tangential notions of Frölicher spaces

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

# Tangential Topics of Frölicher Spaces
{#tangent}

+-- {: .standout}
This section is essentially lifted from [Comparative Smootheology](http://arxiv.org/abs/0802.2225) as I intend to remove it from that article and this seems a good place to put it.
=--

Nearly every construction in differential topology starts with [[tangent bundle|tangent]] or [[cotangent bundles]].
We shall look at how one could define these in the more general setting.
An advantage of the dual nature of the definition of a [[Frölicher space]] is that just about every definition of a tangent or cotangent vector is possible.
Essentially, as we have access to both [[curves]] and [[functionals]] we can consider [[push-forward]]s and [[pull-back]]s.

The ideas in this section can be traced at least as far back as Frölicher's work in [MR842916](http://www.ams.org/mathscinet-getitem?mr=842916).
That there are different orders of tangent and cotangent vectors appears in Kriegl and Michor's book [MR1471480](http://www.ams.org/mathscinet-getitem?mr=1471480), though it doubtless has antecedents.
(However in [MR1471480](http://www.ams.org/mathscinet-getitem?mr=1471480), only operational tangent vectors have higher orders.
This is because the context is that of [[manifolds]] and so non-trivial higher order kinematic tangent vectors do not appear.)

Let us start with _kinematic tangent vectors_.
We start with a notion of what it means for a curve to be flat.

+-- {: mynumdef #Flatness}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space.
 A curve $c \in \mathcal{C}$ is said to be _$k$-flat_ at $t \in \mathbb{R}$ if for each $f \in \mathcal{F}$, $(f c)^{(j)}(t) = 0$ for $1 \le j \le k$.
 It is _flat_ at $t \in \mathbb{R}$ if it is $k$-flat for all $k \in \mathbb{N}$.

 We write $\mathcal{C}_{x,k}$ for the subset of $\mathcal{C}$ consisting of those curves that map $0$ to $x$ and are $(k-1)$-flat at $0$.
=--

For convenience, in the next definition we will say that any curve is $0$-flat.

+-- {: mynumdef #FroTanSet}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space.
 Let $x \in X$ and let $k \in \mathbb{N}$.
 The _$k$th kinematic tangent set_ at $x$, $T_{x,k} X$, is the quotient of $\mathcal{C}_{x,k-1}$ by the equivalence relation $c_1 \simeq c_2$ if
  $(f c_1)^{(k)}(t) = (f c_2)^{(k)}(t)$
 for all $f \in \mathcal{F}$.
=--

We write this as $T_{x,k} X$ rather than $T_x^k X$ to avoid conflict of notation with iterated tangent spaces.
Since any curve can be composed with a suitable smooth map $\mathbb{R} \to \mathbb{R}$ to produce a curve of higher flatness, there are natural inclusions
 $T_{x,k} X \to T_{x,k+1} X$.
For Euclidean spaces with their usual structure, all of these tangent sets coincide.

+-- {: mynumdef #FroKinTan}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space.
 The _full kinematic tangent set_ of $X$ at $x$ is
  $T_{x,\infty} X = \bigcup_k T_{x,k} X$.
=--

It is obvious that each of these kinematic tangent sets is functorial in Frölicher spaces.

One would anticipate that in a given application the only two candidates for a kinematic tangent set will be either the full kinematic tangent set or the first kinematic tangent set.
Note that for a [[manifold with boundary]], the first kinematic tangent set at a point in the boundary is the tangent space of the boundary whereas the full kinematic tangent set is the tangent space of an ambient manifold (including the outward pointing [[normal vector|normal]]).

We have been careful in writing "tangent _set_" rather than "tangent _space_" so as not to imply any particular structure.
Reparametrisation of paths easily defines the structure of an $(\mathbb{R},+)$-[[G-set|set]] on each tangent set (scalar multiplication) but in general one will not be able to add tangent vectors.
Nonetheless, some addition may be possible and addition has nice properties when it is defined.

+-- {: mynumdef #TanSum}
###### Definition
Let $X$ be a Frölicher space, $x \in X$.
For $u,v, w \in T_{x,k} X$ we say that $w$ is a _sum_ of $u$ and $v$ if there are representatives $\alpha$, $\beta$, $\gamma$ such that $(f \alpha)^{(k)} = (f \beta)^{(k)} + (f \gamma)^{(k)}$ for all $f \in F_X$.

More generally, $u_1, \dots, u_k \in T_{x,k} X$ have a sum if there is some $w \in T_{x,k} X$ with $(f \beta)^{(k)} = \sum (f \alpha_i)^{(k)}$.
=--

+-- {: .num_lemma #LemTanSum}
###### Lemma
Sums are unique if they exist.
=--

+-- {: myproof}
###### Proof
If $\beta$ and $\gamma$ both represent sums of $u_1, \cdots, u_k$ then
\[
(f \beta)^{(k)} = \sum (f \alpha_i)^{(k)} = (f \gamma)^{(k)}
\]
so $\beta$ and $\gamma$ represent the same vector in $T_{x,k} X$.
=--

The construction of kinematic tangent vectors suggests a similar construction for cotangent vectors.
As with tangent vectors, we need an auxiliary definition of flatness.

+-- {: mynumdef #FlatFun}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$.
 A functional $f \in \mathcal{F}$ is said to be _$k$-flat_ at $x \in X$ if for each $c \in \mathcal{C}$ with $c(0) = x$, $(f c)^{(j)}(0) = 0$ for $1 \le j \le k$.
 It is _flat_ at $x \in X$ if it is $k$-flat for all $k \in \mathbb{N}$.

 We write $\mathcal{F}_{x,k}$ for the subset of $\mathcal{F}$ of functionals that are $k$-flat at $x$.
=--

Again, for convenience we will say that all functionals are $0$-flat.

+-- {: mynumdef #KinCoTan}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$.
 The _$k$th kinematic cotangent space_ at $x$, $T^*_{x,k} X$, is the quotient of $\mathcal{F}_{x,k-1}$ by the relation $f_1 \simeq f_2$ if
  $(f_1 c)^{(k)}(0) = (f_2 c)^{(k)}(0)$
 for all $c \in \mathcal{C}$ with $c(0) = x$.
=--

The same discussion for kinematic tangent vectors applies to kinematic cotangent vectors except for the fact that cotangent vectors automatically form a [[vector space]]; sums are always defined.

+-- {: mynumdef #KinFullCoTan}
###### Definition
 The _full kinematic cotangent space_ of $X$ at $x$ is
  $T^*_{x,\infty} X = \bigcup T^*_{x,k} X$.
=--

There are obvious pairings between kinematic tangent and cotangent vectors based on evaluation.
One defines a map
 $(f,c) \mapsto (f c)^{(k)}(0)$
and this descends to providing one knows that $(f c)^{(j)}(0)$ vanishes for $j \lt k$.
The simplest case of this is where one of the tangent or cotangent vectors has order $k$.
Thus the highest level pairing,
 $(f, c) \mapsto (f c)'(0)$,
defines a pairing for all tangent and cotangent vectors but one which vanishes unless both are of first order.

We can also define _operational tangent vectors_.
Recall that $\mathcal{F}$ is an algebra of functions.

+-- {: mynumdef #OpTanSet}
###### Definition
 Let $(X, \mathcal{C}, \mathcal{F})$ be a Frölicher space, $x \in X$.
 An _operational tangent vector_ at $x$ is a derivation at $x$ of $\mathcal{F}$.
 That is, a linear map
  $\partial : \mathcal{F} \to \mathbb{R}$
 such that
  $\partial(f g) = f(x) \partial g + g(x) \partial f$.

 We write $D_x X$ for the vector space of operational tangent vectors at $x$ and $D_{x,k} X$ for the space of operational tangent vectors at $x$ that vanish on $\mathcal{F}_{x,k}$.
=--

The order of an operational tangent vector is defined to be the least $k$ for which it lies in $D_{x,k} X$, otherwise it is said to have infinite order.
Again, in a given application it may be preferable to restrict to operational tangent vectors of finite order, or of order $1$.

+-- {: .num_lemma #KinToOp}
###### Lemma
 There are natural maps
  $T_{x, k} X \to D_{x,k} X$
 compatible with the connecting maps on each side.
=--

+-- {: myproof}
###### Proof
 For $c \in \mathcal{C}_{x,k-1}$ and $f \in \mathcal{F}$ we define an operational tangent vector by
  $f \mapsto (f c)^{(k-1)}(0)$.
 This has the required properties.
=--

This map, however, need not be injective and neither need it map to a spanning set.

One can define two more versions of cotangent vectors by taking linear duals of the two versions of tangent vectors (this has to be done with care for the kinematic tangent vectors); however, these definitions may be thought of as one step removed from the smooth structure itself.

[[!redirects tangential notions of Frölicher spaces]]
[[!redirects tangential notions of Frolicher spaces]]
[[!redirects tangential notions of Froelicher spaces]]


tangential structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[smooth manifold]] $X$, by a _tangential structure_ one typically understands (e.g. [GMWT 09, Sec. 5](GMWT09)) a [[lift]] of the [[classifying map]] $X \overset{\vdash}{\longrightarrow} B GL(n)$ of its [[tangent bundle]]  through any prescribed [[map]] $B \overset{f}{\longrightarrow} B GL(n)$ into the [[classifying space]] of the [[general linear group]], up to [[homotopy]]:

$$
  \array{
    && B
    \\
    &
    {}^{
      \mathllap{
        {tangential} \atop {structure}
      }
    }\nearrow
    & \big\downarrow^{ f }
    \\
    X
      &\underset{
        \vdash T X
      }{\longrightarrow}&
    B GL(n)
  }
$$

Since this is all considered (only) for [[homotopy types]] of topological spaces (e.g. via the [[classical model structure on topological spaces]]) and there is a [[weak homotopy equivalence]] $B GL(n) \simeq  B O(n)$ to the [[classifying space]] of the [[orthogonal group]] (the latter being the [[maximal compact subgroup]] of $GL(n)$), authors typically consider the equivalent diagram over $B O(n)$.

Beware that the same kind of lift but understood in [[differentiable stack|differentiable]] [[classifying stacks]] instead of just [[classifying spaces]] is a _[[G-structure]]_ as commonly understood now (for $A = \mathbf{B}G$, the classifying stack/[[delooping]] of a [[Lie group]] $G$).

## Details

### In terms of $(B,f)$-structures
 {#InTermsOfBfStructures}

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

A **$(B,f)$-structure** is

1. for each $n\in \mathbb{N}$ a [[pointed topological space|pointed]] [[CW-complex]] $B_n \in Top_{CW}^{\ast/}$

1. equipped with a pointed [[Serre fibration]]

   $$
     \array{
       B_n
       \\
       \downarrow^{\mathrlap{f_n}}
       \\
       B O(n)
     }
   $$

   to the [[classifying space]] $B O(n)$ ([def.](classifying+space#EOn));

1. for all $n_1 \leq n_2$ a pointed continuous function

   $\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

   which is the identity for $n_1 = n_2$;

such that for all $n_1 \leq n_2 \in \mathbb{N}$ these [[commuting square|squares commute]]

$$
  \array{
    B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2}
    \\
    {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}}
    \\
    B O(n_1) &\longrightarrow& B O(n_2)
  }
  \,,
$$

where the bottom map is the canonical one ([def.](classifying+space#InclusionOfBOnIntoBOnPlusOne)).

The $(B,f)$-structure is **multiplicative** if it is moreover equipped with a system of maps $\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}$ which cover the canonical multiplication maps ([def.](classifying+space#WhitneySumMapOnClassifyingSpaces))

$$
  \array{
    B_{n_1} \times B_{n_2}
      &\overset{\mu_{n_1, n_2}}{\longrightarrow}&
    B_{n_1 + n_2}
    \\
    {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{f_{n_1 + n_2}}}
    \\
    B O(n_1) \times B O(n_2)
     &\longrightarrow&
    B O(n_1 + n_2)
  }
$$

and which satisfy the evident [[associativity]] and [[unitality]], for $B_0 = \ast$ the unit, and, finally, which commute with the maps $\iota$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

$$
  \array{
    B_{n_1} \times B_{n_2}
      &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}&
    B_{n_1} \times B_{n_2 + n_3}
    \\
    {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}}
    \\
    B_{n_1 + n_2}
      &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}&
    B_{n_1 + n_2 + n_3}
  }
$$

and

$$
  \array{
    B_{n_1} \times B_{n_2}
      &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}&
    B_{n_1+n_3} \times B_{n_2 }
    \\
    {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}}
    \\
    B_{n_1 + n_2}
      &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}&
    B_{n_1 + n_2 + n_3}
  }
  \,.
$$

Similarly, an **$S^2$-$(B,f)$-structure** is a compatible system

$$
  f_{2n} \colon B_{2n} \longrightarrow B O(2n)
$$

indexed only on the even natural numbers.

Generally, an **$S^k$-$(B,f)$-structure** for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system

$$
  f_{k n} \colon B_{ k n} \longrightarrow B O(k n)
$$

for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.

=--

([Lashof 63](#Lashof63), [Stong 68, beginning of chapter II](#Stong68), [Kochman 96, section 1.4](#Kochman96))

See also at _[[B-bordism]]_.

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

Examples of $(B,f)$-structures (def. \ref{BfStructure}) include the following:

1. $B_n = B O(n)$ and $f_n = id$ is **orthogonal structure** (or "no structure");

1. $B_n = E O(n)$ and $f_n$ the [[universal principal bundle]]-projection is **[[framing]]-structure**;

1. $B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the [[special orthogonal group]] and $f_n$ the canonical projection is **[[orientation]] structure**;

1. $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the [[spin group]] and $f_n$ the canonical projection is **[[spin structure]]**.

Examples of $S^2$-$(B,f)$-structures include

1. $B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the [[unitary group]], and $f_{2n}$ the canonical projection is **[[almost complex structure]]**.

=--

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

Given a [[smooth manifold]] $X$ of [[dimension]] $n$, and given a $(B,f)$-structure as in def. \ref{BfStructure}, then a **$(B,f)$-structure on the manifold** is an [[equivalence class]] of the following structure:

1. an [[embedding]] $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;

1. a [[homotopy class]] of a  [[lift]] $\hat g$ of the classifying map $g$ of the [[tangent bundle]]

   $$
     \array{
       && B_{n}
       \\
       &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_n}}
       \\
       X &\overset{g}{\longrightarrow}& B O(n)
     }
     \,.
   $$

The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if

1. $k_2 \geq k_1$

1. the second inclusion factors through the first as

   $$
     (i_X)_2
       \;\colon\;
     X
       \overset{(i_X)_1}{\hookrightarrow}
     \mathbb{R}^{k_1}
       \hookrightarrow
     \mathbb{R}^{k_2}
   $$

1. the lift of the classifying map factors accordingly (as homotopy classes)

   $$
     \hat g_2
       \;\colon\;
     X
       \overset{\hat g_1}{\longrightarrow}
     B_{n}
       \longrightarrow
     B_{n}
     \,.
   $$

=--

## Examples

The tangential structures corresponding to lifts through the [Whitehead tower of the orthogonal group](Whitehead+tower#OfTheOrthogonalGroup)

* [[orientation]]

* [[spin structure]]

* [[string structure]]

* [[fivebrane structure]]

* ...

* [[framing]]

## References

The concept of tangential structure originates with [[cobordism theory]], originally under the name _$(B,f)$-structures_:

* {#Lashof63} [[Richard Lashof]], _Poincaré duality and cobordism_, Trans. AMS 109 (1963), 257-277 ([doi:10.1090/S0002-9947-1963-0156357-4](https://doi.org/10.1090/S0002-9947-1963-0156357-4))

* {#Stong68} [[Robert Stong]], beginning of chapter II of: _Notes on Cobordism theory_, 1968,

  reprinted as: Princeton Legacy Library, Princeton University Press 2016 ([ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf))

* {#Kochman96} [[Stanley Kochman]], section 1.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

The terminology "tangential structure" became popular around

* {#GMWT09} [[Søren Galatius]], [[Ib Madsen]], [[Ulrike Tillmann]],[[Michael Weiss]], Section 5 of: _The homotopy type of the cobordism category_, Acta Math. 202 (2009), no. 2, 195--239 ([arXiv:math/0605249](http://arxiv.org/abs/math/0605249))

[[!redirects tangential structures]]

[[!redirects (B,f)-structure]]
[[!redirects (B,f)-structures]]


tangle

[[!redirects tangles]]

#Contents#
* table of contents
{:toc}

## Idea

An *ordinary tangle* is 1-manifold with boundary that is embedded in the 2-cube (or 3-cube), such that the tangle's boundary is embedded in two chosen opposing sides of the cube. Thus, one may think of a tangle as a [[knot]] that was cut at several points and the resulting strands were pulled apart at their endpoints to opposite sides of the cube.

The notion generalizes to that of *$m$-tangles in dimension $n$*, which are $m$-manifolds with corners embedded in the $n$-cube that, such that their corners are appropriately embedded in the $n$-cube's boundary.

## Definition of ordinary tangles

Tangles form a [[category]]. Its objects are finite [[subsets]] of $\mathbf{R}^2$. Morphisms $A\to B$ are embeddings of unions of finitely many closed intervals and circles into $[0,1]\times\mathbf{R}^2$ such that the restriction of the embedding to the endpoints yields a bijection to $A\sqcup B$. Morphisms are composed by gluing two copies of $[0,1]$ together and rescaling.

As usual, this suffers from being associative only up to an ambient isotopy. Thus, one can either take ambient isotopy classes of such embeddings, obtaining a [[1-category]] of tangles, or instead turn tangles into an [[(∞,1)-category]],
in which case morphisms $A\to B$ will encode the whole [[homotopy type]] of the space of embeddings described above.

## Higher-dimensional variants

Higher-dimensional tangles, i.e. $m$-manifold with corners embedded in the $n$-cube, were considered for instance in [Baez and Dolan 95](#BaezDolan95). A "tame" definition of tangles that admit *finite* [[stratified space|stratifications]] by their critical point types was given in [Dorn and Douglas 22](#DornDouglas22).

## Related concepts

* [[generalized tangle hypothesis]]
* [[manifold diagram|manifold diagrams]]

## References

* {#BaezDolan95} [[John Baez]] and [[James Dolan]], _Higher-dimensional Algebra and Topological Quantum Field Theory_ 1995 ([arXiv](http://arxiv.org/abs/q-alg/9503002))

* {#DornDouglas22} Christoph Dorn and Christopher Douglas, _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758))


tangle hypothesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

The _tangle hypothesis_ ([Baez and Dolan 95](#BaezDolan95)) is as follows:
+-- {: .un_prop}
###### Tangle Hypothesis

The $n$-category of framed $n$-[[tangles]] in $n+k$ dimensions is $(n+k)$-equivalent to the free [[k-tuply monoidal n-category]] with duals on one object.
=--

The tangle hypothesis may be seen as a refinement of the [[cobordism hypothesis]] in the sense that the latter arises from the former in the limit $k \to \infty$:
+-- {: .un_prop}
###### [[cobordism hypothesis|Cobordism Hypothesis]]

The $n$-category $n Cob$ of cobordisms is the free stable $n$-category with duals on one object (the point).
=--

## Generalized tangle hypothesis

### Tangles with structure

The tangle hypothesis has been generalized to allow certain structures on the tangles.

The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n + k)$-cube is the [[fundamental category|fundamental]] $(n + k)$-category with duals of $(M G,Z)$.

* $M G$ is the [[Thom space]] of group $G$.
* $G$ can be any group equipped with a homomorphism to $O(k)$. ([comment](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006471))

### Tangles with singularities

The tangle hypothesis can further be generalized to allow for 'tangles with singularities' (also called [[stratified space|stratified]] tangles). The idea is analogous to that of [[cobordism hypothesis#ForCobordismsWithSingularities|cobordisms with singularities]]. The tangle hypothesis for tangles with singularities includes the case of $G$-structured tangles (which requires choosing generators for $M G$, that then yield a datum of singularity types, see ([Lurie 09, Sec. 4.3](#LurieCTFT))).

The generalized tangle hypothesis is closely related to the generalized [[Pontrjagin-Thom collapse map#ReferencesPontrjaginThomConstruction|Thom-Pontryagin construction]] which relates homotopy classes of maps from manifolds $M$ into CW complexes $X$ to cobordism classes of $X$-stratifications of $M$. (The relation was discussed on the $n$Café [here](https://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006196) and [here](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html).)

### Directed variants

[[manifold diagram|Manifold diagrams]] are stratified tangles 'without critical points'. The relation of stratified tangles to presented higher categories with duals is analogous to the relation of manifold diagrams to presented higher categories (without the need for duals). This yields a "directed" version of the generalized tangle hypothesis.

## The tangle hypothesis as a consequence of the cobordism hypothesis

While the tangle hypothesis and its generalizations may be seen as refinements of the cobordism hypothesis and its generalizations, Lurie shows ([Lurie 09, Sec. 4.4](#LurieCTFT)) that the former may be deduced from the latter when expressed in a sufficiently general form (namely, for cobordisms with singularities).

## Related concepts

* [[cobordism hypothesis]]

## References

* {#BaezDolan95} [[John Baez]] and [[James Dolan]], _Higher-dimensional Algebra and Topological Quantum Field Theory_ 1995 ([arXiv](http://arxiv.org/abs/q-alg/9503002))

* {#LurieCTFT}[[Jacob Lurie]], [[On the Classification of Topological Field Theories]]  ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

* {#LurieTQFT}[[Jacob Lurie]], _TQFT and the cobordism hypothesis_, videos of 4 lectures at the [Geometry Research Group](http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html), Mathematics Department, University of Texas Austin. Lecture notes for Lurie's talks are available at the [Geometry Research Group website](http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html).

For a discussion of the generalized tangle hypothesis see

* [n-Category Café] (http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006381), This Week’s Finds in Mathematical Physics (Week 241)
* [n-Category Café](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html), Invitation to geometric higher categories

[[!redirects generalized tangle hypothesis]]

[[!redirects Tangle Hypothesis]]


Taniyama-Shimura conjecture

< [[modularity theorem]]


Tanmoy Das

* [Research Group page](http://www.tanmoydas.com/)

## Selected writings

On [[topological insulators]]:

* [[Tanmoy Das]], *A pedagogic review on designing model topological insulators*, Journal of the Indian Institute of Science **96** 77-106 (2016) $[$[arXiv:1604.07546](https://arxiv.org/abs/1604.07546), [ISSN:0970-4140](http://journal.library.iisc.ernet.in/index.php/iisc/article/view/4606)$]$

category: people


Tannaka duality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Tannaka]] duality or _Tannaka [[reconstruction theorem]]s_ are statements of the form:

if $A$ is a symmetry object (e.g. a [[locally compact topological group]], [[Hopf algebra]]), [[representation|represented]] on objects in a [[category]] $D$, one may _reconstruct_ $A$ from knowledge of the [[endomorphism]]s of the forgetful functor -- the **[[fiber functor]]** --

$$
  F : Rep_D(A) \to D
$$

from the [[category]] $Rep_D(A)$ of [[representation]]s of $A$ on [[object]]s of $D$ that remembers these underlying objects. In a generalization, called mixed Tannakian formalism, not a single [[fiber functor]], but a family of [[fiber functors]] over different bases is needed for a reconstruction.

There is a general-abstract and a concrete aspect to this. The general abstract one says that an algebra $A$ is reconstructible from the [[fiber functor]] on the category of _all_ its modules. The concrete one says that in nice cases it is reconstructible from the category of _dualizable_ (finite dimensional) modules, even if it is itself not finite dimensional.

More precisely, let $V$ be any [[enriched category theory|enriching category]] (a [[locally small category|locally small]] [[closed monoidal category|closed]] [[symmetric monoidal category]] with all [[limit]]s). Then

1. for

   * $A$ a [[monoid]] in $V$;

   * $A Mod$ the $V$-[[enriched category]] of _all_ $A$-[[module]]s in $V$;

   * $F : A Mod \to V$ the [[forgetful functor|forgetful]] _[[fiber functor]]_ ;

   $A$ can be reconstructed as
   the object of [[enriched functor category|enriched endomorphisms]] of $F$,
   given by the [[end]]

   $$
     A
     \simeq
     End(F) := \int_{N \in A Mod} V(F(N), F(N))
     \,.
   $$

   This is just the [[enriched Yoneda lemma]] in a slight disguise.

1. In good cases, this [[end]] is computed already by restriction to the
   [[full subcategory]] $A Mod_{dual}$ of [[dual object|dualizable modules]]

   $$
     \cdots \simeq \int_{N \in A Mod_{dual}} V(F(N), F(N))
     \,.
   $$

## Statement

So far the following examples concern the abstract algebraic aspect of Tannaka duality only, which is narrated here as a consequence of the [[enriched Yoneda lemma]] in [[enriched category theory]]. Some of the Tannaka duality theorems involve
subtle harmonic analysis.

### For $G$-Sets
  {#ForPermutationRepresentations}

A simple case of Tannaka duality is that of [[G-sets]] of a [[group]], i.e. representations on a [[set]]. In this case, Tannaka duality follows entirely from repeated application of the ordinary [[Yoneda lemma]].

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for [[G-sets]])**

Let $G$ be a [[group]], write  $G Set$ for the [[category]] of [[G-sets]] and

$$
  F \colon G Set \longrightarrow Set
$$

for the [[forgetful functor]] that sends a [[G-set]] to its underlying [[set]].

Then there is a canonical [[group homomorphism|group-]][[isomorphism]]

$$
  Aut(F) \;\simeq\; G
  \,.
$$

identifying the [[automorphism group]] of $F$ (the group of [[natural isomorphisms]] from $F$ to itself) with $G$.

=--

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

With a bit of evident abuse of notation, the proof is a one-line sequence of applications of the [[Yoneda lemma]]: we show $End(F) \cong G$, i.e., each endomorphism on $F$ is invertible, so $End(F) = Aut(F) \cong G$.

Write $C \coloneqq Set^G = G Set$.
Observe that the functor $F \colon C \to Set$ is the [[representable functor|representable]] $F = C(G, -)$. Then the argument is

$$
  End(F)
  =
  Set^C(F, F)
    \cong
  Set^C(C(G, -), C(G, -))
    \cong
  C(G, G)
   \cong
  G.
$$

The "$G$" here is used in multiple senses, but each sense is deducible from context.

=--

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

We repeat the same proof, but with more notational details on what the entities involved in each step are precisely.

Let $\mathbf{B}G$ be the [[delooping]] [[groupoid]] of the [[group]] $G$. Then

$$
  G Set \;=\; Func(\mathbf{B}G^{op}, Set)
  \,.
$$

The canonical inclusion $i : {*} \to \mathbf{B}G$ induces the [[fiber functor]]

$$
  Func(i,Set) : G Set \to Set
$$

which evaluates a functor $\rho : \mathbf{B}G^{op} \to Set$ on the unique object of $\mathbf{B}G$. By the [[Yoneda lemma]] this is the same as homming out of the functor [[representable functor|represented by]] that unique object

$$
  Func(i,Set) = Hom_{PSh(\mathbf{B}G)}(Y_{\mathbf{B}G} {*}, -)
  \,,
$$

where $Y_{\mathbf{B}G} : \mathbf{B}G \to PSh(\mathbf{B}G)$ is the [[Yoneda embedding]].

But this way we see that $Func(i,Set) : PSh(\mathbf{B}G) \to Set$  is itself a representable functor in the presheaf category $PSh(PSh(\mathbf{B}G)^{op})$

$$
  Func(i,Set) = Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} *
  \,.
$$

So applying the [[Yoneda lemma]] twice, we find that

$$
  \begin{aligned}
    Aut_{PSh(PSh(\mathbf{B}G)^{op})} Func(i,Set)
    & =
    Aut_{PSh(PSh(\mathbf{B}G)^{op})} Y_{\mathbf{PSh(\mathbf{B}G)^{op}}} Y_{\mathbf{B}G} *
    \\
    & \simeq
    Aut_{PSh(\mathbf{B}G)^{op}} Y_{\mathbf{B}G} *
    \\
    & \simeq
    Aut_{\mathbf{B}G} *
    \\
    & \simeq G
    \,.
  \end{aligned}
$$

=--

Notice that the proof in no way used the fact that $G$ was assumed to be a [[group]], but only that $G$ is a [[monoid]].  So the statement holds just as well for arbitrary monoids.

But moreover, as the long-winded proof above makes manifest, even more abstractly the proof really only depended on the fact that the [[delooping]] $\mathbf{B}G$ is a [[small category]]. It need not have a single object for the proof to go through verbatim. Therefore we immediately obtain the following much more general statement of Tannaka duality for permutation representations of categories:

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for permutation representations of categories)**

Let $C$ be a [[locally small category]] and $Rep_{Set}(C) := Func(C,Set)$ the [[functor category]]. For every object $c \in C$ let $F_c : Rep_{Set}(C) \to Set$ be the fiber-functor that evaluates at $c$.

Then we have a [[natural isomorphism]]

$$
  Hom(F_c,F_{c'}) \simeq Hom_C(c,c')
  \,.
$$

=--

### For $V$-modules
  {#ForVModules}

Let $V$ be a ([[locally small category|locally small]]) [[closed monoidal category|closed]] [[symmetric monoidal category]], so that $V$ is enriched in itself via its [[internal hom]].

Observe that the setup, statement and proof of Tannaka duality for permutation representations given above is the special case for $V = $ [[Set]] of a statement verbatim the same in $V$-[[enriched category theory]], with the ordinary [[functor category]] replaced everywhere by the $V$-[[enriched functor category]]:

Then the $V$-enriched Tannaka duality theorem states that we can reconstruct a monoid $A$ as the monoid of endomorphisms of the forgetful $V$-functor from $A Mod$ to $V$:

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for $V$-modules over $V$-algebras)**

For $A$ a [[monoid]] in $V$ with [[delooping]] $V$-[[enriched category]] $\mathbf{B}A$, and with

$$
  A Mod := [\mathbf{B}A,V]
$$

the [[enriched functor category]] that encodes the $V$-[[module]]s of $A$, we have that the $V$-enriched [[endomorphism]] algebra $End(F) := [F,F]$ of the $V$-[[enriched functor]] $F : Rep(A) \to V$ is [[natural isomorphism|naturally isomorphic]] to $A$

$$
  End(A Mod \stackrel{F}{\to} V) \simeq A
  \,.
$$

=--

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

We can repeat the argument given above for permutation representations, this time employing the [[enriched Yoneda lemma]].

Indeed, we may identify $F$ with the $V$-functor
$$[\bullet]^*\colon\mathbf{Fun}_V(\mathbf{B}A,V)\to\underbrace{\mathbf{Fun}_V(\mathbf{pt}_V,V)}_{\cong V}$$
given by precomposition along the $V$-functor $[\bullet]\colon\mathbf{pt}_V\to\mathbf{B}A$ picking the unique object $\bullet$ of $\mathbf{B}A$.

Now we note that the functor $[\bullet]^*$ (given by sending an $A$-module $M\colon\mathbf{B}A\to V$) to its evaluation $M(\bullet)$ at the unique object $\bullet$ of $\mathbf{B}A$) is $V$-naturally isomorphic to the $V$-functor $\mathbf{Nat}_V(h_\bullet,-)$, since
$$\mathbf{Nat}_V(h_\bullet,M)\cong M(\bullet)$$
by the enriched Yoneda lemma. So in summary we have $F\cong[\bullet]^*\cong\mathbf{Nat}_V(h_\bullet,-)$.

We can then compute $End(F)$ as follows:

$$
  \begin{aligned}
    End(F)
    &\overset{\mathrm{def}}{=}
    \mathbf{Nat}_V(F,F)
    \\
    &\cong
    \mathbf{Nat}_V(\mathbf{Nat}_V(h_{\bullet},-),\mathbf{Nat}_V(h_{\bullet},-))
    \\
    &\cong
    \mathbf{Nat}_V(h_{\bullet},h_{\bullet})
    \\
    &\cong
    \mathbf{Hom}_{\mathbf{B}A}(\bullet,\bullet)
    \\
    &\overset{\mathrm{def}}{=}
    A,
  \end{aligned}
$$

where we have applied the [[enriched Yoneda lemma]] twice.

=--

Notice that the [[endomorphism]] object here is taken in the sense of enriched category theory, as described at [[enriched functor category]]. It is given by the [[end]] expression

$$
  End(F) = \int_{N \in A Mod} V(F(N), F(N))
  \,.
$$

The case of permutation representations is re-obtained by setting $V = $ [[Set]].

As before, the same proof actually shows the following more general statement

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for $V$-modules over $V$-algebroids)**

Let $C$ be a $V$-[[enriched category]] (a "$V$-[[algebroid]]"). Write $C Mod := [C,V]$ for the $V$-[[enriched functor category]]. For every [[object]] $c \in C$ write $F_c : C Mod \to V $ for the [[fiber functor]] that evaluates at $C$. Then we have [[natural isomorphism]]s

$$
  hom(F_c, F_{c'}) \simeq C(c,c')
  \,.
$$

=--

From this statement of Tannaka duality in $V$-enriched category theory
now various special cases of interest follow, by simply choosing suitable enrichment categories $V$.

#### For algebra modules
  {#ForAlgebraModules}

The general case of Tannaka duality for $V$-modules described [above](#ForVModules) restricts to the classical case of Tannaka duality for linear representations by setting $V :=$ [[Vect]], the category of [[vector space]]s over some fixed [[ground field]].

In this case the above says

+-- {: .num_cor}
###### Corollary
**(Tannaka duality for linear modules)**

For $A$ an [[algebra]] and $A Mod$ its category of [[modules]],
and for $F : A Mod \to Vect$ the [[fiber functor]] that sends a module to
its underlying vector space, we have a natural isomorphism

$$
  End( A Mod \to Vect ) \simeq A
$$

in [[Vect]].

=--

Additional structure on the algebra $A$ corresponds to addition structure on its [[category of modules]] as indicated in the following table:

[[!include structure on algebras and their module categories - table]]

#### For linear group representations

Still for the special case $V = Vect$, let now $G$ be a [[group]] and let the algebra in question specifically be its [[group algebra]] $A = k[G]$ . Then the category of linear [[representation]]s of $G$ is

$$
  Rep(G) \simeq k[G] Mod
$$

and we obtain

+-- {: .num_cor}
###### Corollary
**(Tannaka duality for linear group representations)**

There is a natural isomorphism

$$
  End(Rep(G) \to Vect) \simeq k[G]
  \,.
$$

=--

#### For coalgebra comodules {#Coalgebras}

If for $V$ we choose not [[Vect]] but its [[opposite category]] $Vect^{op}$, then a [[monoid]] object $A$ in $V$ is a [[coalgebra]] and $A Mod$ (or $A Mod^{op}$, rather) is the category of comodules over this coalgebra. Again we have a forgetful functor $F : A Mod \to Vect$

In

* [[André Joyal]], [[Ross Street]], _An introduction to Tannaka duality and quantum groups_, [pdf](http://www.math.mq.edu.au/~street/CT90Como.pdf)

([proposition 5, page 40](http://www.maths.mq.edu.au/~street/CT90Como.pdf#page=40))

and

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_

it is shown that $A$ is recovered as the [[coend]]

$$
  \int^{N \in A Mod_{fin}} F(N) \otimes F(N)^*
$$

in [[Vect]], where the coend ranges over finite dimensional modules.

If $A$ itself is finite dimensional then this is yet again just a special case of the enriched Yoneda lemma for $V$-modules, for the case $V = FinDimVect^{op}$ (the [[opposite]] of the [[category]] [[FinDimVect]] of [[finite-dimensional vector spaces]]): this general statement says that $A$ is recovered as the [[end]]

$$
  A = \int_{N \in A Mod_{fin}} V(F(N), F(N))
$$

in $Vect^{op}$. This is equivalently the [[coend]]

$$
  \cdots \simeq \int^{N \in A Mod}( Vect(F(N), F(N)))
$$

in $Vect$. Finally using that $FinDimVect(V,W) \simeq V\otimes W^*$ the above coend expression follows.

As before, more work is required to show that even for $A$ itself not finite dimensional, it is still recovered in terms of the above (co)end over just its finite dimensional modules.

### For Lie groupoids

See [[Tannaka duality for Lie groupoids]].

###  For geometric stacks

See [[Tannaka duality for geometric stacks]].

### In higher category theory
 {#InHigherCategoryTheory}

In as far as the proof of Tannaka duality only depends on the [[Yoneda lemma]], the statement immediately generalizes to [[higher category theory]] whenever a higher generalization of the Yoneda lemma is available.

This is notably the case for [[(∞,1)-category]] theory, where we have the [[(∞,1)-Yoneda lemma]].

#### For permutation $\infty$-representations
 {#ForInfinityPermutations}

By applying the $(\infty,1)$-Yoneda lemma verbatim four times in a row as above [for permutation representations](#ForPermutationRepresentations), we obtain the following statement for [[∞-permutation representations]].

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for $\infty$-permutation representations)**

Let $G$ be an [[∞-group]] and $Rep_{\infty Grpd}(G) := Func(\mathbf{B}G, \infty Grpd)$ the [[category of representations|category of]] [[∞-permutation representations]], the [[(∞,1)-category of (∞,1)-functors]] from its [[delooping]] [[∞-groupoid]] to [[∞Grpd]]. Let $F :  Rep_{\infty Grpd}(G) \to \infty Grpd$ be the fiber functor that remembers the underlying $\infty$-groupoid. Then there is an [[equivalence in a quasi-category]]

$$
  End(Rep_{\infty Grpd}(G) \to \infty Grp) \simeq G
  \,.
$$

=--

As before, this holds immediately even for representations of [[(∞,1)-categories]]

+-- {: .num_theorem}
###### Theorem
**(Tannaka duality for $\infty$-permutation representations)**

Let $c$ be an [[(∞,1)-category]] and $Rep_{\infty Grpd}(C) := Func(C,\infty Grpd)$. For $c \in C$ an [[object]], write $F_c : Rep_{\infty Grpd}(C) \to \infty Grpd$ for the corresponding fiber functor.

Then there is a natural equivalence

$$
  hom(F_c, F_{c'}) \simeq C(c,c')
$$

in [[∞Grpd]].

=--

#### $\infty$-Galois theory
 {#InfinityGaloisTheory}

As a special case of this, we obtain a statement about $\infty$-Galois theory. For details and background see [[homotopy groups in an (∞,1)-topos]]. In that context one finds for a [[locally contractible space]] $X$ that the [[∞-groupoid]] $LConst(X)$ of [[locally constant ∞-stack]]s on $X$  is equivalent to $Rep_{\infty Grpd}(\Pi(X))$, where $\Pi(X)$ is the [[fundamental ∞-groupoid]] of $X$.
For $x \in X$ a point, write $F_x : LConst(X) \to \infty Grpd$ for the corresponding fiber functor.

Then we have

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

For $x \in X$ there is a natural [[weak homotopy equivalence]]

$$
  End(LConst(X) \stackrel{F_x}{\to} \infty Grpd) \simeq
  \mathbf{B} Aut_{\Pi(X)}(x)
  \,.
$$

In particular we have [[natural isomorphisms]] of [[homotopy group]]s

$$
  \pi_n End(LConst(X) \stackrel{F_x}{\to} \infty Grpd)
  \simeq
  \pi_n(X,x)
  \,.
$$

=--

More on this is at [[cohesive (∞,1)-topos -- structures]] in the section Galois theory in a cohesive (∞,1)-topos

## Related concepts

* [[Tannakian category]]

* [[Deligne's theorem on tensor categories]]

## References

* [[André Joyal]], [[Ross Street]], _An introduction to Tannaka duality and quantum groups_, [pdf](http://www.math.mq.edu.au/~street/CT90Como.pdf)

* B.J. Day, _Enriched Tannaka reconstruction_, J. Pure Appl. Algebra __108__ (1996) 17-22, doi

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_

The following paper shortens Deligne's proof

* [[Alexander Rosenberg|Alexander L. Rosenberg]],  _The existence of fiber functors_,  The Gelfand Mathematical Seminars, 1996--1999,  145--154, Birkhäuser, Boston 2000.

Deligne's proof in turn fills the gap in the seminal work with the same title

* N. Saavedra Rivano, "Catégories Tannakiennes." _Bulletin de la Société Mathématique de France_ 100 (1972): 417-430. [EuDML](https://eudml.org/doc/87193)

A revival in algebraic geometry related to the theory of mixed motives was marked by

* [[P. Deligne]], [[J. Milne]], _Tannakian categories_, Springer Lecture Notes in Math. __900__, 1982, pp. 101-228, retyped [pdf](http://jmilne.org/math/xnotes/tc.pdf)

Analogous discussion for [[symmetric monoidal (infinity,1)-categories|symmetric monoidal]] [[stable (infinity,1)-categories]] includes

* {#Wallbridge12} [[James Wallbridge]], _Tannaka duality over ring spectra_ ([arXiv:1204.5787](https://arxiv.org/abs/1204.5787))

* {#Iwanari14} [[Isamu Iwanari]], _Tannaka duality and stable infinity-categories_ ([arXiv:1409.3321](http://arxiv.org/abs/1409.3321))

* [[Jacob Lurie]], [[Spectral Algebraic Geometry]], Chap. 9

Ulbrich made a major contribution at the coalgebra and Hopf algebra level

* K-H. Ulbrich, _On Hopf algebras and rigid monoidal categories_, in special volume, Hopf algebras, Israel J. Math. __72__ (1990), no. 1-2, 252--256, [doi](http://dx.doi.org/10.1007/BF02764622)

This Hopf-direction has been advanced by many authors including

* [[S. L. Woronowicz]], _Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups_, Inventiones Mathematicae __93__, No. 1, 35-76, [doi](http://dx.doi.org/10.1007/BF01393687)

* [[Shahn Majid]], _Foundations of quantum group theory_, chapter 9

* Phung Ho Hai, _Tannaka-Krein duality for Hopf algebroids_, Israel J. Math. __167__ (1):193–225 (2008) [math.QA/0206113](http://arxiv.org/abs/math/0206113)

* [[Volodymyr Lyubashenko|Volodymyr V. Lyubashenko]], _Modular transformations and tensor categories_, J. Pure Appl. Algebra __98__ (1995) 279–327 doi; _Squared Hopf algebras and reconstruction theorems_, Proc. Workshop "Quantum Groups and Quantum Spaces" (Warszawa), Banach Center Publ. __40__, Inst. Math. Polish Acad. Sci. (1997) 111–137, [q-alg/9605035](http://arxiv.org/abs/q-alg/9605035); _Squared Hopf algebras_, Mem. Amer. Math. Soc. __142__ (677):x 180, 1999; Алгебры Хопфа и вектор-симметрии, УМН, 41:5(251) (1986), 185–186, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=2247&what=fullt&option_lang=rus), transl. as: _Hopf algebras and vector symmetries_, Russian Math. Surveys 41(5):153154, 1986.

* A. Bruguières, _Théorie tannakienne non commutative_, Comm. Algebra __22__, 5817–5860, 1994

* K. Szlachanyi, _Fiber functors, monoidal sites and Tannaka duality for bialgebroids_, [arxiv/0907.1578](http://arxiv.org/abs/0907.1578)

* B. Day, R. Street, _Quantum categories, star autonomy, and quantum groupoids_, in
"Galois theory, Hopf algebras, and semiabelian categories", Fields Inst. Comm. __43__
(2004) 187-225

* [[Daniel Schäppi]], _The formal theory of Tannaka duality_, [arxiv/1112.5213](http://arxiv.org/abs/1112.5213), superseding earlier _Tannaka duality for comonoids in cosmoi_, [arXiv:0911.0977](http://arxiv.org/abs/0911.0977)

A generalization of several classical reconstruction theorems with nontrivial [[functional analysis]] is in

* [[Alexander Rosenberg|Alexander L. Rosenberg]],  _[[Reconstruction of Groups|Reconstruction of groups]]_, Selecta Math. (N.S.) __9__, 1 (2003), 101--118, [doi](http://dx.doi.org/10.1007/s00029-003-0322-x).

Categorically oriented notes were written also by Pareigis, emphasising on using [[coend|Coend]] in dual picture. His works can be found [here](http://www.mathematik.uni-muenchen.de/~pareigis/pa_schft.html) but the most important is the chapter 3 of his online book

* Bodo Pareigis, _Quantum groups and noncommutative geometry_, Chapter 3: Representation theory, reconstruction and Tannaka duality, [pdf](http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/98SS/Quantum_Groups/LN3.PDF)

A very neat Tannaka theorem for stacks is proved in

* [[Jacob Lurie]], _Tannaka duality for geometric stacks_, ([arXiv:math.AG/0412266](http://arxiv.org/abs/math/0412266))

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

* [[Bertrand Toen]], [Higher Tannaka duality](http://www.msri.org/publications/ln/msri/2002/hodgetheory/toen/2/index.html), MSRI 2002 (talk, video)

* Moshe Kamensky, _Model theory and the Tannakian formalism_, [arXiv:0908.0604](http://arxiv.org/abs/0908.0604); _Tannakian formalism over fields with operators_, [arxiv/1111.7285](http://arxiv.org/abs/1111.7285)

* H. Fukuyama, I. Iwanari, _Monoidal infinity category of complexes from Tannakian viewpoint_, [arxiv/1004.3087](http://arxiv.org/abs/1004.3087)

* remark of [Ben-Zvi on Tannaka reconstruction for monoidal categories](http://mathoverflow.net/questions/3446/tannakian-formalism/3467#3467)

* [[David Kazhdan]], Michael Larsen, Yakov Varshavsky, _The Tannakian formalism and the Langlands conjectures_, [arxiv/1006.3864](http://arxiv.org/abs/1006.3864)

The classical articles are

* [[Tadao Tannaka]], _Über den Dualitätssatz der nichtkommutativen topologischen Gruppen_, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!), see [[Tannaka-Krein theorem]].

* N. Tatsuuma, _A duality theorem for locally compact groups_, J. Math., Kyoto Univ.
6 (1967), 187–293.

* Nobuhiko Tatsuuma, _Duality theorem for locally compact groups and some related topics_, Algèbres d'opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), 387–408.  Colloq. Internat. CNRS, 274, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1979.  ISBN: 2-222-02441-2.

* M.G. Krein, _A principle of duality for bicompact groups and quadratic block algebras_, Doklady AN SSSR __69__ (1949), 725–728.
* Eiichi Abe, _Dualité de Tannaka des groupes algébriques_, Tohoku Mathematical Journal. Volume 12, Number 2  (1960), 327-332.

The Tannaka-type reconstruction in quantum field theory see [[Doplicher-Roberts reconstruction theorem]].

Tannaka duality in the context of [[(∞,1)-category theory]] is discussed in

* [[James Wallbridge]], _Higher Tannaka duality_, PhD thesis, Adelaide/Toulouse (2011) ([Adelaide University repository](http://digital.library.adelaide.edu.au/dspace/bitstream/2440/69436/1/02whole.pdf), [arXiv:1204.5787](http://arxiv.org/abs/1204.5787))

Tannaka duality for dg-categories is studied in

* J.P.Pridham, _Tannaka duality for enhanced triangulated categories_, [arxiv/1309.0637](http://arxiv.org/abs/1309.0637)

* MathOverflow, [Does the Tannaka-Krein theorem come from an equivalence of 2-categories?](http://mathoverflow.net/questions/30453/does-the-tannaka-krein-theorem-come-from-an-equivalence-of-2-categories)

[[!redirects Tannaka reconstruction]]

[[!redirects Tannaka reconstruction theorem]]


Tannaka duality for geometric stacks

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

#Contents#
* table of contents
{:toc}

## Idea

Under mild conditions, a given [[site]] $C \subset T Alg^{op}$ of formal duals of [[algebra over a Lawvere theory|algebras over an algebraic theory]] admits [[Isbell duality]] exhibited by an [[adjunction]]

$$
  (\mathcal{O} \dashv Spec) : (T Alg^{\Delta})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  Sh_{(\infty,1)}(C)
$$

as described at [[function algebras on ∞-stacks]] Here $\mathcal{O}(X)$ is an $(\infty,1)$-algebra of functions on $X$.

This entry describes for certain [[algebraic stacks]] an analog of this situation where the 1-algebras are replaced by 2-algebras in the form of commutative algebra objects in the [[2-category]] of [[abelian categories]]: abelian [[symmetric monoidal categories]], and where the function algebras $\mathcal{O}(X)$ are replaced with category $QC(X)$ of [[quasicoherent sheaves]].

The replacement of the 1-algebra $\mathcal{O}(X)$ by the 2-algebra $QC(X)$ is the starting point for what is called [[derived noncommutative geometry]].

## Setup

### Ringed toposes

All [[topos]]es that we consider are [[Grothendieck topos]]es. A [[ringed topos]] $(S, \mathcal{O}_S)$ is a topos $S$ equipped with a ring object $\mathcal{O}_S$ -- a [[sheaf]] of rings -- called the [[structure sheaf]] -- on whatever [[site]] $S$ is the [[category of sheaves]] on. We write $\mathcal{O}_S Mod$ for the category of [[module]]s in $S$ (sheaves of modules) over $\mathcal{O}_S$.

We write $RngdTopos$ for the category of ringed toposes.

For $X$ a [[scheme]] or more generally an [[algebraic stack]], write $Sh(X_{et})$ for its [[little etale topos]].

+-- {: .un_def #localWRTEtaleTopology}
###### Definition

A ringed topos $(S,\mathcal{O}_S)$ is a **[[locally ringed topos]]** with respect to the [[étale topology]] if for every object $U \in S$ and every family $\{Spec R_i \to Spec \mathcal{O}_S(U)\}$ of [[étale morphism]]s such that

$$
  \mathcal{O}_S(U) \to \prod_i R_i
$$

is [[faithfully flat]], there exists morphisms $E_i \to E$ in $S$ and factorizations $\mathcal{O}_S(U) \to R_i \to  \mathcal{O}_S(E_i)$ such that

$$
  \coprod_i E_i \to E
$$

is an [[epimorphism]].

=--

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

If $S$ has [[point of a topos|enough points]] then $(S, \mathcal{O}_S)$ is local for the étale topology precisely if the [[stalk]] $\mathcal{O}_S(x)$ at every [[point of a topos|point]] $x : Set \to S$ is a strictly [[Henselian ring|Henselian]] [[local ring]].

=--

This is ([Lurie, remark 4.4](#Lurie)).

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

* The [[little étale topos]] $Sh(X_{et})$ of a [[Deligne-Mumford stack]] $X$ is locally ringed with respect to the étale topology.

=--

### Abelian tensor categories

+-- {: .un_def #AbelianTensorCategory}
###### Definition

An **abelian tensor category** (for the purposes of the present discusission) is a [[symmetric monoidal category]] $(C, \otimes)$ such that

* $C$ is an [[abelian category]];

* for every $x \in C$ the functor $(-) \otimes x : C\to C$ is additive and right-[[exact functor|exact]]: it commutes with finite [[colimit]]s.

A **complete abelian tensor category** is an abelian tensor category such that

* it satisfies the axiom AB5 at [[additive and abelian categories]];

* $(-) \otimes x$ commutes with _all small_ colimits.

  (equivalently, we have a  [[closed monoidal category]]).

An abelian tensor category is called **tame** if for any [[short exact sequence]]

$$
  0 \to M'\to M \to M''\to 0
$$

with $M''$ a _flat object_ (such that $x \mapsto x \otimes M''$ is an [[exact functor]]) and any $N \in C$ also the induced sequence

$$
  0 \to M'\otimes N \to M\otimes N \to M''\otimes N \to 0
$$

is exact.

=--

This appears as ([Lurie, def. 5.2](#Lurie)) together with the paragraph below remark 5.3.

+-- {: .un_def #TCAbTens}
###### Definition

For $C,D$ two [complete abelian tensor categories](#AbelianTensorCategory) write

$$
  Func_\otimes(C,D) \subset Func(C,D)
$$

for the [[core]] of the [[subcategory]] of the [[functor category]] on those [[functor]]s that

* are [[symmetric monoidal functor]]s;

* commute with all small [[colimit]]s (which implies they are [[additive functor|additive]] and [[exact functor|right exact]])

* preserve flat objects and short exact sequences whose last object is flat.

Write

$$
  TCAbTens
$$

for the ([[strict 2-category|strict]]) [[(2,1)-category]] of [tame complete abelian tensor categories](#AbelianTensorCategory) with hom-[[groupoid]]s given by this $Func_\otimes$.

=--

This appears as ([Lurie, def 5.9](#Lurie)) together with the following remarks.

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

For $k$ a [[ring]], write $k Mod$ for its [[abelian category|abelian]] [[symmetric monoidal category]] of [[module]]s

Let $(S,\mathcal{O}_S)$ be a [[ringed topos]]. Then

$$
  \mathcal{O}_S Mod
$$

(the category of sheaves of $\mathcal{O}_S$-[[module]]s) is a tame [complete abelian tensor category](#AbelianTensorCategory).

=--

This is ([Lurie, example 5.7](#Lurie)).

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

For $X$ an [[algebraic stack]], write

$$
  QC(X)
$$

for its category [[quasicoherent sheaves]].

This is a [complete abelian tensor category](#AbelianTensorCategory)

=--

+-- {: .un_lemma}
###### Lemma

If $X$ is a Noetherian geometric stack, then $QC(X)$ is the category of [[ind-object]]s of its full [[subcategory]] $Coh(X) \subset QC(X)$ of [[coherent sheaves]]

$$
  QC(X) \simeq Ind(Coh(X))
  \,.
$$

=--

This appears as ([Lurie, lemma 3.9](#Lurie)).

### Geometric stacks {#GeometricStack}

+-- {: .un_def #geometricStack}
###### Definition

A **[[geometric stack]]** is

* an [[algebraic stack]] $X$ over $Spec \mathbb{Z}$

* that is quasi-compact, in particular there is an [[epimorphism]] $Spec A \to X$;

* with affine and [[representable morphism of stacks|representable]] diagonal $X \to X \times X$.

=--

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

* A quasicompact [[separated scheme]] is a geometric stack.

* The classifying stack of a [[smooth scheme|smooth]] affine [[group object|group]] [[scheme]] is a geometric stack.

=--

The geometricity condition on an algebraic stack implies that there are "enough" [[quasicoherent sheaves]] on it, as formalized by the following statement.

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

If $X$ is a [geometric stack](#geometricStack) then the [[bounded chain complex|bounded-below]] [[derived category]] of [[quasicoherent sheaves]] on $X$ is naturally [[equivalence of categories|equivalent]] to the full [[subcategory]] of the left-bounded derived category of smooth-[[etale site|etale]] $\mathcal{O}_X$-modules whose [[chain cohomology]] sheaves are quasicoherent.

=--

This is ([Lurie, theorem 3.8](#Lurie)).

## Tannaka duality for geometric stacks

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

Let $X$ be a [geometric stack](#GeometricStack).

Then for every ring $A$ there is an [[equivalence of categories]]

$$
  RngdTopos(Sh((Spec A)_{et}),Sh(X_{et})) \simeq Hom_\otimes(QC(X), A Mod)
$$

hence (by the [[2-Yoneda lemma]])

$$
  X(Spec A) \simeq Hom_\otimes(QC(X), QC(Spec A))
  \,.
$$

More generally, for $(S, \mathcal{O}_S)$ any [etale-locally](#localWRTEtaleTopology) [[ringed topos]], we have

$$
  RngdTopos(S,Sh(X_{et})) \simeq Hom_\otimes(QC(X), \mathcal{O}_S Mod)
  \,.
$$

=--

This is ([Lurie, theorem 5.11](#Lurie)) in view of ([Lurie, remark 4.5](#Lurie)).

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

It follows that forming [[quasicoherent sheaves]]  constitutes a [[full and faithful (infinity,1)-functor|full and faithful (2,1)-functor]]

$$
  QC : GeomStacks \to TCAbTens^{op}
$$

from geometric stacks to [tame complete abelian tensor categories](#TCAbTens).

This statement justifies thinking of $QC(X)$ as being the "2-algebra" of functions on $X$. This perspective is the basis for [[derived noncommutative geometry]].

=--

## Related concepts

* [[analytification]]

* [[2-algebraic geometry]]

* [[spectrum of a tensor triangulated category]]

* [[prime spectrum of a symmetric monoidal stable (∞,1)-category]]

* [[Tannaka duality for Lie groupoids]]

* [[Bondal-Orlov reconstruction theorem]]

* [[2-ring]]

## References

The above material is taken from

* {#Lurie} [[Jacob Lurie]], _Tannaka duality for geometric stacks_, ([arXiv:math.AG/0412266](http://arxiv.org/abs/math/0412266))

The generalization to geometric stacks in the context of [[Spectral Schemes]] is in

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

Related discussion is in

* [[Martin Brandenburg]], [[Alexandru Chirvasitu]], _Tensor functors between categories of quasi-coherent sheaves_ ([arXiv:abs/1202.5147](http://arxiv.org/abs/1202.5147))


Tannaka duality for Lie groupoids

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[Tannaka duality]] for [[Lie groupoid]]s.

In ([Trentinaglia](#Trentinaglia)) Tannaka duality for [[proper Lie groupoid]]s is established by using instead of [[vector bundle]]s something more general: not quite [[quasicoherent sheaves]] as in [[Tannaka duality for geometric stacks]], maybe, but something the author calls "smooth Euclidean fields".

## Related concepts

* [[Tannaka duality for geometric stacks]]

## References

* Giorgio Trentinaglia, _Tannaka duality for proper Lie groupoids_ PhD thesis, Journal of Pure and Applied Algebra ([pdf](http://igitur-archive.library.uu.nl/dissertations/2008-0904-200909/trentinaglia.pdf), [arXiv](http://arxiv.org/abs/0809.4423))
 {#Trentinaglia}


Tannaka-Krein theorem

The __Tannaka-Krein theorem__ (in the narrow sense) is a particular [[Tannaka duality]] theorem for compact topological groups.

The original theorem is (in two independent and different formulations) stated and proved in

* [[Tadao Tannaka]], _Über den Dualitätssatz der nichtkommutativen topologischen Gruppen_, Tohoku Math. J. 45 (1938), n. 1, 1–12 (project euclid has only Tohoku new series!)
* M.G. Krein, _A principle of duality for bicompact groups and quadratic block algebras_, Doklady AN SSSR __69__ (1949), 725–728.

For a textbook treatment see

* E. Hewitt, K. A. Ross, _Abstract harmonic analysis_, 2 vols.

See also

* [[Alexander Rosenberg|A. L. Rosenberg]], _[[Reconstruction of groups|Reconstruction of groups]]_ , Selecta Math. N.S. 9:1 (2003) doi

[[!redirects Tannaka-Krein duality]]


Tannakian category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of  _Tannakian category_ is an abstraction of the kind of data that enters the _[[Tannaka reconstruction theorem]]_: it is a suitable [[monoidal category]] $\mathcal{C}$ -- playing the role of a category of [[representations]]/[[actions]] of some algebraic structure over some [[base ring]] $R$ -- and equipped with a [[monoidal functor]] $\mathcal{C} \to R Mod$ to $R$-[[modules]] or similar -- playing the role of the [[forgetful functor]] which forgets the [[action]] (the "[[fiber functor]]").

## Related concepts

* [[Tannaka duality]]

* [[Deligne's theorem on tensor categories]]

## References

* N. Saavedra Rivano, _Catégories Tannakiennes_, Lecture Notes in Mathematics 265, Springer-Verlag, Berlin-New York, (1972).

* [[Pierre Deligne]], _[[Catégories Tannakiennes]]_, The Grothendieck Festschrift, Volume 2, 111--195. Birkhäuser, 1990.

* [[Larry Breen]], _Tannakian categories_, in _Motives_, Proceedings of Symposia in Pure Mathematics, 55, Providence, R.I.: American Mathematical Society, 1994.

The relation to [[motives]] is disucssed in

* [[Pierre Deligne]], [[James Milne]]: *Tannakian categories*,  in: *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics **900**, Springer (1982) [[doi:10.1007/978-3-540-38955-2_4](https://doi.org/10.1007/978-3-540-38955-2_4), [webpage](http://www.jmilne.org/math/xnotes/tc.html)]

* [[Pierre Deligne]], [[James Milne]], A. Ogus, K-y Shih, _Hodge cycles, motives, and Shimura varieties_, Lecture Notes in Mathematics, 900, Springer-Verlag, Berlin-New York, (1982).  [Annotated and corrected version (2012)](http://www.jmilne.org/math/xnotes/tc.pdf) (PDF).

* [[Yves André]], _Une introduction aux motifs (motifs purs, motifs mixtes, périodes)_,  Panoramas et Synthèses, 17, Paris: Société Mathématique de France. (2004)

* R. Sujatha, _Motives from a categorical point of view_, notes from Franco-Asian Conference on Motives, IHES, 2006.  [PDF](http://www.math.tifr.res.in/~sujatha/ihes.pdf)

[[!redirects Tannakian categories]]
[[!redirects Tannaka category]]
[[!redirects Tannaka categories]]


Tapio Simula

* [institute page](https://www.swinburne.edu.au/research/our-research/access-our-research/find-a-researcher-or-supervisor/researcher-profile/?id=tsimula)

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

* [Anyon technologies research group](https://www.swinburne.edu.au/research/centres-groups-clinics/optical-sciences-centre/our-research/anyon-technologies/)

## Selected writings

On [[topological quantum computation]] via [[braid representation|braiding]] of [[anyons]]:

* Bernard Field, [[Tapio Simula]], _Introduction to topological quantum computation with non-Abelian anyons_, Quantum Science and Technology **3** (2018) 045004 $[$[arXiv:1802.06176](https://arxiv.org/abs/1802.06176), [doi:10.1088/2058-9565/aacad2](https://iopscience.iop.org/article/10.1088/2058-9565/aacad2)$]$

On [[vortex]]$\;$[[anyons]] in [[Bose-Einstein condensates]]:

* {#MPSS19} Thomas Mawson, Timothy Petersen, [[Joost Slingerland]], [[Tapio Simula]], *Braiding and fusion of non-Abelian vortex anyons*, Phys. Rev. Lett. **123** (2019) 140404 [[doi:10.1103/PhysRevLett.123.140404](https://doi.org/10.1103/PhysRevLett.123.140404)]

On [[quantum compilation]] with [[su(2)-anyon]] [[braid representation|braid gates]]:

*  [[Emil Génetay-Johansen]], [[Tapio Simula]], Section IV of: *Fibonacci anyons versus Majorana fermions -- A Monte Carlo Approach to the Compilation of Braid Circuits in $SU(2)_k$ Anyon Models*, PRX Quantum **2** 010334 (2021) [[arXiv:2008.10790](https://arxiv.org/abs/2008.10790), [doi:10.1103/PRXQuantum.2.010334](https://doi.org/10.1103/PRXQuantum.2.010334)]

category: people


Taras Panov

* [webpage](http://higeom.math.msu.su/people/taras/english.html)

* [Math-Net page](http://m.mathnet.ru/php/person.phtml?personid=8869&option_lang=eng)

## Selected writings

On [[Calabi-Yau manifolds]] in [[SU-bordism theory]] (see [[Calabi-Yau manifolds in SU-bordism theory]]):

* [[Ivan Limonchenko]], [[Zhi Lu]], [[Taras Panov]], _Calabi-Yau hypersurfaces and SU-bordism_, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 ([arXiv:1712.07350](https://arxiv.org/abs/1712.07350))

* Georgy Chernykh, [[Ivan Limonchenko]], [[Taras Panov]], _$SU$-bordism: structure results and geometric representatives_, Russian Math. Surveys 74 (2019), no. 3, 461-524 ([arXiv:1903.07178](https://arxiv.org/abs/1903.07178))

* [[Taras Panov]], _A geometric view on $SU$-bordism_, talk at Moscow State University 2020 ([webpage](https://www.math.princeton.edu/events/geometric-view-su-bordism-2020-09-17t170000), [pdf](http://higeom.math.msu.su/people/taras/talks/2019SPb-Panov.pdf), [[PanovSU-Bordism.pdf:file]])

category: people


target

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

#Contents#
* table of contents
{:toc}

## Definition

The **target object**, or simply **target**, of a [[morphism]] $f: x \to y$ in some [[category]] $C$ is the [[object]] $y$.  The target of $f$ is also called its **codomain**, since the dual concept (the [[source]]) is also called 'domain'.

Given a [[small category]] $C$ with set of objects $C_0$ and set of morphisms $C_1$, the **target function** of $C$ is the function $t: C_1 \to C_0$ that maps each morphism in $C_1$ to its target object in $C_0$.

Generalising this, given an [[internal category]] $C$ with object of objects $C_0$ and object of morphisms $C_1$, the **target morphism** of $C$ is the morphism $t: C_1 \to C_0$ that is part of the definition of internal category.

## Related concepts

* [[source]], [[domain]]

* [[simplicial identities]]

[[!redirects targets]]

[[!redirects codomain]]
[[!redirects codomains]]

[[!redirects co-domain]]
[[!redirects co-domains]]

[[!redirects target object]]


target space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Generally, for $Maps(\Sigma,X)$ a [[mapping space]], the [[space]] or [[spacetime]] $X$ is the corresponding _target space_ or _target spacetime_.

Specifically, in a [[sigma-model]] [[quantum field theory]] a field configuration over a parameter space $\Sigma$ is a [[morphism]] $\Sigma \to X$ to some [[object]] $X$. This is the _target space_ of the theory. The $\sigma$-model is to be understood as describing the [[quantum mechanics]] of [[brane]]s with trajectories of shape $\Sigma$ propagating in $X$.

## Examples

For [[sigma-models]] that are supposed to describe quantum objects in the observed world (such as the [[relativistic particle]]) the target space $X$ would be our [[spacetime]]. But the target space can be something more general or more exotic. For instance in [[Dijkgraaf-Witten theory]] the target space is the [[classifying space]] of a [[discrete group]].

## Related concepts

* [[sigma model]]

  * [[worldvolume]]

  * **target space**

  * [[background gauge field]]

  * [[genus]]

[[!redirects target spaces]]

[[!redirects target spacetime]]
[[!redirects target spacetimes]]


Tarmo Uustalu

Tarmo Uustalu is a professor at the Dept. of Computer Science of Reykjavik University. He also has  part-time post in the Dept. of Software Science of the Tallinn University of Technology (TUT) as a lead research scientist, taking care of the Lab for High-Assurance Software, in particular the Logic and Semantics Group.

* [Home page](https://www.ioc.ee/~tarmo/)

## Selected writings

On [[comonads in computer science]]:

* {#UustaluVene08}  [[Tarmo Uustalu]], [[Varmo Vene]], *Comonadic Notions of Computation*, Electronic Notes in Theoretical Computer Science **203** 5 (2008) 263-284 [[doi:10.1016/j.entcs.2008.05.029](https://doi.org/10.1016/j.entcs.2008.05.029)]

Introducing [[relative monad|relative]] [[monads in computer science]]:

* [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), doi:10.2168/LMCS-11(1:3)2015]

On ([[comonad|co-]])[[monads in computer science]] (such as the [[writer monad]], [[reader monad]], [[cowriter comonad]], [[coreader comonad]], [[state monad]], ...):

* Marco Gaboardi, [[Shin-ya Katsumata]], [[Dominic Orchard]], Flavien Breuvart, [[Tarmo Uustalu]], *Combining effects and coeffects via grading*, ICFP 2016: Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (2016) 476–489 [[doi:10.1145/2951913.2951939](https://doi.org/10.1145/2951913.2951939), [talk abstract](https://icfp16.sigplan.org/details/icfp-2016-papers/31/Combining-Effects-and-Coeffects-via-Grading), [video rec](https://www.youtube.com/watch?v=l1ZNMT3fQCo)]

  > (also on [[graded modalities]])

* {#KRU20} [[Shin-ya Katsumata]], Exequiel Rivas, [[Tarmo Uustalu]], LICS (2020) 604-618 *Interaction laws of monads and comonads* [[arXiv:1912.13477](https://arxiv.org/abs/1912.13477), [doi:10.1145/3373718.3394808](https://doi.org/10.1145/3373718.3394808)]

* {#AhmanUustalu19} [[Danel Ahman]], [[Tarmo Uustalu]], *Decomposing Comonad Morphisms*, CALCO 2019, Leibniz International Proceedings in Informatics (LIPIcs) **139** (2019) [[doi:10.4230/LIPIcs.CALCO.2019.14](https://doi.org/10.4230/LIPIcs.CALCO.2019.14)]

* [[Tarmo Uustalu]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):

  *Monads and Interaction Lecture 1* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]]

  *Monads and Interaction Lecture 2* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs2.pdf), [[Uustalu-Monads2.pdf:file]]]

  *Monads and Interaction Lecture 3* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs3.pdf), [[Uustalu-Monads3.pdf:file]]]

  *Monads and Interaction Lecture 4* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs4.pdf), [[Uustalu-Monads4.pdf:file]]]

On ([[delta lens|delta]]-)[[lenses in computer science]] and [[monads in computer science|(co)monadic effects]]:

* {#AhmanUustalu17} [[Danel Ahman]], [[Tarmo Uustalu]], *Taking updates seriously*, Proceedings of the Sixth International Workshop on [Bidirectional Transformations (Bx 2017)](https://ceur-ws.org/Vol-1827/), CEUR Workshop Proceedings **1827** (2017) [[pdf](http://ceur-ws.org/Vol-1827/paper11.pdf)]

On [[strong monad|strong]] [[monads in computer science]]:

* {#McDermottUustalu22} [[Dylan McDermott]], [[Tarmo Uustalu]], *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 [[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)]

On [[graded monads]] [[monad in computer science|in computer science]] as [[enriched monad|enriched]] [[relative monads]] (and on [[locally graded categories]]):

* [[Dylan McDermott]], [[Tarmo Uustalu]]. _Flexibly graded monads and graded algebras_, in: *Mathematics of Program Construction MPC 2022*, Lecture Notes in Computer Science **13544**, Springer (2022) [[doi:10.1007/978-3-031-16912-0_4](https://doi.org/10.1007/978-3-031-16912-0_4)]

category: people

[[!redirects T. Uustalu]]


Taro Kimura

* [webpage](https://kimura.perso.math.cnrs.fr/)

## Selected writings

Relation of [[Yang-Mills monopoles]] as [[Dp/D(p+2)-brane intersections]] and [[Yang-Mills instantons]] as [[Dp/D(p+4)-brane intersections]] to the [[K-theory classification of topological phases of matter]] via [[AdS/CFT duality in solid state physics]]:

* [[Koji Hashimoto]], [[Taro Kimura]], _Band spectrum is D-brane_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 1 ([arXiv:1509.04676](https://arxiv.org/abs/1509.04676))

On [[random matrix theory]]:

* [[Bertrand Eynard]], [[Taro Kimura]], Sylvain Ribault, _Random matrices_, lecture notes ([arxiv:1510.04430](http://arxiv.org/abs/1510.04430))

On [[brane webs]] and [[quiver gauge theory]]:

* [[Taro Kimura]], Rui-Dong Zhu, Section 2 and 3  of _Web Construction of ABCDEFG and Affine Quiver Gauge Theories_ ([arXiv:1907.02382](https://arxiv.org/abs/1907.02382))

On [[Yang-Mills instantons]], [[Seiberg-Witten theory]] and [[quiver gauge theory]]:

* [[Taro Kimura]], *Instanton Counting and Localization*, Chapter 1 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

category: people


Tarski group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

A [[mathematical structure]] used to define the [[real numbers]] in [[Alfred Tarski]]’s axioms for the [[real numbers]].

## Definition ##

A [[Tarski group]] is a [[pointed commutative invertible semigroup]] $(G, +, -, 1)$ with a [[dense linear order]] $\lt$ such that $1 \lt 1 + 1$, $a \lt b + c + a$ implies $a \lt b + a$ or $a \lt c + a$, and $a \lt b$ implies $c + a \lt c + b$.

As a result, every Tarski group is an [[abelian group]] with [[identity element]] $0 \coloneqq 1 - 1$, and a [[trivial group|nontrivial]] [[ordered group]].

## Tarski's axioms for the real numbers ##

Tarski's axioms for the real numbers are as follows:

* Axiom 5 says that $\mathbb{R}$ is a [[commutative semigroup]].

* Axiom 6 with axiom 5 together say that $\mathbb{R}$ is a [[commutative invertible semigroup]].

* Axiom 8 says that $\mathbb{R}$ is a [[pointed set]], which with axioms 5 and 6 imply that $\mathbb{R}$ is an [[abelian group]].

* Axiom 1 says that $\mathbb{R}$ has a [[connected relation]] $\lt$.

* Axiom 2 says that $\lt$ is an [[asymmetric relation]] and thus an [[irreflexive relation]].

* The fragment of Axiom 4 that only refers to [[singleton]] [[subsets]] says that $\lt$ is a [[comparison]], making $\lt$ into a [[strict linear order]]. (the full axiom 4 is the [[Dedekind completion|Dedekind completeness]] condition).

* Axiom 3 says that $\lt$ is a [[dense linear order]].

* Axiom 7 says that for all $a, b, c, d \in G$, $a + b \lt c + d$ implies that $a \lt c$ or $b \lt d$, which imply that $\mathbb{R}$ is a linearly [[ordered group]].

* Axiom 9 says that $1 \lt 1 + 1$, which indicates that $\mathbb{R}$ is not [[trivial group|trivial]].

## The archimedean field structure on the Dedekind-complete Tarski group ##

Let us denote the Dedekind-complete Tarski group as $\mathbb{R}$. There is an [[archimedean field]] structure on $\mathbb{R}$.

+-- {: .num_prop}
###### Proposition
$\mathbb{R}$ is a [[Archimedean group|Archimedean]] [[ordered group|ordered]] [[abelian group]].
=--

+-- {: .proof}
###### Proof
Since $\mathbb{R}$ is [[Dedekind-complete]] and a strictly ordered abelian group, $\mathbb{R}$ is [[Archimedean group|Archimedean]], because the Dedekind-completion of any totally ordered abelian group with [[infinity|infinite elements]] or [[infinitesimals]] is not an [[abelian group]], and the Dedekind-completion of any Archimedean ordered abelian group is still Archimedean.
=--

+-- {: .num_prop}
###### Proposition
$\mathbb{R}$ has a [[complete metric]]
=--

+-- {: .proof}
###### Proof
Since $\mathbb{R}$ is strictly ordered, it is a [[totally ordered abelian group]]. As a result, there exist [[maximum]] and [[minimum]] [[binary functions]] $\max:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and $\min:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$, with the [[absolute value]] function defined as $\vert x \vert = \max(x, -x)$.

Since $\mathbb{R}$ is [[Dedekind-complete]], Archimedean, and a totally ordered abelian group, $\mathbb{R}$ is a [[metric space]] with respect to the absolute value $\vert x \vert$ and thus a [[Hausdorff space]], and every [[Cauchy net]] in $\mathbb{R}$ [[limit of a net|converges]] to a unique element of $\mathbb{R}$, and thus the absolute value $\vert x \vert$ is a [[complete metric]] on $\mathbb{R}$.
=--

+-- {: .num_prop}
###### Proposition
$\mathbb{Q}$ embeds in $\mathbb{R}$.
=--

+-- {: .proof}
###### Proof
Since $\mathbb{R}$ is an [[abelian group]], it is a $\mathbb{Z}$-[[module]], and since $\mathbb{R}$ is totally ordered, it is a [[torsion-free module]] and thus a [[torsion-free abelian group]], which means that the integers $\mathbb{Z}$ [[embedding|embed]] in $\mathbb{R}$, with [[injection|injective]] [[group homomorphism]] $f:\mathbb{Z} \to \mathbb{R}$ where $f(0) = 0$ and $f(1) = 1$. As a result, for every integer $a \in \mathbb{Z}$ and $b \in \mathbb{Z}$ the [[affine function]]s $x \mapsto a x + b$ are well defined in $\mathbb{R}$.

Since $\mathbb{R}$ is [[Dedekind-complete]], Archimedean, and a totally ordered abelian group, any [[closed interval]] $[a, b]$ on $\mathbb{R}$ is [[compact space|compact]] and [[connected space|conencted]]. Since $\mathbb{R}$ is also a complete metric space, the [[intermediate value theorem]] is satisfied for every function from a closed interval $[a, b]$ to $\mathbb{R}$. Because $x \mapsto a x + b$ are [[monotonic]] for $a \gt 0$, and for $a \lt 0$ the function is just the negation of a monotonic function, $x \mapsto a x + b$ have a [[root of a function|root]] for $\vert a \vert \gt 0$. Thus $\mathbb{R}$ is a [[divisible group]] and a $\mathbb{Q}$-[[vector space]], with an [[injection|injective]] [[group homomorphism]] $f:\mathbb{Q} \to \mathbb{R}$ where $f(0) = 0$ and $f(1) = 1$, and $\mathbb{Q}$ embeds in $\mathbb{R}$.
=--

+-- {: .num_prop}
###### Proposition
$\mathbb{R}$ is a [[commutative ring]].
=--

+-- {: .proof}
###### Proof
Since every [[Cauchy net]] in $\mathbb{R}$ [[limit of a net|converges]] to a unique element of $\mathbb{R}$, for every directed set $A$ and Cauchy net $(a_i)_{i \in A}$ in the rational numbers, there exists a Cauchy net of [[linear functions]] $(f_i)_{i \in A}$ defined as $f_i(x) = a_i x$. The limit of the Cauchy net $\lim_{i \in A} (f_i)_i$ exists and is a unique function $g(x) = \lim_{i \in A} (a_i)_i x$. Since every real number is the limit of a Cauchy net of rational numbers, there is an $\mathbb{R}$-action $\mu:\mathbb{R} \to (\mathbb{R} \to \mathbb{R})$ which takes a real number $r$ to the linear function $x \mapsto r x$, with $\alpha(1) = x \mapsto x$ being the [[identity function]]. The [[uncurrying]] of $\alpha$ leads to a [[bilinear function]] $(-)(-):\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ called multiplication of the real numbers, defined on the entire domain of the binary function. Since linear functions in the [[function space]] with [[function composition]] and the [[identity function]] is a [[commutative monoid]], $\mathbb{R}$ with multiplication and the multiplicative [[identity element]] $1$ is also [[commutative monoid]], which means that $\mathbb{R}$ is a commutative ring.
=--

+-- {: .num_prop}
###### Proposition
$\mathbb{R}$ is a [[field]]
=--

+-- {: .proof}
###### Proof
Since $\mathbb{R}$ is a commutative ring, [[power series]] are well defined, and because all Cauchy nets converge in $\mathbb{R}$, all Cauchy sequences and all Cauchy power series converge in $\mathbb{R}$. In particular, every [[geometric series]] is a Cauchy power series and the limit of the geometric series
$$\sum_{n=0}^\infty x^n$$
and
$$\sum_{n=0}^\infty (-1)^n x^n$$
[[limit of a sequence|converges]] in the [[open interval]] $(-1, 1)$. Thus let us define functions $f:(-1, 1) \to \mathbb{R}$ and $g:(-1, 1) \to \mathbb{R}$ as

$$f(x) := \sum_{n=0}^\infty x^n$$

$$g(x) := \sum_{n=0}^\infty (-1)^n x^n$$

Let us define the function

$$h(x, a) := (-a) \sum_{n=0}^\infty a^n (x + f(a + 1))^n$$

for $a \lt 0$ and

$$k(x, a) := a \sum_{n=0}^\infty (-a)^n (x + g(a - 1))^n$$

for $a \gt 0$. These are functions which converge on the open interval $(1/a, 0)$for $h(x, a)$ and $(0,1/a)$ for $k(x, a)$, and satisfy the identity $h(x, a) x = 1$ for all $a \lt 0$ and $x \in (1/a, 0)$, and $k(x, a) x = 1$ for all $a \gt 0$ and $x \in (0,1/a)$, by definition of the geometric series.

The [[real reciprocal function]] is piecewise defined as
$$
\frac{1}{x} =
\begin{cases}
\lim_{a \to 0^-} h(x, a)  & \mathrm{if}\; x \lt 0 \\
\lim_{a \to 0^+} k(x, a)  & \mathrm{if}\; x \gt 0 \\
\end{cases}
$$

As [[algebraic limit theorem|limits preserve multiplication]], $\frac{1}{x} x = 1$ for all $x \in \mathbb{R}$. Thus, $\mathbb{R}$ is a field.
=--

## See also ##

* [[abelian group]]

* [[commutative invertible semigroup]]

* [[pointed set]]

* [[dense linear order]]

* [[ordered group]]

* [[real numbers]]

## References ##

* [[Alfred Tarski]],  *Introduction to Logic and to the Methodology of Deductive Sciences* (4th edition). Oxford University Press. (1994) $[$[doi:10.2307/2180610](https://doi.org/10.2307/2180610), ISBN 978-0-19-504472-0$]$

* Ucsnay, Stefanie (Jan 2008), A Note on Tarski's Note. The American Mathematical Monthly, Vol 115 No. 1, pg 66–68. JSTOR 27642393

[[!redirects Tarski groups]]

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