compactification|compactified]] [[moduli stack of elliptic curves]] $\mathcal{M}_{\overline{ell}}$ is equivalently the inclusion of the [[moduli stack of 1-dimensional tori]] $\mathcal{M}_{1dtori} = \mathcal{M}_{\mathbb{G}_m}$ ([Lawson-Naumann 12, Appendix A](#LawsonNaumann12))

$$
  \mathcal{M}_{\mathbb{G}_m}
  \simeq \mathbf{B}\mathbb{Z}_2
 \longrightarrow
  \mathcal{M}_{\overline{ell}}
  \to
  \mathcal{M}_{FG}
$$

and pullback of [[global sections]] of [[Goerss-Hopkins-Miller-Lurie theorem]]-wise $E_\infty$-ring valued structure sheaves yields maps

$$
  KO \longleftarrow \longleftarrow \mathbb{S}
$$

exhibiting [[KO]] $= \Gamma(\mathcal{M}_{\mathbb{G}_m}, \mathcal{O}^{top})$.

At least after [[localization of a ring|2-localization]] the canonical [[double cover]] of the compactification of $\mathcal{M}_{\mathbb{G}_m} \simeq \mathbf{B}\mathbb{Z}_2$ similarly yields under $\Gamma(-,\mathcal{O}^{top})$ the inclusion of $ko$ as the $\mathbb{Z}_2$-[[homotopy fixed points]] of $ku$ (see at _[[KR-theory]]_ for more on this)

$$
  \array{
    ku_{(2)}
    \\
    \uparrow
    \\
    ko_{(2)}
  }
$$

and combined with the above this comes with maps from $tmf$ by restriction along the inclusion of the [[nodal curve]] [[cusp]] as

$$
  \array{
    ku_{(2)} & \longleftarrow & tmf_1(3)_{(2)}
    \\
    \uparrow && \uparrow
    \\
    ko_{(2)} & \longleftarrow & tmf_{(2)}
  }
  \,,
$$

([Lawson-Naumann 12, theorem 1.2](#LawsonNaumann12)), where $tmf_1(3)$ denotes [[topological modular forms]] with [[level structure on an elliptic curve|level-3 structure]] ([Mahowald-Rezk 09](#MahowaldRezk09)).

Moreover, including not just the nodal curve cusp but its [[formal neighbourhood]], which is the [[Tate curve]], there is analogously a canonical map of $E_\infty$-rings

$$
  tmf \longrightarrow KO[ [ q ] ]
$$

to [[Tate K-theory]] (this is originally asserted in [Ando-Hopkins-Strickland 01](#AndoHopkinsStrickland01), details are in [Hill-Lawson 13, appendix A](#HillLawson13)).

### Witten genus and string orientation

The $tmf$-[[spectrum]] is the codomain of the [[Witten genus]], or rather of its refinements to the [[string orientation of tmf]] with value in [[topological modular forms]]

$$
   \sigma : M String \to tmf
  \,.
$$

The original Witten genus is the value of the composite of this with the [map to Tate K-theory](#MapToTateKTheory) on [[homotopy groups]]. ([Ando-Hopkins-Rezk 10](#AndoHopkinsRezk10))

### Chromatic filtration

[[!include chromatic tower examples - table]]

### Anderson self-duality

The [[spectrum]] $Tmf$ is self-dual under [[Anderson duality]], more precisley $Tmf[1/2]$ is Anderson-dual to $\Sigma^{21} Tmf[1/2]$ ([Stojanoska 11, theorem 13.1](#Stojanoska11))

### Modular equivariant versions

See at _[[modular equivariant elliptic cohomology]]_
and at _[[Tmf(n)]]_.

## Related concepts

* [[taf]]

[[!include moduli stack of curves -- table]]

## References ##

The idea of a [[generalized cohomology theory]] with [[coefficients]] the ring of [[topological modular forms]] providing a home for the refined [[Witten genus]] of

* {#Witten87a} [[Edward Witten]],  _Elliptic Genera And Quantum Field Theory_, Commun. Math. Phys. 109 525 (1987) ([euclid.cmp/1104117076](http://projecteuclid.org/euclid.cmp/1104117076), [pdf](https://people.maths.ox.ac.uk/beem/papers/elliptic_genus_witten.pdf))

and produced as a [[homotopy limit]] of [[elliptic cohomology]] theories over the [[moduli stack of elliptic curves]] was originally announced, as joint work with [[Mark Mahowald]] and [[Haynes Miller]], in

* {#Hopkins94} [[Michael Hopkins]], section 9 of _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([[Hopkins_TopModFormsAtICM.pdf:file]], [doi:10.1007/978-3-0348-9078-6_49](https://doi.org/10.1007/978-3-0348-9078-6_49))

(There the spectrum was still called "$eo_2$" instead of "$tmf$".) The details of the definition then appeared in

* {#Hopkins02} [[Michael Hopkins]], section 4 of _Algebraic topology and modular forms_, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

A central tool that goes into the construction is the [[Goerss-Hopkins-Miller theorem]], see there for references on that.

Textbook account:

* {#DFHH14} [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

Expositions include

* {#MazelGee13} [[Aaron Mazel-Gee]], _You could've invented $tmf$_, April 2013 ([pdf slides](http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf), [notes pdf](http://math.berkeley.edu/~aaron/writing/tmf-seminar-talk.pdf))

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

See also

* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:1103.4187](https://arxiv.org/abs/1103.4187), [doi:10.1090/pspum/083](https://doi.org/10.1090/pspum/083))

An actual detailed account of the construction of $tmf$ (via decomposition by [[arithmetic squares]]) is spelled out in

* {#Behrens13} [[Mark Behrens]], _Notes on the construction of $tmf$_, 2013 ([pdf](http://math.mit.edu/~mbehrens/papers/buildTMF.pdf))

A complete account of the computation of the homotopy groups of $tmf$ (following previous unpublished computations) is in

* [[Tilman Bauer]], _Computation of the homotopy groups of the spectrum $tmf$_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/eo2ss.pdf))

A survey of how this works is in

* [[Akhil Mathew]], _The homotopy groups of $TMF$_ ([pdf](http://math.uchicago.edu/~amathew/tmfhomotopy.pdf))

  (This presents as an instructive much simpler but analogous case the construction of [[KO]] in analogy to the construction of $tmf$, more details on this are in [Mathew 13, section 3](#Mathew13).)

and course notes that go through the construction of tmf and the computation of its homotopy groups are here:

*  _Talbot workshop on TMF_ ([web](http://math.mit.edu/conferences/talbot/2007/tmfproc/))

   * {#Hopkins} [[Mike Hopkins]] (talk notes by [[Michael Hill]]), _Stacks and complex oriented cohomology theories_ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter08/MikesTalk1.pdf))

   * {#Henriques} [[André Henriques]], _The homotopy groups of tmf_ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter16/TmfHomotopy.pdf))

   * {#HenriquesModuli} [[André Henriques]], _The moduli stack of elliptic curves_   ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter04/henriques.pdf))

The non-connective version of this is discussed in

* [[Johan Konter]], _The homotopy groups of the spectrum Tmf_ ([arXiv:1212.3656](http://arxiv.org/abs/1212.3656))

Supplementary material graphically displaying parts of these intricate computations is in

* [[André Henriques]],

  the graded ring $tmf^\ast(pt)$ ([pdf](http://www.staff.science.uu.nl/~henri105/PDF/TmfRing.pdf));

  the [[spectral sequence]] used to compute it ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/henriques-tmfSS.pdf));

  whose $E_2$-page is $Ext_{A(2)}(\mathbb{F}_2, \mathbb{F}_2)$ where $A(2)$ is displayed here: [pdf](http://www.staff.science.uu.nl/~henri105/PDF/A2.pdf);

  the spectral sequence that computes $Tmf^\ast(pt)$ ([pdf](http://math.mit.edu/conferences/talbot/2007/tmfproc/EllipticSpectralSequence.pdf))

The $\mathbb{Z}_2$-[[homology]] of $tmf$ is discussed in

* {#Mathew13} [[Akhil Mathew]], _The homology of $tmf$_ ([arXiv:1305.6100](http://arxiv.org/abs/1305.6100))

The refinement of the [[Witten genus]] to a morphism of [[E-∞ rings]] to $tmf$, hence the [[string orientation of tmf]] is due to

* [[Michael Hopkins]], _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Zürich 1994 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850

* [[Michael Hopkins]], _Algebraic topology and modular forms_, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

see also remark 1.4 of

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller and Lurie)_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf)).

and for more on the [[sigma-orientation]] see

* [[Matthew Ando]], _The sigma orientation for analytic circle-equivariant elliptic cohomology_, Geom. Topol. 7 (2003) 91-153 ([arXiv:math/0201092](http://arxiv.org/abs/math/0201092))

Discussion of [[twisted cohomology]]  with [[coefficients]] in $tmf$
is in

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], section 8 of _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

Topological modular forms with _[[level N-structure]]_ -- $tmf(N)$ -- is discussed in

* {#MahowaldRezk09} [[Mark Mahowald]],  [[Charles Rezk]], _Topological modular forms of level 3_, Pure Appl. Math. Quar. 5 (2009) 853-872 ([pdf](http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf))

* {#DavisMahowald10} [[Donald Davis]], [[Mark Mahowald]], _Connective versions of $TMF(3)$_ ([arXiv:1005.3752](http://arxiv.org/abs/1005.3752))

* {#Stojanoska11} [[Vesna Stojanoska]], Duality for Topological Modular Forms ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_, Int. Math. Res. Not. (2013) ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_, Inventiones mathematicae volume 203, pages 359–416 (2016) ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394), [doi:10.1007/s00222-015-0589-5](https://doi.org/10.1007/s00222-015-0589-5))

  > (with [[level structure on an elliptic curve|level structure]])

The self-[[Anderson duality]] of $tmf$ is discussed in ([Stojanoska 11](#Stojanoska11)).

On equivariant topological modular forms (on [[equivariant elliptic cohomology]]):

* [[David Gepner]], [[Lennart Meier]], _On equivariant topological modular forms_, ([arXiv:2004.10254](https://arxiv.org/abs/2004.10254))

On the [[Boardman homomorphism]] (generalized [[Hurewicz homomorphism]]) to [[tmf]]:

* [[Mark Behrens]], [[Mark Mahowald]], J.D. Quigley, _The 2-primary Hurewicz image of $tmf$_ ([arXiv:2011.08956](https://arxiv.org/abs/2011.08956))

More on the role of lifting the [[Witten genus]] to tmf from the point of view of [[heterotic string theory]]:

* [[Yuji Tachikawa]], *Topological modular forms and the absence of a heterotic global anomaly*, Progress of Theoretical and Experimental Physics, **2022** 4 (2022) 04A107 $[$[arXiv:2103.12211](https://arxiv.org/abs/2103.12211), [doi:10.1093/ptep/ptab060](https://doi.org/10.1093/ptep/ptab060)$]$

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Topological modular forms and the absence of all heterotic global anomalies*, Comm. Math. Phys. **402** (2023) 1585-1620 [[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), [doi:10.1007/s00220-023-04761-2](https://doi.org/10.1007/s00220-023-04761-2)]

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras* [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)]

[[!redirects Tmf]]
[[!redirects TMF]]

[[!redirects tmf spectrum]]


Tmf(n)

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

After localization at primes dividing $n \in \mathbb{N}$ the [[covering]] of the [[moduli stack of elliptic curves]] $\mathcal{M}_{{ell}}$ by that of [[elliptic curves with level-n structure]] $\mathcal{M}_{{ell}}[n] \to \mathcal{M}_{{ell}}$ is sufficiently good that the [[Goerss-Hopkins-Miller-Lurie theorem]] may be applied to produce a [[homomorphism]] of [[E-∞ rings]]

$$
  (TMF \to TMF(n))
   =
  \Gamma\left(
    \left(
      \mathcal{M}_{\overline{ell}}[n] \to \mathcal{M}_{\overline{ell}}
    \right),
    \mathcal{O}^{top}
  \right)
$$

exhibiting [[TMF]] (after localization at $n$) as the [[homotopy fixed points]] of a [[modular group]] action by $SL_2(\mathbb{Z}/n\mathbb{Z})$ ([Hill-Lawson 13, p.3](#HillLawson13)).

With a bit more work one obtains analogous statements for the [[Deligne-Mumford compactification|compactified]] [[moduli stack of elliptic curves]] and $Tmf$ instead of $TMF$ ([Hill-Lawson 13, theorem 9.1](#HillLawson13))

This is directly analogous ([Lawson-Naumann 12](#LawsonNaumann12), [Hill-Lawson 13](#HillLawson13)) to how [[KO]] $\to$ [[KU]] exhibits the inclusion of the homotopy fixed points of the $\mathbb{Z}_2$-action on [[complex K-theory]] (which defines [[KR-theory]], see there for more).

See also at _[[modular equivariant elliptic cohomology]]_.

## References

* {#MahowaldRezk09} [[Mark Mahowald]] [[Charles Rezk]], _Topological modular forms of level 3_, Pure Appl. Math. Quar. 5 (2009) 853-872 ([pdf](http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf))

* {#DavisMahowald10} [[Donald Davis]], [[Mark Mahowald]], _Connective versions of $TMF(3)$_ ([arXiv:1005.3752](http://arxiv.org/abs/1005.3752))

* {#Stojanoska11} [[Vesna Stojanoska]], Duality for Topological Modular Forms ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))

* {#LawsonNaumann12} [[Tyler Lawson]], [[Niko Naumann]], _Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2_, Int. Math.
Res. Not. (2013) ([arXiv:1203.1696](http://arxiv.org/abs/1203.1696))

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_ ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394))

[[!redirects tmf(n)]]


tmf0(2)

+-- {: .rightHandSide}
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###Context###
#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Where [[tmf]] is obtained from the [[moduli stack of elliptic curves]], $tmf_0(2)$ (or its versions $Tmf_0(2)$, $TMF_0(2)$) denotes the [[spectrum]] obtained from the cover by the moduli stack of [[elliptic curves with level structure]] for the [[congruence subgroup]] $\Gamma_0(2)$ (the one that preserves the NS-R spin structure on an elliptic curve, see [here](level+structure+on+an+elliptic+curve#RelationToSpinStructures)).

Detailed discussion is in ([Behrens 05](#Behrens05)).

This plays at least roughly the role for the [[Ochanine elliptic genus]] (the [[partition function]] of the [[type II superstring]] in the NS-R sector) as [[tmf]] does for the [[Witten genus]] (the partition function of the [[heterotic superstring]]). See at _[[spin orientation of Ochanine elliptic cohomology]]_ for more.

## Properties

### Relation to Landweber-Ravenel-Stong-Ochanine elliptic spectrum

After inversion of 6, $TMF_0(2)$ is ([Behrens 05](#Behrens05)) the [[elliptic spectrum]] $Ell$ of ([Landweber-Ravenel-Stong 93](#LandweberRavenelStong93)).

### Periodicity

The periodicity of $TMF_0(2)$ is 8.

## Related concepts

* [[modular equivariant elliptic cohomology]]

## References

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et. al. (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

* {#Behrens05} [[Mark Behrens]], _A modular description of the K(2)-local sphere at the prime 3_ ([arXiv:math/0507184](http://arxiv.org/abs/math/0507184))


Tobias Barthel

* [webpage](http://guests.mpim-bonn.mpg.de/tbarthel/)

## Selected writings

A new proof of the [[Strøm model structure]] using [[algebraic weak factorization systems]]:

* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](http://dx.doi.org/10.2140/agt.2013.13.1089))

On [[chromatic homotopy theory]]:

* [[Tobias Barthel]], [[Agnès Beaudry]], *Chromatic structures in stable homotopy theory*, in [[Handbook of Homotopy Theory]], Chapman and Hall/CRC Press (2019) [[arXiv:1901.09004](https://arxiv.org/abs/1901.09004), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

category: people


Tobias Dyckerhoff

* [webpage](https://www.math.uni-hamburg.de/home/dyckerhoff)
* Christopher Brav , Tobias Dyckerhoff, _Relative Calabi–Yau structures_, Compositio Math. 155 (2019) 372--412 [doi](https://doi.org/10.1112/S0010437X1900702); _Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects_, Sel. Math. New Ser. 27, 63 (2021)  [doi](https://doi.org/10.1007/s00029-021-00642-5)
* T. Dyckerhoff, _A categorified Dold-Kan correspondence_, [arXiv:1710.08356](https://arxiv.org/abs/1710.08356)
* Tobias Dyckerhoff, [[Mikhail Kapranov]], _Higher Segal spaces I_, [arXiv:1212.3563](https://arxiv.org/abs/1212.3563), now part of the book Tobias Dyckerhoff, Mikhail Kapranov, _Higher Segal spaces_, Springer LNM 2244 (2019) [doi](https://doi.org/10.1007/978-3-030-27124-4)
* T. Dyckerhoff, _Higher Segal spaces _, talk at Steklov Mathematical Institute (2011) [video](http://www.mathnet.ru/php/presentation.phtml?presentid=3718&option_lang=eng)
* T. Dyckerhoff, _Higher categorical aspects of Hall algebras_, In: Herbera D., Pitsch W., Zarzuela S. (eds) Building Bridges Between Algebra and Topology. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser 2015 [arXiv:1505.06940](https://arxiv.org/abs/1505.06940) [doi](https://doi.org/10.1007/978-3-319-70157-8_1)
* T. Dyckerhoff, _Compact generators in the categories of matrix factorizations_,  Duke Math. J. 159(2): 223--274 (2011) [doi](https://10.1215/00127094-1415869) [arXiv:0904.4713](http://arxiv.org/abs/0904.4713)
* Tobias Dyckerhoff, [[Mikhail Kapranov]], _Triangulated surfaces in triangulated categories_, J. Eur. Math. Soc. __20__:6, 1473--1524 (2018) [doi](https://doi.org/10.4171/JEMS/791)

category: people


Tobias Fritz

* [webpage](http://perimeterinstitute.ca/personal/tfritz/)

## Selected writings

On [[calculus of fractions]] ("[[categories of fractions]]"):

* {#Fritz11} [[Tobias Fritz]], *Categories of Fractions Revisited*, Morfismos **15**  2 (2011) 19-38 [[arXiv:0803.2587](https://arxiv.org/abs/0803.2587), [morfismos:vol15-n2-3](https://morfismos.math.cinvestav.mx/sites/default/files/Upload/vol15-n2-3.pdf)]

On [[homotopy type theory]]:

* [[Cecilia Flori]], [[Tobias Fritz]], *Homotopy Type Theory: Univalent Foundations of Mathematics*, lecture notes (2014) [[pdf](http://tobiasfritz.science/2014/HoTT_lecturenotes.pdf), [webpage](http://tobiasfritz.science/2014/hott.html)]

On [[gs-monoidal categories]] in [[category theory|category theoretic]] [[probability theory]]:

* [[Tobias Fritz]], [[Wendong Liang]], *Free gs-monoidal categories and free Markov categories*, Appl. Categ. Structures **31** 21 (2023)  [[arXiv:2204.02284](https://arxiv.org/abs/2204.02284), [doi:10.1007/s10485-023-09717-0](https://doi.org/10.1007/s10485-023-09717-0)]

category: people


Tobias Hagge

* [GoogleScholar page](https://scholar.google.com/citations?user=RQAtM4kAAAAJ&hl=en)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=151981)

## Selected writings

On [[number theory|number theoretic]] aspects of [[modular tensor categories]]:

* {#DavidovichHaggeWang2013} [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)]

category: people


Tobias Heindel

* [personal page](http://www.informatik.uni-leipzig.de/~heindel/)

## Selected writings

On [[van Kampen colimits]]:

* {#SobocinskiHeindel11} [[Pawel Sobocinski]], [[Tobias Heindel]], *Being Van Kampen is a universal property*, Logical Methods in Computer Science, **7** 1 (2011) [[arXiv:1101.4594](https://arxiv.org/abs/1101.4594), doi:10.2168/LMCS-7(1:14)2011]

## Related entries

* [[strict initial object]]

category: people


Tobias Mistele

Tobias Mistele is a researcher at the [[Frankfurt Institute for Advanced Studies]].

## Publications

* [[Sabine Hossenfelder]], [[Tobias Mistele]], *Strong lensing with superfluid dark matter*. Journal of Cosmology and Astroparticle Physics, Volume 2019, February 2019. ([doi:10.1088/1475-7516/2019/02/001](https://doi.org/10.1088/1475-7516/2019/02/001), [arXiv:1809.00840](https://arxiv.org/abs/1809.00840))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Galactic mass-to-light ratios with superfluid dark matter*, Astronomy and Astrophysics, Volume 664, Article 40, August 2022 ([doi:10.1051/0004-6361/202243216](https://doi.org/10.1051/0004-6361/202243216), [arXiv:2201.07282](https://arxiv.org/abs/2201.07282))

* [[Tobias Mistele]], *Cherenkov radiation from stars constrains hybrid MOND dark matter models*. Journal of Cosmology and Astroparticle Physics, Volume 2022, November 2022 ([doi:10.1088/1475-7516/2022/11/008](https://doi.org/10.1088/1475-7516/2022/11/008), [arXiv:2103.16954](https://arxiv.org/abs/2103.16954))

* [[Tobias Mistele]], *A novel Cherenkov radiation constraint for hybrid MOND dark matter models*, ([arXiv:2208.14308](https://arxiv.org/abs/2208.14308))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Aether Scalar Tensor theory confronted with weak lensing data at small accelerations*, ([arXiv:2301.03499](https://arxiv.org/abs/2301.03499))

* [[Tobias Mistele]], [[Stacy McGaugh]], [[Sabine Hossenfelder]], *Superfluid dark matter in tension with weak gravitational lensing data* ([arXiv:2303.08560](https://arxiv.org/abs/2303.08560))

category: people


Toby Bartels

# Toby Bartels

I am [Toby Bartels](http://tobybartels.name/) (they/them).

My main interests here are the [[foundations]] of mathematics, including [[constructive mathematics]] and how to formalise [[higher category theory]] [[internalization|internally]].  I have a liking for [[centipede mathematics]], which I sometimes take too far, but usually it has applications somewhere.  I also have an aptitude for [[negative thinking]]; my favourite mathematical object is the [[empty set]].  Along these lines, I advocate learning [[order theory]] before [[category theory]], just as one learns category theory before [[higher category theory]]; see [[category theory vs order theory]].

The regulars here may be especially interested in my PhD dissertation, [Higher Gauge Theory I: 2-Bundles](http://tobybartels.name/2bundles/).

I also have [[Toby Bartels:HomePage|my own personal web here]], although not much is on it.

category: people

[[!redirects  Toby Bartels]]
[[!redirects Toby Bartels]]
[[!redirects Toby]]
[[!redirects Ybot Sletrab]]


Toby Bartels, from http://en[dot]wikipedia[dot]org/wiki/File

< [[Toby Bartels]]


Toby Gee

* [personal page](https://www.ma.imperial.ac.uk/~tsg/)

## Selected writings
 {#SelectedWritings}

Introducing the [[Emerton-Gee stack]]:

* {#EmertonGee19} [[Matthew Emerton]], [[Toby Gee]], *Moduli stacks of étale $(\phi, \Gamma)$ modules and the existence of crystalline lifts* ([arXiv:1908.07185](https://arxiv.org/abs/1908.07185))

category: people


Toby St Clere Smithe

* [webpage](https://tsmithe.net/)

### Selected writings

On [[Bayesian inference]] with methods from [[category theory]]:

* [[Toby St Clere Smithe]], _Mathematical Foundations for a Compositional Account of the Bayesian Brain_ [[arXiv:2212.12538](https://arxiv.org/abs/2212.12538)]

category: people


Toda bracket

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Algebraic topology
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[[!include algebraic topology - contents]]
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#Contents#
* table of contents
{:toc}

\linebreak

> the [[stable homotopy groups of spheres]] are not really understood until the Toda bracket structure is revealed ([Iaksen-Wang-Xu 20, p. 17](#IaksenWangXu20))

\linebreak

## Idea
  {#Idea}

Given a [[pointed (∞,1)-category]] $\mathcal{C}$ (such as that [[presentable (∞,1)-category|presented]] by the [[classical model structure on pointed topological spaces]]) or at least a [[(2,1)-category]] (such as its [[homotopy 2-category]]), the _Toda bracket_ ([Toda 62](#Toda62)) is the operation that takes a sequence of 3 [[composition|composable]] [[1-morphisms]] in $\mathcal{C}$

$$
  X_0
    \overset{\;\;f_1\;\;}{\longrightarrow}
  X_1
    \overset{\;\;f_2\;\;}{\longrightarrow}
  X_2
    \overset{\;\;f_3\;\;}{\longrightarrow}
  X_3
  \,,
$$

equipped with a [[pair]] of overlapping [[null homotopies]]

\begin{xymatrix@C=15pt}
  X_0
  \ar[rr]
    |-{ \;f_1\; }
  \ar@/^2.6pc/[rrrr]
    ^-{0}
    _-{\ }="s1"
  &&
  X_1
  \ar[rr]
    |-{ \;f_2\; }
  \ar@/_2.6pc/[rrrr]
    _-{0}
    ^-{\ }="t2"
  &&
  X_2
  \ar[rr]
    |-{ \;f_3\; }
  &&
  X_3
  %
  \ar@{=>}
    "t2"+(0,+6); "t2"
    ^-{ \phi_2 }
  \ar@{=>}
    "s1"; "s1"+(0,-6)
    ^-{ \phi_1 }
\end{xymatrix}

to the [[higher homotopy]]-[[equivalence class]] of their [[pasting diagram|pasting]]-composite, which is a [[2-morphism]] [[loop]] (on the [[zero morphism|zero]] [[1-morphism]]) in the [[(∞,1)-categorical hom space]] $\mathcal{C}(X_0,X_3) \in $ [[∞Groupoids]], hence an element in its [[fundamental group]] $\pi_1 \mathcal{C}(X_0,X_3)$:

\begin{xymatrix@=18pt}
  &&
  &&
  X_0
  \ar[dd]
    _-{ f_1 }
    ^>>{\ }="t1"
  \ar[rr]
    _>>{\ }="s1"
  &&
  0
  \ar[dd]
  \\
  \\
  X_0
  \ar@/^2pc/[rr]
    ^-{ 0 }
    _-{\ }="ss"
  \ar@/_2pc/[rr]
    _-{ 0 }
    ^-{\ }="tt"
  &{\phantom{AAAA}}&
  X_3
  &
  :=
  &
  X_1
  \ar[dd]
    ^>>{\ }="t2"
  \ar[rr]
    |-{ \;f_2\; }
    _>>{\ }="s2"
  &&
  X_2
  \ar[dd]
    ^-{ f_3 }
  &&
  \in
  \;
  \pi_1
  \mathcal{C}
  \big(
    X_0, X_3
  \big)
  \\
  \\
  &&
  &&
  0
  \ar[rr]
  &&
  X_3
  %
  \ar@{=>} "s1"; "t1"
    ^-{
      \color{orange}
      \phi_1
    }
  \ar@{=>} "s2"; "t2"
    ^-{
      \color{orange}
      \phi_2
    }
  \ar@{=>}
    "ss"; "tt"
    |-{
      \mathclap{\phantom{\vert^{\vert}}}
      \scalebox{.6}{$
          \color{orange}
          (\phi_2 \cdot f_1)
          \circ
          (f_3 \cdot \phi_1)
      $}
      \mathclap{\phantom{\vert_{\vert}}}
    }
\end{xymatrix}

Or rather, the Toda bracket is usually and equivalently taken to be the [[homotopy class]] of the [[1-morphism]]
$
  \vdash
  \big(
    (\phi_2 \cdot f_1)
      \circ
    (f_3 \cdot \phi_1)
  \big)
$
that _classifies_ this [[homotopy]], via the [[universal property]] of [[homotopy fibers]]/[[homotopy cofibers]]:

* if $\mathcal{C}$ admits [[finite (∞,1)-colimits]] or at least [[reduced suspensions]] $\Sigma(-)$ ([[homotopy cofibers]] of [[zero morphisms]]), so that this [[2-morphism]] between [[zero morphisms]] corresponds to  a  [[1-morphism]] of the form:

  $$
     \Sigma X_0
      \overset{
        \vdash
        (\phi_2 \cdot f_1)
          \circ
        (f_3 \cdot \phi_1)
      }
      {\longrightarrow}
    X_3
  $$

* if $\mathcal{C}$ admits [[finite (∞,1)-limits]] or at least [[based loop space objects]] $\Omega(-)$ ([[homotopy fibers]] of [[zero morphisms]]), so that this [[2-morphism]] between [[zero morphisms]] corresponds  to a [[1-morphism]] of  the form

  $$
    X_0
      \overset{
        \vdash
        (\phi_2 \cdot f_1)
          \circ
        (f_3 \cdot \phi_1)
      }
      {\longrightarrow}
    \Omega X_3
    \,,
  $$

or both, as we shall assume for ease of notation,
then the Toda bracket is the [[homotopy class]] of this 1-morphism in the [[homotopy category of an (infinity,1)-category|homotopy category]]:

\[
  \label{TheTodaBracketDependingOnTheNullHomotopies}
  \array{
  \big\langle
    f_1, f_2, f_3
  \big\rangle_{(\phi_1,\phi_2)}
  \;\coloneqq\;
  \big[
    \vdash
    (\phi_2 \cdot f_1)
      \circ
    (f_3 \cdot \phi_1)
  \big]
  \;
  \in
  &&
  \pi_0 \mathcal{C}
  \big(
    \Sigma X_0, X_3
  \big)
  \\
  &
  \;\simeq\;
  &
  \pi_0 \mathcal{C}
  \big(
    X_0, \Omega X_3
  \big)
  \\
  &
  \;
  \simeq
  \;
  &
  \pi_1 \mathcal{C}
  \big(
    X_0, X_3
  \big)
  }
  \,.
\]

\begin{imagefromfile}
    "file_name": "TodaBracketHomotopyDiagrammatically.jpg",
    "width": 650,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}

Here the last [[pasting diagram]] on the bottom right shows the [[homotopy cofiber]]-construction equivalently realized via [[mapping cones]] (ordinary [[cofiber coproducts]] after [[cofibrant resolution|resolving]] points to [[cones]]), by which one may present the top [[homotopy coherent diagram]] in, for instance, any [[pointed model category|pointed]] [[cofibration category]]- or [[model category]]-[[presentable (infinity,1)-category|presentation]] of the [[pointed (∞,1)-category]] $\mathcal{C}$.

It is in this last form, by "consecutively extending maps over cones", that the Toda bracket was introduced in [Toda 62](#Toda62), and in which it is still presented in most references to date.

The more abstract formulation shown at the top, via [[homotopy coherent diagram|homotopy]]-[[pasting diagrams]], is made more explicit in [Hardie-Kamps-Kieboom 99, (0.2)-(0.3)](#HardieKampsKieboom99), [Hardie-Marcum-Oda 01](#HardieMarcumOda01),  [Hardie-Kamps-Marcum 02, (2.2)](#HardieKampsMarcum02) (formalized there inside a [[homotopy 2-category]]).

{#GroupActionOnChoicesOfTodaBrackets} From this abstract homotopy-pasting perspective it is manifest that the set of choices of refined Toda brackets (eq:TheTodaBracketDependingOnTheNullHomotopies) for given maps $(f_1, f_2, f_2)$ is, if [[inhabited set|inhabited]], a [[torsor]] over the [[direct product group]]

\[
  \label{SymmetryGroupOfTodaBracket}
  \begin{aligned}
  G_{(X_0, X_1, X_2, X_3)}
  &
  \coloneqq
  \;
  \pi_1 \mathcal{C}(X_0,X_2) \times \pi_1 \mathcal{C}(X_1, X_3)^{op}
  \\
  &
  =
  \;
  \pi_0 \mathcal{C}(X_0, \Omega X_2) \times \pi_0 \mathcal{C}(X_1, \Omega X_3)^{op}
  \\
  & = \;
  \pi_0 \mathcal{C}(\Sigma X_0, X_2) \times \pi_0 \mathcal{C}(\Sigma X_1, X_3)^{op}
  \,,
  \end{aligned}
\]

whose [[action]] is given by the evident [[composition]] of [[2-morphisms]]:

\begin{imagefromfile}
    "file_name": "GroupActionOnTodaBracket.jpg",
    "width": 750,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [SS21](https://ncatlab.org/schreiber/show/M-Theory+as+Mf-Theory)"
\end{imagefromfile}

The _plain Toda bracket_ is meant to be independent of the choice of [[null homotopies]] $(\phi_1,\phi_2)$ and thus taken to be the image of (eq:TheTodaBracketDependingOnTheNullHomotopies) in the [[quotient set]] by this action

$$
  \big\langle
    f_1, f_2, f_3
  \big\rangle
  \;\;
    \in
  \;\;
  Ho(\mathcal{C})
  \big(
    X_0, \Omega X_3
  \big)
  \big/
  G_{X_0, X_1, X_2, X_3}
  \,.
$$

With elements in the quotient set identified with the [[orbit]]-sets, the plain Toda bracket may be taken to *be* the [[orbit]] of this action, hence the [[subset]]

\[
  \label{PlainTodaBracket}
  \array{
  \big\langle
    f_1, f_2, f_3
  \big\rangle
  \;\coloneqq\;
  \left\{
    \big\langle
      f_1, f_2, f_3
    \big\rangle_{(\phi_1,\phi_2)}
    \;
    \left\vert
    \;
    \array{
      0
        &\overset{\phi_1}{\Rightarrow}&
      f_2 \circ f_1
      \\
      f_3 \circ f_2
        &\underset{\phi_2}{\Rightarrow}&
      0
    }
    \right.
  \right\}
  \;
  \subset
  &&
  \pi_0 \mathcal{C}
  \big(
    \Sigma X_0, X_3
  \big)
  \\
  &
  \;\simeq\;
  &
  \pi_0 \mathcal{C}
  \big(
    X_0, \Omega X_3
  \big)
  \\
  &
  \;
  \simeq
  \;
  &
  \pi_1 \mathcal{C}
  \big(
    X_0, X_3
  \big)
  }
\]

of all the classes (eq:TheTodaBracketDependingOnTheNullHomotopies) as one varies the [[null homotopies]] $(\phi_1,\phi_2)$.

{#TheTodaBundle} This means that the plain Toda bracket (eq:PlainTodaBracket) is not a function with values, but a [[bundle]] with [[fibers]]  (whose elements are the refined Toda brackets (eq:TheTodaBracketDependingOnTheNullHomotopies)):

\begin{xymatrix@C=17pt@R=10pt}
    {\phantom{AAAA}}
    \overset{
      \mathclap{
      \raisebox{6pt}{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          element of fiber of Toda bundle:
          \\
          a refined Toda bracket
        \end{tabular}
      }
      }
    }{
      \langle
        f_1, f_2, f_3
      \rangle_{(\phi_1,\phi_2)}
    }
    \ar@{}[rr]|-{ \in }
    &&
    \overset{
      \mathclap{
      \raisebox{6pt}{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          fiber of Toda bundle:
          \\
          a Toda bracket
        \end{tabular}
      }
      }
    }{
      \langle
        f_1, f_2, f_3
      \rangle
    }
    \ar@{}[ddrr]
      |-{ \mbox{\tiny(pb)} }
    \;
    \ar@{^{(}->}[rr]
    \ar[dd]
    &&
    \overset{
      \raisebox{3pt}{
        \tiny
        \color{olive}
        \bf
        the Toda bundle
      }
    }{
      \mathrm{Toda}(X_0, X_1, X_2, X_3)
    }
    \ar[dd]
    \\
    \\
    &&
    \underset{
      \mathclap{
      \raisebox{-3pt}{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          a triple of classes of maps
        \end{tabular}
      }
      }
    }{
      \big\{
        \big(
          [f_1], [f_2], [f_3]
        \big)
      \big\}
    }
    \;
    \ar@{^{(}->}[rr]
    &&
    \underset{
      \mathclap{
      \raisebox{-3pt}{
        \tiny
        \color{olive}
        \bf
        \begin{tabular}{c}
          all possible triples of
          \\
          classes of consecutive maps
        \end{tabular}
      }
      }
    }{
      \underset{i \in \{0,1,2\}}{\prod}
      \pi_0
      \mathcal{C}
      (X_i,X_{i+1})
    }
\end{xymatrix}

In any case, the Toda bracket may be thought of as being in [[homotopy theory]] what the [[Massey product]] is in [[cohomology theory]]: It is a "[[secondary characteristic class|secondary]] invariant", which exists (has [[inhabited set|inhabited]] [[fibers]]) when/since "primary invariants" -- namely the [[homotopy classes]] of the morphisms $f_2 \circ f_1$ and $f_3\circ f_2$ -- vanish, as witnessed by the [[null homotopies]].

A generalization of the Toda bracket produces an invariant for sequences of morphisms, equipped with consecutive pair-wise null-homotopies, which may contain possibly more than three morphisms; this _higher Toda bracket_ was maybe first considered in [Cohen 68, Sec. 2](#Cohen68)

\linebreak

## Preliminaries

Consider a sequence of maps $ A_0 \to A_1 \to A_2 \to A_3 $.  If the composites $A_0 \to A_2 $ and $ A_1\to A_3$ are nulhomotopic, then one has a diagram
$$\begin{array}{ccccc}
A_0 & \to & A_1 & \to & * \\
\downarrow & & \downarrow & & \downarrow \\
* & \to & A_2 & \to & A_3
\end{array} $$
any choice of homotopies in the two squares gives a map $ \Sigma A_0 \to A_3 $.

Define $C$ and $D$ to be the cofibers of $A_0 \to A_1$ and $A_1\to A_2$, respectively. A choice of homotopy $ A_0 \to A_2 \sim 0 $ corresponds to a choice of factorization $ A_1 \to C \to A_2 $, which gives a diagram of pushout squares

$$\begin{array}{ccccccc}
A_0 & \to & A_1 & \to & * \\
\downarrow & & \downarrow & & \downarrow \\
* & \to & C &\to & \Sigma A_0  & \to & *\\
 & & \downarrow & & \downarrow & & \downarrow \\
 & & A_2 & \to & D & \to & C'
\end{array}$$

It is to be noted that the map $ \Sigma A_0 \to D $ and possibly the object $C'$ depend on the choice of factor $ C \to A_2 $, but that $A_2 \to D$ does not, in any meaningful sense, so depend: this is just the structure map of the cofiber of $A_1\to A_2$.  Note that the cofiber $C'$ of $C\to A_2$ is thus equivalent to that of $\Sigma A_0 \to D$; but again the role of choices must be studied.

## Definitions

A sequence of maps $A_0 \to A_1 \to \cdots \to A_n$ will be called *a bracket sequence* (a novel phrase for the purposes of this entry) in either of two cases:

* $n = 3$ and the composites $A_0 \to A_2$ and $A_1 \to A_3$ are nulhomotopic; OR
* $n \gt 3$, and (using the preceding notations), there are choices of factor $C\to A_2$ and $ D \to A_3 $ such that the induced sequence $ \Sigma A_0 \to D \to A_3 \to \cdots \to A_n$ is a bracket sequence.

In all cases, a bracket sequence leads to a three-map sequence
$$ \Sigma^m A_0 \to D_m \to A_{m+2} \to A_{m+3} $$
in which consecutive maps compose trivially, and so there are induced choices of maps
$$ \Sigma^{m+1} A_0 \to A_{m+3} .$$

The collection of all such maps, taking all compatible variations, is the **Toda Bracket** of the bracket sequence.

Among the bracket sequences, a particular family arises which here will be called _null-bracket_ (again, a novel phrase).  A sequence will be called null-bracket if

* $n=2$ and $A_0 \to A_2$ is trivial, OR
* $n \gt 2$, and there is a choice of factorization $ A_1 \to C \to A_2 $ such that the sequence $ C \to A_2 \to \cdots \to A_n $ is null-bracket.

If the Toda bracket for a bracket sequence includes the trivial map $\Sigma^{m+1} A_0 \to A_{m+3}$ then the sequence is null-bracket.

## Applications

By definition, if a sequence is a bracket sequence AND NOT a null-bracket sequence, it follows that all the relevant maps $\Sigma^{k} A_0 \to A_n$ are nontrivial.  Things like these Toda brackets have been studied by many _(FIXME: references later)_ and especially the length-three brackets used by H. Toda to describe most of $\pi_k \mathbb{S}^n$ for $k \lt 31$ or so.

In ([Cohen, 1968](#Cohen68)) is given a criterion for stable maps of spheres to inhabit non-null Toda brackets; this turns out to be most of the [[stable homotopy groups of spheres]] $\pi_* \mathbb{S}$, and furthermore the maps in the bracket sequences can be chosen from a very small set (_FIXME_: be more precise! degree maps $n \iota$, [[Hopf map]]s $\eta, \theta,\sigma$, and $\alpha_p$... )

## Related concepts

* [[Massey product]]

* [[e-invariant]]

## References

The concept of Toda brackets is due to:

* {#Toda62} [[Hirosi Toda]], _Composition Methods in Homotopy Groups of Spheres_, Annals of Mathematics Studies Volume 49, Princeton University Press (1962) ([jstor:j.ctt1bgzb5t](https://www.jstor.org/stable/j.ctt1bgzb5t))

and the concept of higher Toda brackets appears around

* {#Cohen68} Joel Cohen, Section 2 of: _The decomposition of stable homotopy_, Annals of Mathematics (2) 87 (2): 305–320 (1968) ([doi:10.2307/1970586](https://doi.org/10.2307/1970586))

{#ReferencesTodaBracketInHomotopy2Category} The conceptualization of Toda brackets a [[homotopy coherent diagram|homotopy coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]] (regarded as a [[strict (2,1)-category]]) is made explicit in:

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

* {#HardieMarcumOda01} [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

* [[Howard Marcum]], [[Nobuyuki Oda]], _Long Box Bracket Operations in Homotopy Theory_, Appl Categor Struct 19, 137–173 ([doi:10.1007/s10485-009-9186-3](https://doi.org/10.1007/s10485-009-9186-3))

Application of Toda brackets to the concrete computations of [[stable homotopy groups of spheres]]:

* [Toda 62](#Toda62)

* [Hardie-Marcum-Oda 01, Sec. 7](#HardieMarcumOda01)

* {#IaksenWangXu20} [[Daniel Isaksen]], [[Guozhen Wang]], [[Zhouli Xu]], _More stable stems_ ([arXiv:2001.04511](https://arxiv.org/abs/2001.04511))

See also:

* [[Hans-Joachim Baues]], _On the cohomology of categories, universal Toda brackets and homotopy pairs_, K-Theory __11__:3, April 1997, pp. 259-285 (27) ([doi:10.1023/A:1007796409912](http://dx.doi.org/10.1023/A:1007796409912))

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

Discussion for [[ring spectra]]:

* {#Kochmann96} [[Stanley Kochmann]], section 5.7 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* Boryana Dimitrova, _Universal Toda brackets of commutative ring spectra_, poster, Bonn 2010 ([pdf](http://www.math.uni-bonn.de/people/grk1150/YWT2010/YWT_Dimitrova.pdf))

* [[Constanze Roitzheim]], [[Sarah Whitehouse]], _Uniqueness of $A_\infty$-structures and Hochschild cohomology_ ([arxiv/0909.3222](http://arxiv.org/abs/0909.3222))

* [[Steffen Sagave]], _Universal Toda brackets of ring spectra_, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008 ([math.KT/0611808](http://arxiv.org/abs/math/0611808))

[[!redirects Toda Brackets]]

[[!redirects Toda brackets]]


Toda lattice

## Overview

_Toda lattice_ is a particular completely [[integrable system]], a model of a 1-dimensional crystal. It can be generalized to general root systems.

## Relation to other models

Toda lattice can be obtained by reduction from Wess-Zumino models.

## Literature

* wikipedia: [Toda lattice](https://en.wikipedia.org/wiki/Toda_lattice)
* Morikazu Toda, _Vibration of a chain with a non-linear interaction", J. Phys. Soc. Jpn., 22 (2) (1967) 431-436 [doi](https://doi.org/10.1143/JPSJ.22.431)
* [[Bertram Kostant]], _The solution to a generalized Toda lattice and representation theory_, Adv. Math. __34__, 195--338 (1998) [pdf](https://core.ac.uk/download/pdf/82787789.pdf)
* L. O'Raifeartaigh, V. V. Sreedhar, _Path integral formulation of the conformal Wess-Zumino-Witten $\to$ Toda reductions_, Nuclear Physics B 529:3 (1998) 547-566 doi


toda-smith complex

##Informal definition##

Toda-Smith complexes are [[finite spectra]] characterized by having a particularly simple homology, and are used in stable homotopy theory.

Toda-Smith complexes provided examples of [[periodic map]]s. Thus, they led to the construction of the [[nilpotent and periodicity theorems]], which provided the first organization of the [[stable homotopy groups of spheres]] (localized at a prime) into families of maps via [[chromatic homotopy theory]].

## Mathematical Context ##
The story begins with the degree p map on $S^1$  (as a circle in the complex plane):

$$S^1 \to S^1$$
$$z \mapsto z^p$$

The degree p map is well defined for $S^k$ in general, where $k \in math{N}$.
If we apply the infinite suspension functor to this map, $\Sigma^{\infty}S^1 \to \Sigma^{\infty}S^1 =: \mathbb{S}^1 \to \mathbb{S}^1$ and we take the cofiber of the resulting map:

$$S \xrightarrow{p} S \to S/p$$

We find that $S/p$ has the remarkable property of coming from a Moore space (i.e., a designer (co)homology space: $H^n(X) \simeq Z/p$, and $\tilde{H}^*(X)$ is trivial for all $* \neq n$).

## Formal definition ##
The $n$th Toda-Smith complex, $V(n)$ where $n \in \{-1, 0, 1, 2, 3, ... \}$, is a finite spectrum which satisfies the property that its BP-homology, $BP_*(V(n)) \text{:=} [\mathbb{S}^0, BP \wedge V(n)]$, is isomorphic to $BP_*/(p, ..., v_n)$.

That is, Toda-Smith complexes are completely characterized by their $BP$-local properties, and are defined as any object $V(n)$ satisfying one of the following equations:

* $BP_*(V(-1)) \simeq BP_*$
* $BP_*(V(0)) \simeq BP_*/p$
* $BP_*(V(1)) \simeq BP_*/(p, v_1)$
*  etc.

## Examples of Toda-Smith complexes ##

* the sphere spectrum, $BP_*(S^0) \simeq BP_*$, which is $V(-1)$.
* the mod p Moore spectrum, $BP_*(S/p) \simeq BP_*/p$, which is $V(0)$

## Relevance to stable homotopy theory ##

It may help the reader to recall that that $BP_* = \mathbb{Z}_p[v_1, v_2, ...]$, $|v_i| = 2(p^i-1)$.

The periodic maps, $\alpha_t, \beta_t,$ and $\gamma_t$, come from degree maps between the Toda-Smith complexes, $V(0)_k, V(1)_k,$ and $V(2)_k$ respectively.

(... add in their relation to the $v_2$ periodic elements, and the reasoning behind the shift $2(p^i-1)$)


Todd class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[characteristic class]].

By the [[Hirzebruch-Riemann-Roch theorem]] the [[index]] of the [[Dolbeault operator]] is the Todd genus (e.g. [Gilkey 95, section 5.2](#Gilkey95) (more generally so for the [[Spin^c Dirac operator]]).

## Definition

The [characteristic series](genus#LogarithmAndCharacteristicSeries) of the Todd genus is

$$
  x \,\mapsto\, \frac{x}{ 1 - e^{-x} }
  \,.
$$

This means that as a [[formal power series]] in [[Chern classes]] $c_i$ the Todd class starts out as

$$
  td
    \;=\;
  1
    +
  \tfrac{1}{2} c_1
    +
  \tfrac{1}{12}
  \big(
    c_1^2 + c_2
  \big)
    +
  \tfrac{1}{24}
  c_1 c_2
    +
  \mathcal{O}(deg \geq 8)
  \,.
$$

## Properties

### Relation between Todd class and $\hat A$-genus
  {#RelationBetweenToddClassAndAHatGenus}

On an [[almost complex manifold]] $M_{\mathrm{U}}$, the [[Todd class]] coincides with the [[A-hat genus|A-hat class]] up to the [[exponential]] of half the [[first Chern class]]:

$$
  Td(M_{\mathrm{U}})
   \;=\;
  \big(e^{c_1/2} \hat A\big)(M_{\mathrm{U}})
  \,.
$$

(e.g. [Freed 87 (1.1.14)](#Freed87)).

In particular, on manifolds $M_{S\mathrm{U}}$ with [[special unitary group|SU]]-[[G-structure|structure]], where $c_1 = 0$, the Todd class is actually equal to the [[A-hat genus|A-hat class]]:

$$
  Td(M_{S\mathrm{U}})
   \;=\;
  \hat A(M_{S\mathrm{U}})
  \,.
$$

### Relation to Thom class and Chern character

+-- {: .num_prop #RationalToddClassIsChernCharacterOfThomClass}
###### Proposition
**([[rational Todd class is Chern character of Thom class]])**

Let $V \to X$ be a [[complex vector bundle]] over a [[compact topological space]]. Then the [[Todd class]] $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in [[rational cohomology]] equals the [[Chern character]] $ch$ of the [[Thom class]] $th(V) \,\in\, K\big( Th(V) \big)$ in the [[complex topological K-theory]] of the [[Thom space]] $Th(V)$, when both are compared via the [[Thom isomorphisms]] $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$$
  \phi_{H\mathbb{Q}}
  \big(
    Td(V)
  \big)
  \;=\;
  ch\big( th(V) \big)
  \,.
$$

More generally , for $x \in K(X)$ any class, we have

$$
  \phi_{H\mathbb{Q}}
  \big(
    ch(x)
    \cup
    Td(V)
  \big)
  \;=\;
  ch\big( \phi_{K}(x) \big)
  \,,
$$

which specializes to the previous statement for $x = 1$.

=--

([Karoubi 78, Chapter V, Theorem 4.4](#Karoubi78))

### Relation to the Adams e-invariant
 {#RelationToAdamsEInvariant}

We discuss how the [[e-invariant]] in its [[Q/Z]]-incarnation ([this Def.](e-invariant#eInvariantInComplexTopologicalKtheoryAsRationalNumberModuloIntegers)) has a natural formulation in [[cobordism theory]] ([Conner-Floyd 66](#ConnerFloyd66)), by evaluating Todd classes on [[coboundary|cobounding]] [[(U,fr)-manifolds]].

This is Prop. \ref{EInvariantIsToddClassOnCoboundingUFrManifold} below; but first to recall some background:

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

In generalization to how the [[U-bordism ring]] $\Omega^U_{2k}$ is [[Brown representability theorem|represented]] by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MU]], so the [[(U,fr)-bordism ring]] $\Omega^{U,fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):

\[
  \label{InTermsOfHomotopyGroupsOfQuotientedThomSpace}
  \Omega^{(U,fr)}_\bullet
  \;=\;
  \pi_{\bullet + 2k}
  \big(
    MU_{2k}/S^{2k}
  \big)
  \,,
  \;\;\;\;\;
  \text{for any}
  \;
  2k \geq \bullet + 2
  \,.
\]

=--

([Conner-Floyd 66, p. 97](#ConnerFloyd66))

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

The [[bordism rings]] for [[MU]], [[MUFr]] and [[MFr]] sit in a [[short exact sequence]] of the form

\[
  \label{ShortExactSequenceOfUFrBordismRings}
  0
  \to
  \Omega^U_{\bullet+1}
  \overset{i}{\longrightarrow}
  \Omega^{U,f}_{\bullet+1}
  \overset{\partial}{
    \longrightarrow
  }
  \Omega^{fr}_\bullet
  \to
  0
  \,,
\]

where $i$ is the evident inclusion, while $\partial$ is restriction to the [[boundary]].

In particular, this means that $\partial$ is [[surjective function|surjective]], hence that every $Fr$-manifold is the boundary of a [[(U,fr)-manifold]].

=--

+-- {: .num_prop #EInvariantIsToddClassOnCoboundingUFrManifold}
###### Proposition
**([[e-invariant is Todd class of cobounding (U,fr)-manifold]])**

Evaluation of the [[Todd class]] on [[(U,fr)-manifolds]] yields [[rational numbers]] which are [[integers]] on actual $U$-manifolds. It follows with the [[short exact sequence]] (eq:ShortExactSequenceOfUFrBordismRings) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-manifolds yields a well-defined element in [[Q/Z]].

Under the [[Pontrjagin-Thom isomorphism]] between the [[framed bordism ring]] and the [[stable homotopy group of spheres]] $\pi^s_\bullet$, this assignment coincides with the [[Adams e-invariant]] in [its Q/Z-incarnation](Adams+e-invariant#TheEInvariantAsAnElementOfQModZ):

\[
  \label{ToddClassesOnShortExactSequenceOfUFrBordismRings}
  \array{
  0
  \to
  &
  \Omega^U_{\bullet+1}
  &
  \overset{i}{\longrightarrow}
  &
  \Omega^{U,f}_{\bullet+1}
  &
  \overset{\partial}{
    \longrightarrow
  }
  &
  \Omega^{fr}_\bullet
  &
  \simeq
  &
  \pi^s_\bullet
  \\
  &
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{\mathrlap{Td}}
  &&
  \big\downarrow{}^{}
  &&
  \big\downarrow{}^{e}
  \\
  0
  \to
  &
  \mathbb{Z}
  &\overset{\;\;\;\;\;}{\hookrightarrow}&
  \mathbb{Q}
  &\overset{\;\;\;\;}{\longrightarrow}&
  \mathbb{Q}/\mathbb{Z}
  &=&
  \mathbb{Q}/\mathbb{Z}
  }
  \,,
\]

=--

([Conner-Floyd 66, Theorem 16.2](#ConnerFloyd66))

## Related concepts

* [[genus]]

* [[Grothendieck-Riemann-Roch theorem]]

[[!include genera and partition functions - table]]

## References

Named after [[John Arthur Todd]].

Original articles:

* {#Hirzebruch56} [[Friedrich Hirzebruch]], Section 3 of: _Neue topologische Methoden in der Algebraischen Geometrie_, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 ([doi:10.1007/978-3-662-41083-7](https://www.springer.com/de/book/9783662406052))

  English translation: Section 3 of _Topological Methods in Algebraic Topology_, Classics in Mathematics, vol 131. Springer 1995 ([doi:10.1007/978-3-642-62018-8_4](https://doi.org/10.1007/978-3-642-62018-8_4))

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Sections 12, 13 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#Karoubi78} [[Max Karoubi]], Chapter V.4 of: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

Review:

* {#Freed87} [[Daniel Freed]], Sections 1.1, 1.2 of: _Geometry of Dirac operators_, 1987 ([pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]])

On the Todd character:

* [[Pierre Conner]], [[Larry Smith]], Section 7 of: _On the complex bordism of finite complexes. II_, J. Differential Geom. Volume 6, Number 2 (1971), 135-174 ([euclid:euclid.ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

* [[Larry Smith]], Section 1 of: _The Todd character and the integrality theorem for the Chern character_, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 ([euclid:ijm/1256051760](https://projecteuclid.org/euclid.ijm/1256051760))

* [[Larry Smith]], _The Todd character and cohomology operations_, Advances in Mathematics Volume 11, Issue 1, August 1973, Pages 72-92 (doi:10.1016/0001-8708(73)90003-0)

Review:

* {#Gilkey95} [[Peter Gilkey]], Section 5.2 of: _Invariance Theory: The Heat Equation and the Atiyah-Singer Index Theorem_, 1995 ([pdf](https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf))

See also:

* Wikipedia, _[Todd class](http://en.wikipedia.org/wiki/Todd_class)_

Discussion of Todd classes for [[noncommutative topology]]/in [[KK-theory]] is in

* Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_, Commun. Math. Phys. 277:643-706,2008 ([arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020))

Review of the Todd genus with an eye towards generalization to the [[Witten genus]] is in the introduction of

* [[Pokman Cheung]], _The Witten genus and vertex algebras_ ([arXiv:0811.1418](http://arxiv.org/abs/0811.1418))

[[!redirects Todd classes]]

[[!redirects Todd genus]]

[[!redirects Todd character]]
[[!redirects Todd characters]]


Todd Trimble

I am a [[mathematics|mathematician]] specializing in [[category theory]]. My PhD was completed in 1994 under the direction of [[Myles Tierney]], and centered on categorical [[coherence relation|coherence]] problems and their relation to [[categorical logic]]. Subsequently I spent a few years in Sydney working with [[Ross Street]], mainly on [[higher-dimensional categories]], and then spent a few years in Chicago; the Chicago category seminar at the University of Chicago was founded during my stay.

In 2001 I left academe to become a full-time parent, continuing to do mathematics as time allows. Part of those activities include participating in the [$n$-Category Café](http://golem.ph.utexas.edu/category/) blog, and at one time coauthoring the mathematics blog [Topological Musings](http://topologicalmusings.wordpress.com). Since late 2013 I have also been a moderator at [MathOverflow](http://mathoverflow.net/questions).

Starting in January 2019 I hold a temporary appointment as Assistant Professor at Western Connecticut State University.

A few of my older scribblings can be found [here](http://math.ucr.edu/home/baez/trimble); some more recent ones are at my nLab web [here](https://ncatlab.org/toddtrimble/published/HomePage).

I can be reached at topological-dot-musings-at (insert gmail extension).

category: people

[[!redirects Todd Trimble]]


Toeplitz operator

[[!redirects Toeplitz operators]]

## Related concepts

* [[Fredholm operator]]

* [[odd Chern character]]

## References

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


Tohru Eguchi

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

* [webpage](http://www2.yukawa.kyoto-u.ac.jp/~tohru.eguchi/)

* [obituary](http://scgp.stonybrook.edu/archives/28517)

## Selected writings

Introducing [[gravitational instantons]]:

* {#EguchiFreund67} [[Tohru Eguchi]], [[Peter Freund]], _Quantum Gravity and World Topology_, Phys. Rev. Lett. 37, 1251 (1967) ([doi:10.1103/PhysRevLett.37.1251](https://doi.org/10.1103/PhysRevLett.37.1251))

* [[Tohru Eguchi]], [[Andrew Hanson]], _Gravitational instantons_, General Relativity and Gravitation volume 11, pages 315–320 (1979) ([doi:10.1007/BF00759271](https://doi.org/10.1007/BF00759271))

On [[topology]]/[[homotopy theory]] and [[differential geometry]] in [[gauge theory]] and [[gravity]] (such as in [[fiber bundles in physics]]):

* [[Tohru Eguchi]], [[Peter Gilkey]], [[Andrew Hanson]], _Gravitation, gauge theories and differential geometry_, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (doi:10.1016/0370-1573(80)90130-1)

On the [[large N limit]] and a precursor to [[holographic QCD]] and [[matrix models]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

On [[gravitation]], [[gauge theory]] and [[differential geometry]]:

* [[Tohru Eguchi]], Peter Gilkey, Andrew Hanson, _Gravitation, gauge theories and differential geometry_, Physics Reports __66__:6 (1980) 213—393 ([pdf](http://empg.maths.ed.ac.uk/Activities/GT/EGH.pdf))

On [[E-strings]]:

* [[Tohru Eguchi]], Kazuhiro Sakai, _Seiberg-Witten Curve for the E-String Theory_, JHEP 0205:058, 2002 ([arXiv:hep-th/0203025](https://arxiv.org/abs/hep-th/0203025))

category: people


tolerance relation

A binary relation on a set $S$ is a __tolerance relation__ if it is symmetric and reflexive.

This is a wide weakening of the concept of equality. For example, every equivalence relation is a tolerance relation. Equivalence relations are used to make equivalence classes, quotient sets and quotient groupoids, as well as associated algebras. Some modifications of operator algebras related to quotients may be studied even in generality of tolerance relations.

* [[Alain Connes]], Walter D. van Suijlekom, _Tolerance relations and operator systems_, [arXiv:2111.02903](https://arxiv.org/abs/2111.02903)

* F. D’Andrea, G. Landi, F. Lizzi, _Tolerance relations and quantization_, Lett Math Phys __112__, 65 (2022) [doi](https://doi.org/10.1007/s11005-022-01553-y)


Tom Apostol

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

## Selected writings

On [[analysis]]:

* [[Tom Apostol]], *Mathematical Analysis* (1973) [ISBN:0201002884, [pdf](http://webpages.iust.ac.ir/amtehrani/files/Addison%20Wesley%20-%20Mathematical%20Analysis%20_%20Apostol%20%285Th%20Ed%29%20%281981%29.pdf)]

category: people


Tom Avery

## Selected writings

On the [[codensity monad]] and the [[Giry monad]]:

* {#Avery14} [[Tom Avery]], _Codensity and the Giry monad_, Journal of Pure and Applied Algebra **220** 3 (2016) 1229-1251 [[arXiv:1410.4432](https://arxiv.org/abs/1410.4432), [doi:10.1016/j.jpaa.2015.08.017](https://doi.org/10.1016/j.jpaa.2015.08.017)]

On [[Isbell duality]] and [[reflexive completion]]:

* [[Tom Avery]], [[Tom Leinster]]. _Isbell conjugacy and the reflexive completion_. Theory and Applications of Categories, **36** 12 (2021) 306-347 [[tac:36-12](http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf)]

category: people


Tom Bachmann

* [webpage](http://tom-bachmann.com/)

## related $n$Lab entries

* [[motivic homotopy theory]]

* [[geometric fixed point spectrum]]

category: people