trace alphabet

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

##Idea##
When discussing [[language|languages]], say, in the theory of [[finite automata]], the usual starting point is a set, $\Sigma$, of symbols, which is taken to be the _alphabet_ for the language.  The next step is to form the [[free monoid]], $\Sigma^*$, on the alphabet, and then a language is simply a subset of $\Sigma^*$, and these can be used in a number of situations to describe the workings of a 'system', the symbols being used to represent certain operations or actions of the system, and the words in $\Sigma^*$ thus corresponding to sequences of operations. Usually the system will not allow arbitrary sequences of operations to be performed, and so the system determines a language, namely the allowable execution sequences. Describing the language, helps describe and determine the overall 'system'.

If the system being studied has some concurrency in it, so that certain of the operations are (causally) independent of each other, then one way of modelling such a system is to use a _(Mazurkiewicz) trace alphabet_, that is one in which the independence is explicitly taken into consideration.

+--{: .un-definition}
##Definition##
A _(Mazurkiewicz) trace alphabet_ is a pair $(\Sigma, I)$ where $\Sigma$ is a (usually finite) set of _actions_ and $I \subseteq \Sigma \times \Sigma$ is an _irreflexive_ and _symmetric_ relation.
=--
The relation $I$ is usually referred to as *independence*.  Its complement $D = (\Sigma \times \Sigma) - I$ is called the _dependency_ relation.  It will be reflexive and symmetric. (Such a sense of 'dependency relation' is interpreted in a related but slightly different way from that in the use of [dependency graphs](http://en.wikipedia.org/wiki/Dependency_graph) in project planning.)

+--{: .un-example}
##Example##

Consider the following 'dependency diagram, with some vertices (states) and labelled edges (actions):

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(arrow go from left to right but are not showing up ...  bother!)

We take $\Sigma = \{ a,b,c,d,e\}$ and note that, starting at the left, $b$ depends on $a$ (written $a\lt b$, since we cannot start action $b$ until action $a$ has been completed. This gives a labelled [[poset]].

Here the first information to encode in $D$ is the poset structure of dependency

$$\{(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,d), etc.\},$$

then symmetrise this, and take the complement to get

$$I = \{(b,c),(c,b),(d,e),(e,d),(b,e), (e,b), (c,d),(d,c)\}.$$

Note this encoding does not involve the relationship given by adjacency, and hence ignore if an 'execution sequence' is 'allowable' (i.e., realistic) so $(c,d)$ is in $I$ even though we could not first 'do' $c$ and then 'do' $d$ (unless of course $b$ had already been done).

The language ([[trace language]]) corresponding to the model will be a subset of the [[trace monoid]] of the trace alphabet $(\Sigma, I)$ given here.  It will encode allowable execution sequences  but also will consider the allowable sequences ( _traces_ ): $a b c d e$, $a c b d e$, $a b d c e$, $a c e b d$ etc. to be identified.

=--
##Reference

*  A Mazurkiewicz, _Trace theory_ in 'Advances in Petri Nets', volume 255 of LNCS, Springer (1987).

[[!redirects trace alphabets]]
[[!redirects Mazurkiewicz trace alphabet]]


trace monoid

# Contents
* automatic table of contents goes here
{:toc}
##Idea##
The discussion in [[trace alphabet]] describes how in a state based system where there is a notion of (causal) independence on certain actions, the usual use of the free monoid on the set of actions is better replaced by a construction taking the independence into account. That construction gives the **trace monoid**.

Let $(\Sigma,I)$ be a [[trace alphabet]]:

+--{: .un-definition}
##Definition##
The **trace monoid**, $M(\Sigma,I)$, of the trace alphabet, $(\Sigma,I)$, is the monoid, with presentation $\langle \Sigma\mid I\rangle$, where to an independent pair $(a,b)$ corresponds the pair of words (relations), $(ab,ba)$.
 The resulting monoid is thus formed by taking $\Sigma^*/\equiv$, where $\equiv$ is the congruence relation generated by the relations. The elements of $M(\Sigma,I)$ are usually called **traces**.
=--

##References##

* [Wikipedia](http://en.wikipedia.org/wiki/Trace_monoid)


trace of a category

# The trace of a category
* table of contents
{: toc}

## Idea

The trace of a [[category]] (or more generally of an endo[[bimodule]] or endo[[profunctor]]) is a [[categorification]] of the [[trace]] of a linear [[endomorphism]] on a finite dimensional [[vector space]] (that is a [[matrix]]).

A notion of trace is generally definable for maps in a [[compact closed category]] (even more generally in a [[traced monoidal category]]), and here the idea is to categorify this to the context of compact closed [[bicategories]], in particular the bicategory of bimodules between small categories.

As an instance of the [[microcosm principle]], the trace of a category is the recipient of the *universal trace* function for morphisms *in* that category, and also supplies the necessary structure to define [[trace in a bicategory|bicategorical traces]] in the bicategory [[Prof]].

## Definition

Let $C$ be a compact closed [[symmetric monoidal category]], with monoidal product $\otimes$ and monoidal unit $1$. The **trace** of an endomorphism $f: c \to c$ is the composite

$$1 \overset{\eta}{\to} c^* \otimes c \overset{1_{c^*} \otimes f}{\to} c^* \otimes c \overset{\varepsilon}{\to} 1$$

where $\eta$ is a unit and $\varepsilon$ is a counit of appropriate [[adjunctions]] (note that the symmetry makes the dual $c^*$ both a right and left adjoint of $c$: the adjunctions are ambidextrous). In the classical case where $C$ is the category of finite-dimensional vector spaces with its usual monoidal structure, this gives the usual trace of an endomorphism; in particular, for $f = 1_c$, this defines $dim(c) \in hom(1, 1)$.

The same idea applies to compact closed symmetric monoidal bicategories. In particular, it applies to the bicategory [[Prof]] whose objects are small categories, whose 1-morphisms are [[profunctors]] $C \to D$, i.e., functors

$$R: D^{op} \times C \to Set$$

and whose 2-morphisms are [[natural transformations]] between profunctors. The bicategory $Prof$ is a [[cartesian bicategory]] and hence symmetric monoidal under $\times$, and is also compact closed: the dual of a category $C$ in this case is just the opposite category $C^{op}$, and the unit and counit profunctors

$$\eta: 1 \to C^{op} \times C, \, \varepsilon: C^{op} \times C \to 1$$

are given by $hom_{C^{op}}$ and $hom_C$. Composing these (according to the [[coend]] formula for profunctor composition) yields

$$\int^{c, c' \in Ob(C)} hom(c, c') \times hom(c', c) \cong \int^c hom(c, c)$$

and this is the trace of the identity $1_C$ in $Prof$ (which as a functor is also given by $hom_C$); this coend is called the **trace** of the category $C$. It could also reasonably be called, by analogy with the vector space case, the **dimension** of the category $C$. The trace of a general endoprofunctor $F$ on $C$ is the coend

$$\int^{c \in Ob(C)} F(c, c)$$

which generalizes the trace of linear functions:

$$Tr(f) = \sum_i f_{i i}$$

(where the matrix entries $f_{i j}$ are computed with respect to any basis).

The foregoing discussion can be generalized to the case of bimodules between small categories enriched in a [[cocomplete category|cocomplete]] symmetric [[monoidal closed category]] $V$, where the dimension of a small $V$-category $C$ is the object of $V$ given by the enriched coend

$$\int^c hom(c, c)$$

## Example

We calculate the trace or dimension of [[FinSet]], the category of finite sets. The calculation is quite down-to-earth: the relevant coend is just the quotient of the set of all endofunctions $h: c \to c$ between finite sets, modulo the equivalence relation $\sim$ generated by the stipulation $f \circ g \sim g \circ f$ whenever $f: c \to d$ and $g: d \to c$ are functions between finite sets.

Let $h: c \to c$ be a finite endofunction, and let

$$c \overset{p}{\to} d \overset{i}{\to} c$$

be its epi-mono factorization. Then $h = (i \circ p) \sim (p \circ i)$; if we think of $d$ as the image $h(c)$, then $p \circ i$ can be viewed as the restriction

$${h|}\colon h(c) \to h(c)$$

and this process iterates. The sequence of epis

$$h(c) \overset{h|}{\to} h^{(2)}(c) \overset{h|}{\to} \ldots$$

eventually stabilizes (after finitely many steps) to a finite set $h^{(\infty)}(c)$ on which $h$ restricts to a [[surjection|surjective]] [[endofunction]], which is a bijection since we are dealing with finite sets. Thus every $h: c \to c$ is $\sim$-equivalent to a permutation $\sigma: d \to d$.

Furthermore, given two permutations $\sigma: c \to c$ and $\tau: d \to d$ such that $\sigma \sim \tau$ is witnessed by a chain of function equalities

$$\sigma = g_1 f_1, \; f_1 g_1 = h_1 = g_2 f_2, \; \ldots, \; f_{n-1} g_{n-1} = h_{n-1} = g_n f_n, \; f_n g_n = \tau$$

one may show (using $g_k h_k^j f_k = g_k (f_k g_k)^j f_k = (g_k f_k)^{j+1} = h_{k-1}^{j+1}$ and similarly $f_k h_{k-1}^j g_k = h_k^{j+1}$) that

$$g_1 g_2 \ldots g_{n-1} g_n f_n f_{n-1} \ldots f_2 f_1 = \sigma^n, \qquad f_n f_{n-1} \ldots f_2 f_1 g_1 g_2 \ldots g_{n-1} g_n = \tau^n$$

which implies that $f = f_n \ldots f_1$ is invertible and $\tau f = f \sigma$, or $\tau = f \sigma f^{-1}$. Conversely, if $\tau = f \sigma f^{-1}$, then for $g = \sigma f^{-1}$ we have $\sigma = g f$ and $f g = \tau$, so $\sigma \sim \tau$.

In other words, the trace of the category of finite sets is isomorphic to the trace of the [[core|underlying groupoid]] of finite sets and bijections, where the equivalence classes with respect to $\sim$ are the conjugacy classes of [[permutation]]s, given by cycle types. In this way, the trace of $FinSet$ is naturally identified with the class of finite [[Young diagram]]s.

+-- {: .num_remark}
###### Remark
Furthermore, the functorial operations

$$\hom(x, x) \times \hom(y, y) \to \hom(x \times x, y \times y)\qquad \text{and}\qquad \hom(x, x) \times \hom(y, y) \to \hom(x + y, x + y)$$

induce operations $\cdot: Tr(FinSet) \times Tr(FinSet) \to Tr(FinSet)$ and $+: Tr(FinSet) \times Tr(FinSet) \to Tr(FinSet)$. Since $\times$ distributes over $+$ in $FinSet$, we obtain a [[rig]] structure on $Tr(FinSet)$, namely the [[Burnside rig]] of $\mathbb{Z}$.
=--

## References

* [[Harald Lindner]]. _Center and trace_, Archiv der Mathematik 35 (1980): 476-496. ([doi:10.1007/BF01235372](https://doi.org/10.1007/BF01235372))

[[!redirects trace of a category]]
[[!redirects traces of categories]]
[[!redirects universal trace]]
[[!redirects shadow]]


trace of horizontal to round chord diagrams

[[!redirects tracing horizontal to round chord diagrams]]

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

#Contents#
* table of contents
{:toc}

## Idea

In [[knot theory]], a certain [[trace]]-like operation turns [[horizontal chord diagrams]] into [[round chord diagrams]]:

Given a [[horizontal chord diagram]] on $n$ strands and given any choice of [[cyclic permutation]] of $n$ elements, the trace is the [[round chord diagram]] obtained by gluing the ends of the strands according to the cyclic permutation, and retaining the chords in the evident way.

The following graphics shows basic examples of the trace operation for [[cyclic permutation]] of strands one step to the right:

## Properties

### Relation to 4T relations

The trace operation sends the [[4T relations]] on [[horizontal chord diagrams]] to those for [[round chord diagrams]]:

## References

The trace operation is implicit in

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

It is made a little more explicit in

* {#Kohno02} [[Toshitake Kohno]], around Figure 2 of: _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))


trace-class map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
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# Contents
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## Definition ##

Given a [[closed monoidal (infinity,1)-category|closed monoidal $(\infty, 1)$-category]] $\mathcal{C}$, with unit $I \in \mathcal{C}$, [[tensor product]] $(-)\otimes (-):\mathcal{C} \times \mathcal{C} \to \mathcal{C}$, and [[internal hom]] $[(-), (-)]:\mathcal{C} \times \mathcal{C} \to \mathcal{C}$, let $a \in \mathcal{C}$ and $b \in \mathcal{C}$ be [[objects]] in $\mathcal{C}$. Let $f:a \to b$ be a map. $f$ is a **trace-class map** or **nuclear map** if $f$ lies in the [[image]] of the natural map $\mathrm{Hom}(I, [a, I] \otimes b) \to \mathrm{Hom}(a, b)$.

## See also ##

* [[closed monoidal (infinity,1)-category]]

* [[condensed mathematics]]

## References ##

* [[Dustin Clausen]], [[Peter Scholze]], _Condensed Mathematics and Complex Geometry_, [pdf](https://people.mpim-bonn.mpg.de/scholze/Complex.pdf)

[[!redirects trace-class maps]]
[[!redirects nuclear map]]
[[!redirects nuclear maps]]


trace-class operator

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## Definition

A [[compact operator|compact]] [[linear operator]] is said to be **trace class** if its [[trace]] exists.

[[!redirects trace class]]
[[!redirects trace class operator]]
[[!redirects trace class operators]]
[[!redirects trace-class operator]]
[[!redirects trace-class operators]]


traced monoidal category

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## Idea

The concept of _traced monoidal category_ axiomatizes the structure on a [[monoidal category]] for it to have a sensible notion of ([[partial trace|partial]]) [[trace]] the way it exists canonically in [[compact closed categories]]. Traced monoidal categories sometimes only have _right_ or _left_ traces, and are referred to as _right traced_ or _left traced_ respectively. A category with compatible right and left traces is called a planar traced category. If the right and left traces agree, then we call a planar traced category _spherical_.

Graphically, the trace of an endomorphism $f:A\to A$ is represented by the closed loop

\begin{imagefromfile}
"file_name": "trace.png",
"width": 150,
"unit": "px"
\end{imagefromfile}

The ambiguity in whether to close the loop to the right or to the left is codified in the difference between left and right traced categories. Every [[pivotal category]] can be canonically made into a planar traced category by using its implicit distinguished morphisms to close loops. Pivotal categories which induce spherical traced structures are known as [[spherical categories]]. In this way, traced monoidal categories generalize the trace operation to categories which do not necessarily have duals.

In [[denotational semantics]] of programming languages, the trace is used to model [[recursion]], though if the language is not [[substructural]], then this can be simplified to a [[fixed point operator]].

## Definition

The original definition due to ([Joyal-Street-Verity 96](#JoyalStreetVerity96)) is stated in the setting of [[balanced monoidal categories]]. Here we present a very general formulation, as given in [Selinger's survey](#Selinger2011).

A [[monoidal category]] $\mathscr{C}$ is said to be _right traced_ if it is equipped with family of operations

$$\tr^X_R:\text{Hom}(A\otimes X,B\otimes X)\to \text{hom}(A,B)$$

for all $A,B,X\in \mathscr{X}$, satisfying the following four axioms:

 * **Tightening** (naturality in $A,B$) For all $A,B,C,D,X\in\mathscr{C}$, $g:A\to B$, $f:B\otimes X\to C\otimes X$, $h:C\to D$

$$\tr_R^X((g\otimes \text{id}_X)\circ f\circ (h\otimes \text{id}_X))=g\circ \tr_R^X(f)\circ h$$

 * **Sliding** (naturality in $X$) for all $A,B,X,Y\in $, $f:A\otimes X\to B\otimes Y$, $g:Y\to X$

$$\tr_R^{X}((\text{id}_B\otimes g)\circ f)=\tr_R^{Y}(f\circ (\text{id}_A\otimes g))$$

 * **Vanishing** For all $A,B,X,Y\in \mathscr{C}$,

$$\tr^1_R(f)=f,\,\, \forall \, f:A\to B$$

and

$$\tr^{X\otimes Y}_R(f)=\tr_R^X(\tr_R^Y(f))$$

for all $f:A\otimes X\otimes Y\to B\otimes X\otimes Y$.

 * **Strength** For all $A,B,C,D,X\in \mathscr{C}$, $f:C\otimes X\to D\otimes X$, $g:A\to B$

$$\tr_R^X(g\otimes f)=g\otimes \tr_R^X(f)$$

Graphically, we fix the notation for right trace

\begin{imagefromfile}
"file_name": "right-trace-graphical.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

The above axioms are described with the following pictures:

\begin{imagefromfile}
"file_name": "selinger-diagram-tracing.png",
"width": 550,
"unit": "px"
\end{imagefromfile}

In the presence of a [[braided monoidal category | braiding]] (such as in a [[symmetric monoidal category]]), we additionally require a _yanking_ axioms:

* **Yanking:** $Tr^X(\beta_{X,X}) = id_X$ for all $X\in\mathscr{C}$

where $\beta:X\otimes X\to X\otimes X$ is the briading.

A left traced category is defined similarly, with a family of morphisms

$$\tr_L^X:\text{Hom}(X\otimes A,X\otimes B)\to \text{Hom}(A,B)$$

satisfying symmetric axioms. We write the left trace graphically as

\begin{imagefromfile}
"file_name": "left-trace-graphical.png",
"width": 300,
"unit": "px"
\end{imagefromfile}

A planar traced category is a monoidal category $\mathscr{C}$ which is simultaneously left and right traced, such that the following axioms are satisfied:

* **Interchange** For all $A,B,X,Y\in \mathscr{C}$, $f:Y\otimes A\otimes X\to Y\otimes B\otimes X$

$$\tr_R^X(\tr_L^Y(f))=\tr_L^Y(\tr_R^X(f)).$$

* **Left pivoting** For all $A,B\in\mathscr{C}$, $f:1\to A\otimes B$

$$\tr_R^B(\text{id}_B\otimes f)=\tr_L^A(f\otimes \text{id}_A).$$

* **Right pivoting** For all $A,B\in\mathscr{C}$, $f:A\otimes B\to 1$

$$\tr_R^B(\text{id}_B\otimes f)=\tr_L^A(f\otimes \text{id}_A).$$

Graphically, these axioms can be stated as below:

\begin{imagefromfile}
"file_name": "pivoting-axioms.png",
"width": 500,
"unit": "px"
\end{imagefromfile}

It is from these axioms that [[pivotal categories]] get their name.

A spherical trace category is a planar traced category in which the left and right partial traces agree. That is, for any endomorphism $f:A\to A$ in $\mathscr{C}$ we have that

$$\tr_R^A(f)=\tr_L^A(f).$$

## Examples

* The category of [[finite-dimensional vector spaces]] is traced monoidal. In skeletal form, the objects are natural numbers regarded as dimensions, morphisms are matrices regarded as linear maps, and tensor product is multiplication of the dimensions. The trace is a minor generalization of matrix trace:
$$ Tr_{m,n}^p((a_{(i,k),(j,k')})_{(i,k),(j,k')\in m\times p\times n\times p})_{i,j}=
\sum_{k=1}^p a_{(i,k),(j,k)}.$$
This is an instance of a trace in a compact closed category.

* The category of [[profunctors]] is also compact closed, hence traced monoidal (actually a bicategory, though). The trace of a profunctor $H:(B\times X)^{op} \times A\times X\to \mathbf{Set}$ is the profunctor $Tr_{A,B}^X(H):B^{op}\times A\to \mathbf{Set}$ given by the [[coend]] formula:
$$ Tr(H)(b,a)=\int^x H(b,x,a,x) .$$

* The category of sets and [[partial functions]] with the coproduct monoidal structure is traced, if we let the trace of a partial function
$f:A+X\to B+X$ be defined as follows. First, let $f_{A,B}:A\to B$, $f_{X,X}:X\to X$ and so on be the evident partial functions that come from restricting the domain and codomain of $f$. Then define
$$Tr_{A,B}^X(f)(a)=b$$
if either $f_{A,B}(a)=b$ or if there exists $n$ such that $b=f_{X,B}(f_{X,X}^n(f_{A,X}(a)))$.
See for instance Section 5.2 of ([Abramsky, Haghverdi and Scott](#AbramskyHaghverdiScott02)). This is an example of a partially additive category.

  This is a basis for a categorical semantics of "while loops" in programming languages: we keep applying $f$ _while_ it is returning in $X$.

* Consider the category of pointed [[cpo]]'s and continuous functions. A pointed cpo is a [[poset]] with a bottom element $\bot$ and least upper bounds of $\omega$-chains. The morphisms preserve least upper bounds of chains but are not required to preserve bottom elements. This is a [[cartesian monoidal category]], and with this structure it is traced, if we let the trace of a continuous function $f:A\times X\to B\times X$ be given as follows.
First, for fixed $a\in A$, let $x_a\in X$ be the least [[fixed point]] of
$(\pi_2\cdot f(a,-)):X\to X$, writing $\pi_2$ for the second product projection. Here the least fixed point exists by Tarski's fixed point theorem, indeed
$$x_a=\bigvee_n(\pi_2\cdot f(a,-))^n(\bot).$$
Then we let
$$Tr_{A,B}^X(f)(a)=\pi_1(f(a,x_a))\in B$$
In [[Haskell]] notation, where 'let' allows recursive definitions, we can more swiftly just write
$$Tr_{A,B}^X(f)(a)\ \ \ = \ \ \ let\ (b,x)=f(a,x)\ in\ b$$
(except that the built-in product types in Haskell are lifted products rather than categorical products: $(A,B)$ in Haskell is like $A\times B$ but with an extra bottom element added).

  This is an example of a cartesian category with a parameterized fixed point operator ([Hasegawa 1997](#Hasegawa97)).

## Properties

### Relation to compact closed categories
 {#RelationToCompactClosedCategories}

Every [[compact closed category]] is equipped with a canonical trace defined by

$$ Tr_{A,B}^X(f) = A \overset{id\otimes \eta}{\to} A \otimes X \otimes X^* \overset{f \otimes id}{\to} B \otimes X \otimes X^* \overset{id \otimes \varepsilon'}{\to} B$$

where $\eta$ is a unit and $\varepsilon'$ is a counit of appropriate [[adjunctions]] (note that the symmetry makes the dual $X^*$ both a right and left adjoint of $X$: the adjunctions are ambidextrous).

Conversely, given a traced monoidal category $\mathcal{C}$, there is a [[free construction]] completion of it to a [[compact closed category]] $Int(\mathcal{C})$ ([Joyal-Street-Verity 96](#JoyalStreetVerity96)):

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and [[composition]] of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by [[trace|tracing out]] $B^-$ from the composite
$$ A^+ \otimes B^- \otimes C^- \xrightarrow{f\otimes 1} A^- \otimes B^+ \otimes C^- \xrightarrow{1\otimes g} A^- \otimes B^- \otimes C^+. $$
Note that $\mathcal{C}$ embeds [[fully faithful functor|fully-faithfully]] in $Int(\mathcal{C})$ by sending $A$ to $(A,I)$, where $I$ is the unit object of the monoidal structure.

Furthermore, a traced monoidal category that is $\ast$-[[star-autonomous category|autonomous]] must already be compact closed; see [HH13](#HH13).

### In cartesian monoidal categories

For a [[cartesian monoidal category]], the existence of a trace operator is equivalent to the existence of a "parameterized" [[fixed point]] operator satisfying certain properties ([Hasegawa 1997](#Hasegawa97)). When the category is a [[Lawvere theory]], this is related to the notion of [[iteration theory]].

### Adding traces

Since any full monoidal subcategory of a traced monoidal category inherits a trace, not every monoidal category can be fully embedded into a traced monoidal category, and hence also not into a compact closed category.  In fact, Plotkin observed that there are monoidal categories that cannot even be *faithfully* mapped into a traced monoidal category.  This can be seen from the fact that a traced monoidal category has the "cancellation property" that if $f\otimes id_{X\otimes X} = g \otimes id_{X\otimes X}$ then $f\otimes id_X = g\otimes \id_X$ (since the latter is a trace of the former with one of the copies of $X\otimes X$ transposed), but not all monoidal categories have this property.

### Relation to feedback categories

[[feedback categories|Feedback categories]]  [KSW02](#KatisSabadiniWalters02) are a weakening of the axioms of traced monoidal categories. Feedback categories do not satisfy the yanking axiom and they only satisfy a weaker form of naturality on the object to be traced. Examples include some categories of automata.

### Categorical semantics

Traced monoidal categories serve as an "operational" [[categorical semantics]] for [[linear logic]], known as _[[Geometry of Interactions]]_. See there for more.

In this context the free compact closure $Int(\mathcal{C})$ from [above](#RelationToCompactClosedCategories) is sometimes called the _Geometry of Interaction construction_ and denoted $\mathcal{G}(\mathcal{C})$ ([Abramsky-Haghverdi-Scott 02, def. 2.6](#AbramskyHaghverdiScott02)).

## References

The concept was introduced in

* {#JoyalStreetVerity96} [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. **119** (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), [doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

A good exposition is found in

* {#Selinger2011} [[Peter Selinger]], _A survey of graphical languages for monoidal categories_, New structures for physics ([pdf](https://www.mscs.dal.ca/~selinger/papers/graphical-2up.pdf))

A characterization of trace structures on cartesian monoidal categories is given in

* {#Hasegawa97} [[Masahito Hasegawa]], _Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi_, Proc. 3rd International Conference on Typed Lambda Calculi and Applications (TLCA 1997). Springer LNCS1210, 1997 ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.31))

The equivalence between traces and parameterized fixed point operators appears as Theorem 3.1 (the author notes that this theorem was also proved independently by [[Martin Hyland]]).

Comprehensive discussion as a source for [[categorical semantics]] of the [[Geometry of Interactions]] is in

* {#AbramskyHaghverdiScott02} [[Samson Abramsky]], [[Esfandir Haghverdi]],  [[Philip Scott]], _Geometry of Interaction and Linear Combinatory Algebras_. MSCS, vol. 12(5), 2002, 625-665, CUP (2002) ([citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.7818))

The combination with [[star-autonomous category|star-autonomous]] structure was discussed in

* Tamás Hajgató and [[Masahito Hasegawa]], *Traced $\ast$-autonomous categories are compact closed*, [TAC](http://www.tac.mta.ca/tac/volumes/28/7/28-07abs.html), 2013
 {#HH13}

Feedback categories and their connection to traced monoidal categories were discussed in

* {#KatisSabadiniWalters02} Piergiulio Katis, Nicoletta Sabadini, [[Bob Walters]].  Feedback, trace and fixed-point semantics, 2002 ([Numdam](http://www.numdam.org/item/ITA_2002__36_2_181_0/)).

[[!redirects traced monoidal categories]]


tractable model category

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According to definition 1.21 and remark 1.23 in

* [[Clark Barwick]] , _On left and right model categories and left and right Bousfield localization_ ([pdf](http://www.math.harvard.edu/~clarkbar/complete.pdf))

a [[model category]] is called **tractable** if it is [[combinatorial model category|combinatorial]] and it has sets $I$ and $J$ of generating cofibrations and acyclic cofibrations, respectively, whose sources are all [[cofibrant object|cofibrant]].


Tractatus Logico-Philosophicus

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This page collects links related to the text

* [[Ludwig Wittgenstein]],

  _Logisch-Philosophische Abhandlung_ (1921), published in English as

  _Tractatus Logico-Philosophicus_ (1922)

  ([Wikipedia entry](http://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicus)).

The text deals with [[ontology]] and [[formal logic]] but, arguably, also is concerned with ethics as pointed out by ([Cora Diamond](#Diamond00)) and [Wittgenstein himself](#Wittgenstein19):

> der Sinn des Buches ist ein Ethischer. Ich wollte einmal in das Vorwort einen Satz geben, der nun tatsächlich nicht darin steht, den ich Ihnen aber jetzt schreibe, weil er Ihnen vielleicht ein Schlüssel sein wird: Ich wollte nämlich schreiben, mein Werk bestehe aus zwei Teilen:
aus dem, der hier vorliegt, und aus alledem, was ich _nicht_ geschrieben habe. Und gerade dieser zweite Teil ist der Wichtige. Es wird nämlich das Ethische durch mein Buch gleichsam von Innen her begrenzt; und ich bin überzeugt, dass es, _streng, nur so_ zu begrenzen ist.

> the book's point is ethical. I once meant to include in the preface a sentence which is not in fact there now, but which I will write out for you here, because it will perhaps be a key for you. What I meant to write then was this: my work consists of two parts: the one presented here plus all that I have _not_ written. And it is precisely this second part that is the important one. For the ethical gets its limit draw from the inside, as it were, by my book; and I am convinced that this is the _only rigorous_ way of drawing that limit.

This quote gives a taste of the problems encountered by any attempt to give an in-depth interpretation of Wittgenstein and the Tractatus in particular.
It therefore may not seem surprising that the Tractatus has been related to [[mysticism]], see ([McGuinnes66](#McGuinnes66)).

## Contents

The book states seven "propositions" (with sub-propositions):

1. The world is everything that is the case.

1. What is the case (a fact) is the existence of states of affairs.

1. A logical picture of facts is a thought.

1. A thought is a proposition with a sense.

   {#4.11} 4.11 The totality of true propositions is the whole of natural science (or the whole corpus of the natural sciences).

1. A proposition is a truth-function of elementary propositions. (An elementary proposition is a truth-function of itself.)

1. The general form of a proposition is the general form of a truth function, ...

1. Whereof one cannot speak, thereof one must be silent.

## Formal Logic

Wittgenstein suggest a new formal language that he puts in contrast to Russell's type theoretic system elaborated in the famous [[Principia Mathematica]], which he criticizes.
Russell in turn took Wittgenstein's suggestions serious giving an interpretation of them, first, in his introduction to the English translation of the Tractatus, which in the Anglo-Saxon reception has become almost a part of early Wittgenstein's magnum opus itself, and, second, in an appendix to a later edition of the Principia Mathematica.

The meaning of Wittgenstein's formal system remains disputed to this day, e.g. ([Wehmeier11](#Wehmeier11)).

## References

An expandable/collapsable online version (Pears/McGuinness English translation) is at

* [tractatuslogico-philosophicus.com](http://www.tractatuslogico-philosophicus.com/)

A side-by-side version of the German original and the Ogden/Ramsey and Pears/McGuinness English translations is at

* [Ludwig Wittgenstein: Tractatus Logico-Philosophicus Side-by-Side-by-Side Edition](http://people.umass.edu/klement/tlp/)

An edition which emphasises the tree structure of the work is

* [The Tractatus According to Its Own Form](http://www.lulu.com/spotlight/wittgensteinL) edited by Lucianno Bazzocchi,its [preface](http://www.bazzocchi.net/wittgenstein/tractatus/eng/preface.pdf) by Giovanna Corsi, and an [online version](http://www.bazzocchi.net/wittgenstein/tractatus/eng/index.htm).

* {#Diamond00}Cora Diamond, _Ethics, imagination and the method of Wittgenstein's Tractatus_

{#Wittgenstein19}Wittgenstein wrote that in a letter to Ludwig von Ficker in 1919, quoted from

* Joachim Schulte, _Wittgenstein -- Eine Einführung_, 1989

* Brian McGuinnes, _The mysticism of the tractatus_, 1966, published several times (including a German translation) for instance more recently as a chapter _Mysticism_ in

* Brian McGuinnes, _Approaches to Wittgenstein_, 2002

* {#Russell10}Bertrand Russell, _[[Principia Mathematica]]_, 1910

* Kai F. Wehmeier, _Wittgensteinian Predicate Logic_ , Notre Dame J. Formal  Logic **45** no.1 (2004) pp.1-11. ([projecteuclid](http://projecteuclid.org/euclid.ndjfl/1094155275))

* {#Wehmeier11}Wehmeier, Vortrag _The First-Order Logic of the Tractatus_, [Video](https://www.youtube.com/watch?v=e75U1fgBC6g)

category: reference


tractor bundle

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## Idea

Tractor bundles are certain [[associated bundles]] considered in
[[conformal geometry]] and more generally in [[parabolic geometry]].

Even more generally, given a [[subgroup]] inclusion $H \hookrightarrow G$ of [[Lie groups]] and given a linear [[representation]] of $G$ on some [[vector space]] $V$, with [[restricted representation]] by $H$, then the operation of forming [[associated bundles]]

$$
  P \mapsto P \underset{H}{\times} V
$$

to $H$-[[principal bundles]] $P$ constitutes a construction of [[natural vector bundles]] on the category of $(H\to G)$-[[Cartan geometries]]. For the case that $H \hookrightarrow G$ is a [[parabolic subgroup]], hence for the case of [[parabolic geometry]], and the case that $P \to X$ is the $H$-[[frame bundle]] of an $(H \to G)$ [[Cartan geometry]] on a manifold $X$, then these [[associated bundles]] $P \underset{H}{\times} V \to X$ are called _tractor bundles_ (e.g [Čap-Souček 07, p. 11](#CapSoucek07)), often denoted

$$
  V X \to X
  \,.
$$

These tractor bundles carry special [[connection on a bundle|connections]] and as such are called _tractor connections_ ([Čap-Gover 02](#CapGover02)).

Some $G$-representations are special and hence some tractor bundles are singled out (e.g. [Čap-Slovák 09, 1.5.7](#CapSlovak09)):

* For the case that $V = \mathfrak{g}$ is the [[Lie algebra]] of $G$ equipped with its [[adjoint action]], then

  $$
    \mathcal{A}X \coloneqq P \underset{H}{\times} \mathfrak{g}
  $$

  is called the _adjoint tractor bundle_. A key point is that for [[parabolic geometries]] the Lie algebra structure is retained by this bundle in that $\mathcal{A}X$ is a bundle of [[Lie algebras]], hence its [[space of sections]] $\Gamma(\mathcal{A}X)$ carries a [[Lie bracket]] $\{-,-\}$ which makes each [[fiber]] [[isomorphism|isomorphic]] to $\mathfrak{g}$ not just as a vector space, but as [[Lie algebra]]. In particular therefore for any representation $V$ then the sections of the associated tractor bundle $V X$ carry an [[action]] by sections of $\mathcal{A}X$.

* By the axioms on [[Cartan geometries]] (see also at [[Cartan connection]]) the tractor bundle corresponding to the [[quotient]] $\mathfrak{g}/\mathfrak{h}$ is [[isomorphism|isomorphic]] to the [[tangent bundle]] $T X$.

* The inclusion $\mathfrak{p}\hookrightarrow \mathfrak{g}$ of the [[Lie algebra]] of the [[parabolic subgroup]] induces a tractor subbundle $P \underset{H}{\times} \mathfrak{p} \hookrightarrow P \underset{H}{\times}\mathfrak{g}$ denoted

  $$
    \mathcal{A}^0 X \hookrightarrow \mathcal{A}X
  $$

  (e.g [Čap-Souček 07, p. 5](#CapSoucek07)). The corresponding [[quotient]] is again the [[tangent bundle]]

  $$
    \mathcal{A}X/\mathcal{A}^0 X = P\underset{H}{\times} \mathfrak{g}/\mathfrak{p} \simeq T X
  $$

  (e.g [Čap-Souček 07, p. 12](#CapSoucek07)).

* Dually, the tractor bundle associated to $(\mathfrak{g}/\mathfrak{p})^\ast \simeq \mathfrak{p}_+$ is $\mathcal{A}^1 X $ which is [[isomorphism|isomorphic]] to the [[cotangent bundle]] $T^\ast X$.

  $$
    \mathcal{A}^1 X \simeq T^\ast X
    \,.
  $$

* From this one gets that [[differential forms]] with values in the tractor bundle of $V$ are equivalently sections of the tractor bundle given by the [[tensor product]] representation $\wedge^k \mathfrak{p}_+ \otimes V$

  $$
    \Omega^k(X, V X) \simeq \Gamma( P \underset{H}{\times} \wedge^k \mathfrak{p}_+ \otimes V )
    \,.
  $$

* The [[differential]] in the [[Lie algebra homology]] of $\mathfrak{p}_+$ with [[coefficients]] in $V$ is a [[natural transformation]]

  $$
    \wedge^{k+1} \mathfrak{p}_+ \otimes V \longrightarrow \wedge^{k} \mathfrak{p}_+ \otimes V
  $$

  (often called the _Kostant codifferential_ $\partial^\ast$ is this parabolic context) and hence gives homomorphisms of natural bundles

  $$
    \partial^\ast_k \colon \Lambda^{k+1} T^\ast X \otimes_X V X \longrightarrow \Lambda^{k} T^\ast X \otimes_X V X
  $$

  This yields a [[complex]] of bundles whose [[chain homology]] is fiberwise the [[Lie algebra homology]], hence $ker(\partial^\ast)/im(\partial^\ast)$ is the bundle associated to the [[Lie algebra homology]] group of $\mathfrak{p}_+$ regarded as an $H$-representation:

  $$
    ker(\partial^\ast_{k-1})/im(\partial^\ast_k)
    \simeq
    P \underset{H}{\times} H_k(\mathfrak{p}_+, V)
  $$

  (e.g [Čap-Souček 07, p. 17](#CapSoucek07))

  For homogeneous Cartan geometries $X = G/H$ this gives a geometric construction  of [[BGG resolutions]] of representations, for general Cartan geometries it gives a curved generalization of that.

## Related concepts

* [[Fefferman-Graham ambient construction]]

## References

* {#CapGover02} [[Andreas Čap]], A. Rod Gover, _Tractor calculi for parabolic geometries_, Trans. Amer. Math. Soc. 354 (2002), 1511–1548.

* {#CapSoucek07} [[Andreas Čap]], [[Vladimír Souček]], _Curved Casimir operators and the BGG machinery_, SIGMA __3__, 2007, 111 ([arxiv:0708.3180](http://arxiv.org/abs/0708.3180) [doi](http://dx.doi.org/10.3842/SIGMA.2007.111))

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]],  _Parabolic Geometries I -- Background and General Theory_, AMS 2009

* Sean Curry, A. Rod Gover, _An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity_, 2014 ([arXiv:1412.7559](http://arxiv.org/abs/1412.7559))

* Wikipedia, _[Tractor bundle](http://en.wikipedia.org/wiki/Tractor_bundle)_

[[!redirects tractor bundles]]


traditional orbifold cohomology -- references

### Traditional orbifold cohomology

Traditionally, the [[cohomology]] of [[orbifolds]]  has, by and large, been taken to be simply the [[ordinary cohomology]] of (the plain [[homotopy type]] of) the [[geometric realization of simplicial topological spaces|geometric realization]] of the [[topological groupoid|topological]]/[[Lie groupoid]]  corresponding to the orbifold.

For the [[global quotient orbifold]] of a [[topological G-space|G-space]] $X$, this is the [[ordinary cohomology]] of (the bare [[homotopy type]] of) the  [[Borel construction]] $X \!\sslash\! G \;\simeq\; X \times_G E G $, hence is _[[Borel cohomology]]_ (as opposed to finer versions of [[equivariant cohomology]] such as [[Bredon cohomology]]).

A dedicated account of this Borel cohomology of orbifolds, in the generality of [[twisted cohomology]] (i.e. with [[local coefficients]]) is in:

* {#MoerdijkPronk99} [[Ieke Moerdijk]], [[Dorette Pronk]], _Simplicial cohomology of orbifolds_, Indagationes Mathematicae Volume 10, Issue 2, 1999, Pages 269-293 (doi:10.1016/S0019-3577(99)80021-4)

Moreover, since the orbifold's [[isotropy groups]] $G_x$ are, by definition, [[finite groups]], their [[classifying spaces]] $\ast \!\sslash\! G \simeq B G$ have purely [[torsion subgroup|torsion]] [[integral cohomology]] in [[positive number|positive]] degrees, and hence become indistinguishable from the [[point]] in [[rational cohomology]] (and more generally whenever their [[order of a group|order]] is [[unit|invertible]] in the [[coefficient]] [[ring]]).

Therefore, in the special case of [[rational cohomology|rational]]/[[real cohomology|real]]/[[complex cohomology|complex]] [[coefficients]], the traditional orbifold Borel cohomology reduces further to an invariant of just (the [[homotopy type]] of) the naive quotient underlying an orbifold. For global quotient orbifolds  this is the [[topological quotient space]] $X/G$.

In this form, as an invariant of just $X/G$, the [[real cohomology|real]]/[[complex cohomology|complex]]/[[de Rham cohomology]] of orbifolds was originally introduced in

* {#Satake56} [[Ichiro Satake]], _On a generalisation of the notion of manifold_, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 ([doi:10.1073/pnas.42.6.359](https://doi.org/10.1073/pnas.42.6.359))

following analogous constructions in

* [[Walter Lewis Baily]], _On the quotient of an analytic manifold by a group of analytic homeomorphisms_, PNAS 40 (9) 804-808 (1954) ([doi:10.1073/pnas.40.9.804](https://doi.org/10.1073/pnas.40.9.804))

Since this traditional [[rational cohomology]] of orbifolds does, hence, not actually reflect the specific nature of orbifolds, a proposal for a finer notion of orbifold cohomology was famously introduced (motivated from orbifolds as [[target spaces]] in [[string theory]], hence from orbifolding of [[2d CFTs]]) in

* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))

However, [[Chen-Ruan cohomology]] of an orbifold $\mathcal{X}$ turns out to be just Borel cohomology with rational coefficients, hence is just Satake's coarse cohomology -- but applied to the [[inertia orbifold]] of $\mathcal{X}$. A review that makes this nicely explicit is (see p. 4 and 7):

* {#Clader14} Emily Clader, _Orbifolds and orbifold cohomology_, 2014 ([pdf](http://www-personal.umich.edu/~eclader/OctLect1.pdf))

Hence [[Chen-Ruan cohomology]] of a global quotient orbifold is equivalently the [[rational cohomology]] ([[real cohomology]],  [[complex cohomology]]) for the [[topological quotient space]] $AutMor(X\!\sslash\!G)/G$ of the space of [[automorphisms]] in the [[action groupoid]] by the $G$-[[conjugation]] [[action]].

On the other hand, it was observed in (see p. 18)

* {#Moerdijk02} [[Ieke Moerdijk]], _Orbifolds as Groupoids: an Introduction_, [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]] (eds.) _[[Orbifolds in Mathematics and Physics]]_, Contemporary Math 310 , AMS (2002), 205–222 ([arXiv:math.DG/0203100](http://arxiv.org/abs/math.DG/0203100))

that for global quotient orbifolds Chen-Ruan cohomology indeed is equivalent to a $G$-equivariant [[Bredon cohomology]] of $X$ -- for one specific choice of [[equivariant cohomology|equivariant]] [[coefficient]] system ([[abelian sheaf]] on the [[orbit category]] of $G$), namely for $G/H \mapsto ClassFunctions(H)$.

Or rather, [Moerdijk 02, p. 18](#Moerdijk02) observes that the Chen-Ruan cohomology of a global quotient orbifold is equivalently the [[abelian sheaf cohomology]] of the naive quotient space $X/G$ with [[coefficients]] in the [[abelian sheaf]] whose [[stalk]] at $[x] \in X/G$ is the [[ring]] of [[class functions]] of the [[isotropy group]] at $x$; and then appeals to Theorem 5.5 in

* {#Honkasalu90} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology for actions of compact Lie groups_, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 ([jstor:24492569](https://www.jstor.org/stable/24492569))

for the followup statement that the [[abelian sheaf cohomology]] of $X/G$ with coefficient sheaf $\underline{A}$ being "[[locally constant sheaf|locally constant]] except for dependence on [[isotropy groups]]" is equivalently [[Bredon cohomology]] of $X$ with coefficients in $G/H \mapsto \underline{A}_x$ for $Isotr_x = H$. This identification of the coefficient systems is Prop. 6.5 b) in:

* {#Honkasalo88} [[Hannu Honkasalo]], _Equivariant Alexander-Spanier cohomology_,  Mathematica Scandinavia,  63, 179-195, 1988 ([doi:10.7146/math.scand.a-12232](https://doi.org/10.7146/math.scand.a-12232))

See also Section 4.3 of

* {#SzaboValentino07} [[Richard Szabo]], [[Alessandro Valentino]], _Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory_, Commun. Math. Phys. 294:647-702, 2010 ([arXiv:0710.2773](https://arxiv.org/abs/0710.2773))

In summary:

* Traditional orbifold cohomology theory is [[Borel cohomology]] of underlying [[Borel construction]]-spaces, and reduces [[rational cohomology|rationally]] further to the [[rational cohomology]] of underlying naive [[quotient spaces]].

* [[Chen-Ruan cohomology]] is just the latter rational cohomology but applied after passage to the [[inertia orbifold]]. This is equivalent to the [[Bredon cohomology]] of the original orbifold, for one specific [[equivariant cohomology|equivariant coefficient]]-system.

This suggests, of course, that more of [[proper equivariant homotopy theory|proper]] [[equivariant cohomology]] should be brought to bear on a theory of orbifold cohomology. A partial way to achieve this is to prove for  a given [[equivariant cohomology]]-theory that it descends from an invariant of [[topological G-spaces]] to one of the associated [[global quotient orbifolds]].

For [[topological K-theory|topological]] [[equivariant K-theory]] this is the case, by

* {#PronkScull07} [[Dorette Pronk]], [[Laura Scull]], Prop. 4.1 in: _Translation Groupoids and Orbifold Bredon Cohomology_, Canad. J. Math. 62(2010), 614-645 ([arXiv:0705.3249](https://arxiv.org/abs/0705.3249), [doi:10.4153/CJM-2010-024-1](https://doi.org/10.4153/CJM-2010-024-1))

Therefore it makes sense to _define_ [[orbifold K-theory]] for [[orbifolds]] $\mathcal{X}$ which are equivalent to a [[global quotient orbifold]]
$
  \mathcal{X} \simeq \prec(X \!\sslash\! G)
$
to be the $G$-[[equivariant K-theory]] of $X$:
$
  K^\bullet(\mathcal{X})
  \;\coloneqq\;
  K_G^\bullet(X)
  \,.
$

This is the approach taken in

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], Section 3 of: _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))

Exposition and review of traditional orbifold cohomology, with an emphasis on [[Chen-Ruan cohomology]] and [[orbifold K-theory]], is in:

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))


transcendence degree

Let $k$, $E$ be [[fields]], with $k \hookrightarrow E$ a [[field extension]], also written as $E/k$. The _transcendence degree_ of $E/k$ is the [[cardinality]] of a maximal set of elements of $E$ that are [[matroid|algebraically independent]] over $k$.

The transcendence degree is well-defined, i.e., independent of which maximal set of algebraically independent elements is used. This is often proven by invoking a Mac Lane-Steinitz *exchange condition*; see [[matroid]] for a general argument.


transcendental element

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## Definition

Given a [[field]] $K$ and a [[field extension]] $K \subseteq F$, an element $\alpha \in F$ is **transcendental** if for all [[polynomial functions]] $f$ with [[coefficients]] in $K$, if $f(\alpha) = 0$ then $f$ is the zero polynomial.

Alternatively, an element $\alpha \in F$ is **transcendental** if the [[field extension]] $K(\alpha) \subseteq F$ is isomorphic to the [[field of fractions]] $K(x)$ of the generic [[polynomial ring]] $K[x]$.

### In constructive mathematics

These two notions coincide in [[classical mathematics]]. They also coincide in [[constructive mathematics]] if $F$ and $K$ are both [[discrete fields]]. However, they are different in [[constructive mathematics]] if it is not the case that $F$ and $K$ are both discrete fields. If $K$ is only a [[Heyting field]], the first notion is weaker and so is called **weakly transcendental**, while the second notion is stronger and so is called **strongly transcendental** or **strictly transcendental**.

## Examples

* If $K$ is the rational numbers $\mathbb{Q}$ and $F$ is the complex numbers $\mathbb{C}$, then the transcendental elements in $\mathbb{C}$ are precisely the [[transcendental numbers]].

* If $K$ is the rational numbers $\mathbb{Q}$ and $F$ is the p-adic complex numbers $\mathbb{C}_p$, then the transcendental elements in $\mathbb{C}_p$ are precisely the transcendental [[p-adic numbers]].

## Related concepts

* [[algebraic element]]

* [[transcendental number]]

* [[transcendental extension]]

## References

See also

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

[[!redirects transcendental element]]
[[!redirects transcendental elements]]

[[!redirects weakly transcendental element]]
[[!redirects weakly transcendental elements]]

[[!redirects strongly transcendental element]]
[[!redirects strongly transcendental elements]]

[[!redirects strictly transcendental element]]
[[!redirects strictly transcendental elements]]


transcendental extension

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## Idea

A kind of [[field extension]]

## Related concepts

* [[algebraic extension]]

## References

* Wikipedia, _[Transcendental extension](http://en.wikipedia.org/wiki/Transcendental_extension)_

[[!redirects transcendental extension]]
[[!redirects transcendental extensions]]


transcendental grammar

[[!redirects Transcendental grammar]]

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**Transcendental grammar** is the name given to applications of ideas inspired by Kant's trancendental philosophy to the theory of grammar. In a looser sense "transcendental linguistics" could be applied to any theory of language in the sphere around Kantian philosophy e.g. the language based criticism of Kant's account of pure reason by Hamann and Herder, the theory of language of [[Wilhelm von Humboldt]] or even the linguistic turn of Neokantianism in the writings of [[Ernst Cassirer]] and the early Walter Benjamin.

## Related entries

* [[Critique of Pure Reason]]

* [[Immanuel Kant]]

* [[Stephan Wannowski]]

* [[linguistics]]

* [[transcendental syntax]]

## References

* August Ferdinand Bernhardi, _Anfangsgründe der Sprachwissenschaft_ , Frölich Berlin 1805.

* [[Johann Fichte|Johann Gottlieb Fichte]], _Von der Sprachfähigkeit und dem Ursprung der Sprache_ , pp.301-341 in I. H. Fichte (ed.), _Fichte's Werke *VII*_ , de Gruyter Berlin 1971.

* Johann Georg Hamann, _Metakritik über den Purismum der Vernunft_ , pp.205-212 in Majetschak (ed.), _Vom Magus im Norden - Ein Hamann-Brevier_ , dtv München 1988.

* Jochem Hennigfeld, _Schellings Philosophie der Sprache_ , Philosophisches Jahrbuch **91** (1984) pp.16–29. ([pdf](http://philosophisches-jahrbuch.de/wp-content/uploads/2019/03/PJ91_S16-29_Hennigfeld_Schellings-Philosophie-der-Sprache.pdf))

* Wolfram Hogrebe, _Kant und das Problem einer transzendentalen Semantik_ , Fink München 1974.

* [[Immanuel Kant]], _Prolegomena zu einer jeden künftigen Metaphysik_ , Meiner Hamburg 2001[1783]. (§39, p.323)

* Wilhelm Lütterfels, _Kant in der gegenwärtigen Sprachtheorie_ , pp.150-176 in Heidemann, Engelhard (eds.), _Warum Kant heute?_ , de Gruyter Berlin 2003.

* [[Salomon Maimon]], _Versuch über die Transzendentalphilosophie_ , Meiner Hamburg 2004[1790]. (pp.263-333)

* [[Karl Leonhard Reinhold]], _Das menschliche Erkenntnißvermögen, aus dem Gesichtspunkte des durch die Wortsprache vermittelten Zusammenhangs zwischen Sinnlichkeit und dem Denkvermögen_ , Akademische Buchhandlung Kiel 1816. ([wikisource](https://de.wikisource.org/wiki/Karl_Leonhard_Reinhold))

* Manfred Riedel, _Sprache und Vernunft. Grundmodell der transzendentalen Grammatik in Kants Lehre vom Kategoriengebrauch_ , pp.44-60 in Riedel, _Urteilskraft und Vernunft_ , Suhrkamp Frankfurt am Main 1989.

* Jürgen Villiers, _Kant und das Problem der Sprache - Die historischen und systematischen Gründe für die Sprachlosigkeit der Transzendentalphilosophie_ , Verlag am Hockgraben Konstanz 1997.

* Anonymous (i.e. [[Stephan Wannowski]]), _Philosophische Principien einer allgemeinen Sprachlehre nach Kant und Sacy, in einer ausführlichen Recension der Grundsätze des Letztern_ , Nicolovius Königsberg 1805. ([online version Bayrische Staatsbibliothek](https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10583079_00005.html))


transcendental ideal

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## Idea

The **transcendental ideal** is the embodiment of all predicative possibilities - _Inbegriff aller Möglichkeiten_ - in Kant's transcendental philosophy and as such it reconstructs the role of the concept of 'ens perfectissimum' in the ontological proof of the existence of god in the rational theology of Leibniz and Wolff.

Kant's innovation with respect to the tradition is that he suspends existential import and substantiality of the concept and gives a thoroughly (meta)logical derivation instead which exhibits the functional role of the ideal as a 'guideline' in all reasoning.

This relocation of theology from ontology to logic proved to be consequential in the philosophy of the world age of the mature Schelling where it becomes a key concept ([Hogrebe 1989](#Hogrebe89)), as well as (indirectly) on the overall architectonic structure of Hegel's system as a self determination of the idea.

## Related entries

* [[Critique of Pure Reason]]

* [[infinite judgement]]

* [[absolute conclusion]]

* [[construction in philosophy]]

* [[Aufhebung]]

* [[Science of Logic]]

## References

* P. Baumanns, _Kants vierte Antinomie und das Ideal der Vernunft_, Kant-Studien **79** (1988) pp.183-200.

* J. Ferrari, _Das Ideal der reinen Vernunft_, pp.391-424 in Mohr, Willaschek (eds.), _Klassiker Auslegen - Immanuel Kant: Kritik der reinen Vernunft_, Akademie Verlag Berlin 1998.

* {#Hoeffe03} O. Höffe, _Kants Kritik der reinen Vernunft_, Beck München 2003. (pp.258-60)

* {#Hogrebe89} W. Hogrebe, _Prädikation und Genesis_, Suhrkamp Frankfurt a. M. 1989.

* {#KdrV1} [[Immanuel Kant|I. Kant]], _[[Critique of Pure Reason|Kritik der reinen Vernunft 2]]_, Suhrkamp Frankfurt a. M. 1985[1781, rev. 1787]. (B 600-11/ A 572-84)

* S. Kovač, _The totality of predicates and the possibility of the most real being_ , Journal of Applied Logics—IfCoLog Journal of Logics and their Applications **5** no. 7 (2018) pp.1523-1552. ([pdf](http://www.collegepublications.co.uk/downloads/ifcolog00027.pdf))

* P. Rohs, _Kants Prinzip der durchgängigen Bestimmung alles Seienden_, Kant-Studien **69** (1978) pp.170-180.

* {#Rohs96}P. Rohs, _Wahrnehmungsurteile und Erfahrungsurteile_, pp.166-189 in Schönrich, Kato (eds.), _Kant in der Diskussion der Moderne_, Suhrkamp Frankfurt a. M. 1996. (pp.182ff)

* F. W. J. Schelling, _Zweites Buch. Philosophische Einleitung in die Philosophie der Mythologie oder Darstellung der reinrationalen Philosophie (Zwischen 1847 und 1852)_, pp.263-582 in Schelling, _Ausgewählte Werke Band 5_, Suhrkamp Frankfurt a. M. 1985. (Lecture 12, pp.294ff)

* H. Sidgwick, _Lectures on the Philosophy of Kant and Other Philosophical Lectures and Essays_, Macmillan London 1905.

* N. Stang, _Kant on Complete Determination and Infinite Judgement_, Brit. J. Hist. Phil. **20** no. 6 (2012) pp.1117-1139.


transcendental number

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## Idea

A **transcendental number** is a [[complex number]] for which the only [[polynomial]] with [[rational number|rational]] [[coefficients]] that has the number as a [[root]] is the zero polynomial. Equivalently in [[classical mathematics]], it is a number which is not a root of a non-zero polynomial with rational coefficients, or a number which is not equal to any [[algebraic number]].

## Definition

An complex number $\alpha \in \mathbb{C}$ is **weakly transcendental** if the only [[polynomial function]] with [[rational number|rational]] [[coefficients]] $f \in \mathbb{Q}[x] \subseteq \mathbb{C}^\mathbb{C}$ such that $f(\alpha)$ is equal to zero is the constant polynomial function at zero.

An complex number $\alpha \in \mathbb{C}$ is **strictly transcendental** if the field extension $\mathbb{Q}(\alpha) \subseteq \mathbb{C}$ is [[isomorphic]] to the [[field of fractions]] $\mathbb{Q}(x)$ of the generic [[polynomial ring]] $\mathbb{Q}[x]$. Equivalently, $\alpha \in F$ is **strictly transcendental** if the complex absolute value of the difference of $\alpha$ and every algebraic number is positive:

$$\mathrm{isTranscendental}(\alpha) \coloneqq \forall \beta \in \mathbb{C}.(\beta \in \overline{\mathbb{Q}}) \Rightarrow (\vert \alpha - \beta \vert \gt 0)$$

These two notions coincide in [[classical mathematics]], with both concepts just called **transcendental**; however, they are different in [[constructive mathematics]].

## Examples

Famous examples are the base ($\mathrm{e} = 2.7\ldots $) and period ($2 \pi \mathrm{i} = 6.28\ldots \mathrm{i}$, or equivalently $\pi = 3.14\ldots $) of the natural [[logarithm]] in the [[complex numbers]] with its [[Archimedean absolute value]].

## Related concepts

* [[absolute value]]

* [[algebraic number]]

* [[irrational number]]

* [[transcendental element]]

## References

See also

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

[[!redirects transcendental number]]
[[!redirects transcendental numbers]]

[[!redirects weakly transcendental number]]
[[!redirects weakly transcendental numbers]]

[[!redirects strictly transcendental number]]
[[!redirects strictly transcendental numbers]]


transcendental syntax

#Contents#
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{:toc}

## Idea

"Transcendental syntax" ([Girard 13](#Girard13)) is the name of a proposal (or maybe a pamphlete) by [[Jean-Yves Girard]] which means to rethink fundamental aspects of [[formal logic]], of [[syntax]]/[[semantics]]. According to Girard, _[[linear logic]]_ and _[[Geometry of Interaction]]_ are but exercises in transcendental syntax ([Girard 13b](#Girard13b)).

While Girard's prose is notoriously demanding, exegesis may be found in ([Abrusci-Pistone 12](#AbrusciPistone12)) for the philosophical side and ([Rouleau 13](#Rouleau13)) which was written before the more modern developements of transcendental syntax by means of stars and constellations ([Girard 13b](#Girard13b)).

The positionning of transcendental syntax in logic is summarised by Girard's four levels of understanding of proofs, in which the last level corresponds to the transcendental syntax:

> Girard describes four levels of [[semantics]]: alethic, functional, interactive, and deontic. They descend into the depths of meaning, and thus are numbered from -1 to -4. The negatively first, alethic, is the layer of truth or models. The negatively second, functional, is the layer of functions or categories. The negatively third, interaction, is the layer of games or game semantics. The negatively fourth, deontic, is the layer of normativity or formatting. ([Equivalent eXchange](http://equivalentexchange.wordpress.com/2012/03/07/j-y-girards-transcendental-syntax/))

### Philosophical interpretation

The transcendental syntax starts from the strong claim that [[formal logic]] makes a lot of rather arbitrary choices such as the so-called *rules of logic* (see [[deductive systems]]) and that it should be possible to study the mechanisms of reasoning without these assumptions (to which Girard refers to as being the "prejudices" of logic).

The solution presented by the transcendental syntax can be understood as an update of [[Kant]]'s epistemology by taking *computation* into account through recent works in [[linear logic]] and the [[Curry-Howard correspondence | Curry-Howard-Lambek correspondence]]. It suggests the division of the whole logical activity into four concepts ([Girard 16](#Girard13b)):

|                             | Analytic / Answers | Synthetic / Questions |
|-----------------------------|--------------------|-----------------------|
| **A posteriori / Explicit** | Constat            | Usine                 |
| **A priori / Implicit**     | Performance        | Usage                 |

The point of view of the transcendental syntax is that logic studies the relationship between questions (formulas) which are subjective and their answers (proofs) which are objective, by means of finite objects (because of the intuition that reasoning should be finite and verifiable). In order to do so with as less assumptions about logic as possible, we should start by defining the *answers*, seen as neutral and objective. Then we study the sufficient conditions making the logical concepts (proofs, formulas, logical correctness) emerge from the meaningless. In reference to [[Kant]], Girard uses the expression ``conditions of possibility of language".

The **answers** corresponds to a space of general computational objects called *epistates* which can be evaluated (performance) into a [[normal form]] (constat). This is the space where proofs are formulated but not yet considered logically meaningful or correct.

The space of **questions** considers two alternative definitions of *meaning* corresponding to [[formula | formulas]] or [[type | types]] called *dichology*:

- the use (usage): it corresponds to [[Ludwig Wittgenstein]] meaning-as-use. The computational behavior of epistates and their potential interaction with other ones induces a classification into dichologies.

- the factory (usine): we want our computational objects to pass a specific set of *tests* in order to accept them in the corresponding type.

This is related to the distinction between existentialism and essentialism in philosophy. The **logical certainty** corresponds to the adequacy between usine and usage: we would like the usine to be finite and sufficient to ensure what we consider a right use of epistates. According to Girard [Girard 16](#Girard16), the separation between constat and performance comes from undecidability (as in the halting problem) because the potential of programs cannot always be reduced to their result (which is not always defined). As for the separation between usine and usage, it would come from Gödel's [[incompleteness theorem]] which intuitively means that the possible uses of a logical object may go beyond what we expect from our definition (our formatting of the concept of logic).

### Technical interpretation

Technically speaking, the transcendental syntax is basically a finitary improvement of the [[Geometry of Interaction]]. Some preliminary works include *flows* ([Girard 95](#Girard95), [Bagnol 14](#Bagnol14)) or *interaction graphs* ([Seiller 16](#Seiller16)). The transcendental syntax generalizes and extends both by taking into account [[proof net | proof nets']] logical correctness in a more general form.

The elementary objects we consider are *constellations*, defined as multisets of clauses containing first-order terms. Two constellations can interact by using the dynamics of Robinson's resolution ([Robinson 65](#Robinson65)). We use constellations to encode [[proof net | proof structures]], their [[cut rule | cut elimination]] but also the correction graphs asserting logical correctness.

We can obtain two alternative notions of typing:

- By using [[realizability]] techniques for linear logic as in [[ludics]] or Seiller's interaction graphs, we can define [[formula | formulas]] called *behaviors*. This corresponds to types "à la Curry" where untyped computational objects are typed. In this case, types are *designed* and not defined as primitive objects.

- Inspired by the logical correctness of [[proof net | proof nets]], we can define types as a finite sets of tests (encoded as constellations). If a constellation passes all tests, it is considered as part of the corresponding type. This corresponds to types "à la Church" where types exist before the computational objects being typed. In the context of [[proof net | proof nets]], it corresponds to testing proof structures against correction graphs.

### Getting rid of semantics

An important ambition of the transcendental syntax is to make assumptions computationally explicit, and to "put everything on the table". In the transcendental syntax, it is no more possible to write a proof as usual. A proof necessarily comes as a hybrid object $(\Phi, \mathcal{T})$ where $\Phi$ is a constellation (elementary computational object) and $\mathcal{T}$ is a set of constellations corresponding to tests asserting the logical correctness of $\Phi$. The constellation $\Phi$ has to satisfy a given criterion of orthogonality when interacting with $\Phi' \in \mathcal{T}$, written $\Phi \bot \Phi'$. Logical objects only exist as computationally justified constructions.

Following this approach and by choosing a right notion of orthogonality, it is possible to construct axiom-free systems. In transcendental syntax's fourth paper ([Girard 20](#Girard20)), Girard sketches a reconstruction of [[Peano arithmetic]]. An essential point which allows us to actually get rid of semantic explanations is that both tests and tested are of the same kind, have the same role and interact in a symmetric way. The "boundaries" of logic then rely on a given notion of orthogonality and not on a semantic system deciding what is logical or not.

This might be compared with other approaches in [[type theory]] such as [[computational type theory]] or [[cubical type theory]] where some logical constructions are not primitive anymore but reconstructed. A major difference seems to be the starting point and the primitives considered.

All these approaches, including Girard transcendental syntax start from the assumption that computation can fully explain logic and are thus extensions of the [[computational trinitarianism | Curry-Howard-Lambek correspondence]].

### Transcendental syntax as an instance of realizability

Although never explicitly mentioned by Girard, at the technical level, the transcendental syntax is indeed very close to [[realizability | realizability theory]]. In realizability theory, types are designed from a model of computation (for instance, natural numbers and recursive functions or lambda-terms with beta-reduction). But while realizability theory is more focussed on speaking about logic with computation, the transcendental syntax describes computation with logic instead. Moreover, the transcendental syntax can speak about logical correctness and is able to reconstruct linear logic in a convenient way. Note that other alternative approaches for the reconstruction of linear logic exist such as Beffara's concurrent realizability ([Beffara 06](#Beffara06)).

### Transcendental syntax as a toolbox for computer science

The transcendental syntax generalizes, extends or is simply related to several important aspects of computer science (although no direct link has been studied yet):

- in [[descriptive complexity]], we are interested in capturing classes with fragments of logic. This is something which can be considered in the transcendental syntax. Moreover, Girard's third paper on transcendental syntax ([Girard 18](#Girard18)), a reconstruction of predicate and second order logic are sketched, which are essential in descriptive complexity.

- In [[certified programming | model checking and program specification]], we are interested in designing types as specifications or capturing properties of a model of computation with formulas in order to reason about a system. Girard's stars and constellations used in the transcendental syntax actually generalizes state machines and types can be constructed with realizability techniques. A logical system can then be extracted from these techniques.

- In *unit testing*, we are interested in testing a program against a finite number of values in order to assert that the program has a specific behavior and that it is sound. By generalizing the idea of correctness criterion coming from the theory of proof-nets, in the transcendental syntax, it is possible encode tests and programs as objects of the same kind. Both can be typed and testing extends to any model of computation.

## Related entries

* [[realizability]]

* [[proof net]]

* [[geometry of interaction]]

* [[transcendental grammar]]

* [[Critique of Pure Reason]]

* [[computational type theory]]

* [[cubical type theory]]

* [[type theory]]

* [[ludics]]

* [[computational trinitarianism]]

* [[certified programming]]

## References

* {#Robinson65} [[JA Robinson]], _A machine-oriented logic based on the resolution principle_, 1965.

* {#Girard95} [[Jean-Yves Girard]], _Geometry of interaction III: accommodating the additives_, 1995 ([pdf](https://girard.perso.math.cnrs.fr/GOI3.pdf)).

* {#Bagnol14} [[Marc Bagnol]], _On the Resolution Semiring_, PhD Thesis 2014 ([pdf](https://www.normalesup.org/~bagnol/phd/these_screen.pdf))

* {#Seiller16} [[Thomas Seiller]], _Interaction Graphs: Full Linear Logic_, 2016 ([pdf](https://www.seiller.org/documents/articles/IG-Full.pdf))

* {#Girard13b} [[Jean-Yves Girard]], _Geometry of Interaction VI: a Blueprint for Transcendental Syntax_, 2013 ([CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.303.7141))

* {#Girard13} [[Jean-Yves Girard]], _Transcendental syntax 2.0_, 2013 ([pdf](https://pdfs.semanticscholar.org/8548/a157279b27de84d1effd772b683c7b9d7701.pdf))

* {#Girard16} [[Jean-Yves Girard]], _Transcendental syntax I: deterministic case_, 2016 ([pdf](http://girard.perso.math.cnrs.fr/trsy1.pdf))

* {#Girard16b} [[Jean-Yves Girard]], _Transcendental syntax II: non-deterministic case_, 2016 ([pdf](http://girard.perso.math.cnrs.fr/trsy2.pdf))

* {#Girard18} [[Jean-Yves Girard]], _Transcendental syntax III: equality_, 2018 ([pdf](http://girard.perso.math.cnrs.fr/trsy3.pdf))

* {#Girard20} [[Jean-Yves Girard]], _Transcendental syntax IV: logic without systems_, 2020 ([pdf](http://girard.perso.math.cnrs.fr/trsy4.pdf))

* {#AbrusciPistone12} [Vito Michele Abrusci](http://www.uniroma3.it/persona.php?persona=v9rdc8l%2Fv34h57mDvlwB1jjlJFqAH%2B3Or9wKcvkvgtI%3D&tab=2), Paolo Pistone, _On Transcendental syntax: a Kantian program for logic?_, 2012 ([pdf](https://www.academia.edu/10495057/On_Trascendental_syntax_a_Kantian_program_for_logic))

* {#Rouleau13} [[Vincent Laurence Rouleau]], _Towards an understanding of Girard's transcendental syntax: Syntax by testing_, PhD thesis 2013  ([pdf](https://ruor.uottawa.ca/bitstream/10393/23680/3/Laurence_Rouleau_Vincent_2013_thesis.pdf))

* {#Beffara06} [[Emmanuel Beffara]], _A Concurrent Model for Linear Logic_, PhD thesis 2013  ([pdf](https://www.sciencedirect.com/science/article/pii/S1571066106001927))

The beginnings of an attempt to formalize Transcendental Syntax is given in

* [[Boris Eng]], [[Thomas Seiller]], _Multiplicative Linear Logic from Logic Programs and Tilings_ ([hal-02895111](https://hal.archives-ouvertes.fr/hal-02895111))

* [[Boris Eng]], _A gentle introduction to Girard's Transcendental Syntax for the linear logician_ ([hal-02977750](https://hal.archives-ouvertes.fr/hal-02977750))

[[!redirects Transcendental Syntax]]


transchromatic character

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of generalization of [[group characters]] to [[chromatic homotopy theory]].

Let $X  \sslash G$ be a [[topological stack|topological]] [[quotient stack]]. Its [[free loop stack]] $\mathcal{L}(X\sslash G) =  hom_{Stacks}(S^1, X//G)$ restricts to those loops that are constant as continuous maps, and contain only possibly the transition data with values in $X \sslash G$ (i.e. the "[[twisted loop space]]" retaining "twisted sector" data), this is the [[inertia stack]]

$$
  \mathcal{L}_{const}( X \sslash G )
  \simeq
  hom_{Stacks}(\mathbf{B}\mathbb{Z}, X \sslash G)
  \,.
$$

The [[geometric realization of simplicial topological spaces|geometric realization]] of $\mathcal{L}_{const}(X \sslash G)$ is denoted $Fix(X)$ in ([Hopkins-Kuhn-Ravenel 00](#HopkinsKuhnRavenel00), [Stapleton 13, p. 2](#Stapleton13)).

Regarding $S^1$ as the [[circle group]], there is a canonical $S^1$ [[infinity-action]] on any free loop space $\mathcal{L}(-)$ (by rigid rotation of loops), and it restricts to $\mathcal{L}_{const}(-)$. Hence there is the [[homotopy quotient]] stack

$$
  \mathcal{L}_{const}( X \sslash G ) \sslash S^1
$$

The [[geometric realization of simplicial topological spaces|geometric realization]] of $\mathcal{L}_{const}(X  \sslash G) \sslash S^1$ is denoted $Twist(X)$ in ([Stapleton 13, p. 2](#Stapleton13)).

Now with $n \in \mathbb{N}$ and given some prime, write $E_n$ for the $n$th [[Morava E-theory]] and $K(t)$ for the $t$th [[Morava K-theory]]. Then there is a homomorphism of Borel [[equivariant cohomology theories]]

$$
  E_n^\bullet(X//G)
    \longrightarrow
  B_{n-1}^\ast
   \underset{L_{K(n-1)} E_n^\bullet(B \mathbb{Q}_p/\mathbb{Z}_p)}{\otimes}
  L_{K(n-1)} E^\bullet_n( \mathcal{L}_{const}( X//G ) )// S^1)
  \,,
$$

where $L_{(-)}$ denotes [[Bousfield localization of spectra|Bousfield localization]]. This is the _twisted transchromatic character map_ ([Stapleton 13, p. 5](#Stapleton13)), shown here for the special case $t = n-1$, in the notation there.

Here $B_{n-1}$ is a ring such that... ([Stapleton 13, p. 3](#Stapleton13))

## Related concepts

* [[Chern character]]

* [[higher chromatic Chern character]]

* [[Morava E-theoretic Chern character]]

* [[free loop stack]], [[Huan's inertia orbifold]]

* [[free loop space of classifying space]]

## References

Introduction and review:

* [[Nathaniel Stapleton]], _An Introduction to HKR Character Theory_ ([pdf](http://homepages.uni-regensburg.de/~stn30788/papers/Characters.pdf))

* {#Raksit15} [[Arpon Raksit]], _Characters in global equivariant homotopy theory_, 2015 ([pdf](https://www.math.harvard.edu/media/raksit.pdf), [[Raksit_Characters.pdf:file]])

Original articles:

* {#HopkinsKuhnRavenel00} [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], _Generalized group characters and complex oriented cohomology theories_, J. Amer. Math. Soc. 13 (2000), 553-594 ([publisher](http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf))

* {#Stapleton11} [[Nathaniel Stapleton]], _Transchromatic generalized character maps_, Algebr. Geom. Topol. 13 (2013) 171-203 ([arXiv:1110.3346](https://arxiv.org/abs/1110.3346), [euclid:agt/1513715495](https://projecteuclid.org/euclid.agt/1513715495))

* {#Stapleton13} [[Nathaniel Stapleton]], _Transchromatic twisted character maps_,  J. Homotopy Relat. Struct. 10, 29–61 (2015). ([arXiv:1304.5194](https://arxiv.org/abs/1304.5194), [doi:10.1007/s40062-013-0040-9](https://doi.org/10.1007/s40062-013-0040-9))

* [[Nathaniel Stapleton]], _Transchromatic generalized character maps (and more!)_ ([pdf](http://etale.site/livetex/saft/stapleton.pdf))

* [[Tomer Schlank]], [[Nathaniel Stapleton]], _A transchromatic proof of Strickland's theorem_ ([arXiv:1404.0717](http://arxiv.org/abs/1404.0717))

* [[Takeshi Torii]], _HKR characters, p-divisible groups and the generalized Chern character_, ([pdf](http://www.ams.org/journals/tran/2010-362-11/S0002-9947-2010-05194-3/S0002-9947-2010-05194-3.pdf))

[[!redirects transchromatic characters]]

[[!redirects tanschromatic character]]

[[!redirects transchromatic character map]]
[[!redirects transchromatic character maps]]


transfer context

#Contents#
* table of contents
{:toc}

## Idea

A _transfer context_ is an abstract context of [[linear homotopy-type theory]] in which the concept of [[Becker-Gottlieb transfer]] makes sense.

## Definition

A _transfer context_ according to ([Haugseng 13, def. 4.9](#Haugseng13)) is a [[Wirthmüller context]] $(f_! \dashv f^\ast \dashv f_\ast)$ of [[six operations]] in which not only $f_!$ but also $f_\ast$ satisfies its [[projection formula]].

This induces an abstract [[Becker-Gottlieb transfer]]/[[Umkehr map]] by ([Haugseng 13, prop.  4.14](#Haugseng13))

## Related concepts

* [[sheaf with transfer]]

## References

* {#Haugseng13} [[Rune Haugseng]], _The Becker-Gottlieb Transfer Is Functorial_ ([arXiv:1310.6321](http://arxiv.org/abs/1310.6321))

[[!redirects transfer contexts]]