terminal geometric morphism

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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> For the time being, the following speaks in [[(infinity,1)-topos theory|$\infty$-topos theory]], for definiteness; but all statements and proofs apply verbatim also to [[Grothendieck toposes]], since they just depend on general abstract category-theoretic properties.

## Idea

For $\mathbf{H}$ an [[(infinity,1)-topos|$\infty$-topos]] over the [[base (infinity,1)-topos|base]] [[Infinity-Grpd|$Grpd_\infty$]], it has an essentially unique [[(infinity,1)-geometric morphism|geometric morphism]] to [[Infinity-Grpd|$Grpd_\infty$]]

\[
  \label{TheGeometricMorphism}
  \mathbf{H}
  \underoverset
    {\underset{\Gamma}{\longrightarrow}}
    {\overset{LConst}{\longleftarrow}}
    {\;\;\;\;\bot\;\;\;\;}
  Grpd_\infty
\]

where:

* the [[left adjoint]] has the interpretation of constructing [[locally constant infinity-stacks|locally constant $\infty$-stacks]],

* the [[right adjoint]] has the interpretation of forming [[global sections]].

In other words, [[Infinity-Grpd|$Grpd_\infty$]] is the [[terminal object in an (infinity,1)-category|terminal object]] in the [[(infinity,1)-category|$\infty$-category]] [[(infinity,1)Topos|$Topos_\infty$]] of [[infinity-stack (infinity,1)-toposes|$\infty$-stack $(\infty,1)$-toposes]] and [[(infinity,1)-geometric morphisms|$\infty$-geometric morphisms]] between these.

## Properties

\begin{proposition}
  The geometric morphism (eq:TheGeometricMorphism) exists essentially uniquely.
\end{proposition}
(e.g. [Lurie 2009, HTT 6.3.4.1](#Lurie09))
\begin{proof}
  Since every [[infinity-groupoid|$\infty$-groupoid]] is an [[(infinity,1)-colimit|$\infty$-colimit]] (over itself) of the [[point]] (see [there](infinity1-limit#Tensoring)):

\[
  \label{InfinityGroupoidIsColimitOfPointOverItself}
  S \,\simeq\,
  \underset{\underset{S}{\longrightarrow}}{\lim} \,\ast
\]
and since the [[inverse image]] of a geometric morphism [[preserved limit|preserves]] [[finite limits]] (by definition), such as the [[terminal object]], and all [[(infinity,1)-colimits|$\infty$-colimits]] (since [[left adjoints preserve colimits]]), we have that $LConst$ must be given by forming the corresponding $\infty$-colimit of copies of the terminal object $\ast_{\mathbf{H}}$ in $\mathbf{H}$, which does exist:

$$
  \begin{aligned}
    LConst(S)
    &
    \;\simeq\;
    LConst
    \Big(
      \underset{
        \underset{S}{\longrightarrow}
      }{\lim}
      \,
      \ast
    \Big)
    \\
    &
    \;\simeq\;
    \underset{
      \underset{S}{\longrightarrow}
    }{\lim}
    \,
    LConst
    \big(
      \ast
    \big)
    \\
    &
    \;\simeq\;
    \underset{
      \underset{S}{\longrightarrow}
    }{\lim}
    \,
    \ast_{\mathbf{H}}
  \end{aligned}
$$
\end{proof}

\begin{proposition}
\label{DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject}
The [[direct image]] of the terminal geometric morphism (eq:TheGeometricMorphism) is given by the [[(infinity,1)-categorical hom-space|hom-space]] out of the [[terminal object]], in that
for $X \,\in\, \mathbf{H}$ there is a [[natural equivalence]]
$$
  \Gamma (X)
  \;\simeq\;
  \mathbf{H}(\ast_{\mathbf{H}} ,\, X)
  \,,
$$
where $\ast_{\mathbf{H}} \,\in\, \mathbf{H}$ denotes the [[terminal object in an (infinity,1)-category|terminal object]].
\end{proposition}
\begin{proof}
For all $S \,\in\, Grpd_\infty$ we have the following sequence of [[natural equivalences]]:
$$
  \begin{array}{lll}
    Grpd_\infty
    \big(
      S
      ,\,
      \mathbf{H}(\ast_{\mathbf{H}} ,\, X)
    \big)
    &
    \;\simeq\;
    Grpd_\infty
    \Big(
      \underset{\underset{S}{\longrightarrow}}{\lim} \ast
      ,\,
      \mathbf{H}(\ast_{\mathbf{H}} ,\, X)
    \Big)
    \\
    &
    \;\simeq\;
    \underset{\underset{S}{\longleftarrow}}{\lim}
    \,
    Grpd_\infty
    \big(
      \ast
      ,\,
      \mathbf{H}(\ast_{\mathbf{H}} ,\, X)
    \big)
    \\
    &
    \;\simeq\;
    \underset{\underset{S}{\longleftarrow}}{\lim}
    \,
    \mathbf{H}(\ast_{\mathbf{H}} ,\, X)
    \\
    &
    \;\simeq\;
    \mathbf{H}
    \Big(
      \underset{\underset{S}{\longleftarrow}}{\lim}
      \ast_{\mathbf{H}}
       ,\,
       X
    \Big)
    \\
    &
    \;\simeq\;
    \mathbf{H}
    \Big(
      \underset{\underset{S}{\longleftarrow}}{\lim}
      \,
      LConst(\ast)
       ,\,
       X
    \Big)
    \\
    &
    \;\simeq\;
    \mathbf{H}
    \Big(
      LConst
      \big(
        \underset{\underset{S}{\longleftarrow}}{\lim}
        \ast
      \big)
      ,\,
      X
    \Big)
    \\
    &
    \;\simeq\;
    \mathbf{H}
    \big(
      LConst(S)
      ,\,
      X
    \big)
  \end{array}
$$
(Here we used (eq:InfinityGroupoidIsColimitOfPointOverItself) and that [[hom-functor preserves limits]] and that [[left adjoints preserve colimits]] and that $LConst$ [[preserved limit|preserves]] [[finite limits]] such as the [[terminal object]], by definition.)

But this is a hom-equivalence which exhibits (see [here](adjoint+infinity1-functor#CharacterizationInTermsOfHomEquivalences)) $\mathbf{H}(\ast,-)$ as a [[right adjoint|right]] [[adjoint (infinity,1)-functor|adjoint $\infty$-functor]] to $LConst$. This implies the claim by essential uniqueness of adjoints.
\end{proof}

## Literature

For discussion in plain [[topos theory]] see any of the references listed there.

Discussion in [[(infinity,1)-topos theory|$\infty$-topos theory]] includes:

* {#Lurie09} [[Jacob Lurie]], Section 6.3.4 in: *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

[[!redirects terminal geometric morphisms]]

[[!redirects terminal (infinity,1)-geometric morphism]]
[[!redirects terminal (infinity,1)-geometric morphisms]]


terminal object

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### Context
#### Category theory
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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# Contents
* table of contents
{:toc}

## Definition

A **terminal object** in a [[category]] $C$ is an [[object]] $1$ of $C$ satisfying the following [[universal property]]:

for every object $x$ of $C$, there exists a unique [[morphism]] $!:x\to 1$.  The terminal object of any category, if it exists, is unique up to unique [[isomorphism]].  If the terminal object is also [[initial object|initial]], it is called a [[zero object]].

## Remarks

* Less usual synonyms are _final object_ and _terminator_.

* A terminal object is often written $1$, since in [[Set]] it is a 1-element set, and also because it is the unit for the cartesian [[product]].  Other notations for a terminal object include $*$ and $pt$.

## Properties

\begin{remark}\label{TerminalObjectIsEmptyLimit}
A terminal object may also be viewed as a [[limit]] over the [[empty diagram]].
\end{remark}

* Conversely, a [[limit]] over any diagram is a terminal cone over that diagram.

* For any object $x$ in a category with terminal object $1$, the categorical [[product]] $x\times 1$ and the [[exponential object]] $x^1$ both exist and are canonically isomorphic to $x$.

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

Let $\mathcal{C}$ be a [[category]].

1. The following are equivalent:

   1. $\mathcal{C}$ has a [[terminal object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[right adjoint]]

      $$
        \ast
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \mathcal{C}
      $$

    Under this equivalence, the [[terminal object]] is identified with the image under the right adjoint of the unique object of the [[terminal category]].

1. Dually, the following are equivalent:

   1. $\mathcal{C}$ has an [[initial object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[left adjoint]]

      $$
        \mathcal{C}
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \ast
      $$

    Under this equivalence, the [[initial object]] is identified with the image under the left adjoint of the unique object of the [[terminal category]].

=--

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

Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|the]] [[singleton]], the hom-isomorphism characterizing the [[adjoint functors]] is directly the [[universal property]] of an [[initial object]] in $\mathcal{C}$

$$
  Hom_{\mathcal{C}}( L(\ast) , X )
  \;\simeq\;
  Hom_{\ast}( \ast, R(X) )
  =
  \ast
$$

or of a [[terminal object]]

$$
  Hom_{\mathcal{C}}( X , R(\ast) )
  \;\simeq\;
  Hom_{\ast}( L(X), \ast ) = \ast
  \,,
$$

respectively.

=--

## Examples

Some examples of terminal objects in notable categories follow:

* The terminal object of a [[partial order|poset]] is its [[top element]], if it exists.

* Any [[one-element set]] is a terminal object in the category [[Set]].

* The terminal object of [[Top]] is the [[point space]].

* The [[trivial group]] is the terminal object of [[Grp]] and, as an [[abelian group]], of [[Ab]].

* The terminal object of [[Ring]] is the [[zero ring]].  (Note however that if [[ring|rings]] have unities and ring [[homomorphism|homomorphisms]] must preserve them, then the zero ring is not a [[zero object]] of [[Ring]].)

* Including most of the above, the terminal object of an [[algebraic category]] is its [[trivial algebra]].

* The terminal object of [[Cat]] is the [[discrete category]] with just one object, the [[trivial category]].

* The terminal object of a [[over category|slice category]] $C/x$ is the [[identity morphism]] $x \to x$.

## Related concepts

* [[strict terminal object]]

* [[initial object]]

* [[unit type]]

* [[terminal object in a 2-category]]

* [[terminal object in an (infinity,1)-category|terminal object in an $(\infty,1)$-category]]

## References

Textbook account:

* [[Francis Borceux]], Section 2.3 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

[[!redirects terminator]]
[[!redirects terminal object]]
[[!redirects terminal objects]]
[[!redirects terminal]]
[[!redirects final object]]
[[!redirects final objects]]


terminal object in a quasi-category

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#### Limits and colimits
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# Contents#
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{: toc}

## Idea

In a [[quasi-category]] the notion of [[terminal object]] known from ordinary [[category theory]] is relaxed in the [[homotopy theory|homotopy theoretic]] sense to the suitable notion in [[(∞,1)-category theory]]:

instead of demanding that from any other object there is a _unique_ morphism into the terminal object, in a quasi-category there is a _[[contractible space]]_ of such morphisms, i.e. the morphism to the terminal object is unique _up to [[homotopy]]_.

##Definition

Let $C$ be a [[quasi-category]] and $c \in C$ one of its [[object]]s (a vertex in the corresponding [[simplicial set]]). The object $c$ is a **terminal object** in $C$ if the following equivalent conditions hold:

* The projection from the [[over quasi-category]] $C_{/c} \to C$ is a [[model structure on simplicial sets|trivial fibration of simplicial sets]].

* For every object $d$ of $C$ the right hom Kan-complex into $d$ is contractible:

$$
  Hom_C^R(d,c) \simeq {*}
  \,.
$$

## Related concepts

* [[initial object]], [[terminal object]]

* [[h-initial object]]

* [[terminal object in a 2-category]]

* [[initial object in a 2-category]]

##References#

A quick survey is on page 159 of

* [[André Joyal]], _The theory of quasicategories and its applications_, lectures at [Simplicial Methods in Higher Categories](http://www.crm.es/HigherCategories/), ([pdf](https://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf))

For more details see [definition 1.2.12.3, p. 46](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=46) in

*  [[Jacob Lurie]], _[[Higher Topos Theory]]_

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[[!redirects terminal object in a quasicategory]]
[[!redirects terminal objects in quasicategories]]

[[!redirects terminal object in an (∞,1)-category]]
[[!redirects terminal objects in an (∞,1)-category]]
[[!redirects terminal objects in (∞,1)-categories]]

[[!redirects terminal object in an (infinity,1)-category]]
[[!redirects terminal objects in an (infinity,1)-category]]
[[!redirects terminal objects in (infinity,1)-categories]]

[[!redirects initial object in an (∞,1)-category]]
[[!redirects initial objects in an (∞,1)-category]]
[[!redirects initial object in an (infinity,1)-category]]
[[!redirects initial objects in an (infinity,1)-category]]


termination

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### Context
#### Type theory
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#### Constructivism, Realizability, Computability
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#Contents#
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## Idea

A [[rewriting system]] is terminating if we cannot perform a never-ending computation.

## Definition

Let $(X,\rightarrow)$ be an [[abstract rewriting system]]. Then $\rightarrow$ is terminating, strongly normalizing or noetherian iff there doesn't exist any infinite reduction $x_{1} \rightarrow x_{2} \rightarrow ... \rightarrow x_{n} \rightarrow ... $

## Example

The cut elimination of [[second order linear logic]] is strongly normalizable, see [Pagani & Tortora de Falco (2009)](#PaganiTortora09) where they prove it using notably Girard's [[reducibility candidates]].

## Related entries

* It is possible that only one term is strongly normalizable but not the entire rewriting system: see [[normal form]].

* [[abstract rewriting system]]

* [[rewriting system]]

## References

On [[strong normalization]] of second order linear logic:

* {#PaganiTortora09} [[Michele Pagani]], [[Lorenzo Tortora de Falco]], *Strong Normalization Property for Second Order Linear Logic* (2009) [[pdf](https://www.irif.fr/~michele/snllTCS-1.pdf)]

[[!redirects strongly normalizing]]
[[!redirects strong normalization]]


ternary factorization system

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# Ternary factorisation systems
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## Idea

Just as an (orthogonal/unique) [[orthogonal factorization system|factorization system]] $(E,M)$ on a [[category]] $C$ gives a way to factor every [[morphism]] of $C$ as an $E$-map followed by an $M$-map, a *ternary (orthogonal) factorization system* $(E,F,M)$ gives a way to factor every map of $C$ as an $E$-map followed by an $F$-map followed by an $M$-map.

This is a special case of a notion of [[k-ary factorization system]].

## Definition

It turns out that a convenient way to state the definition is in terms of a pair of ordinary (orthogonal) factorization systems.  We define a **ternary factorization system** on $C$ to consist of a pair $(L_1,R_1)$ and $(L_2,R_2)$ of ordinary orthogonal factorization systems such that $L_1 \subseteq L_2$ (or equivalently $R_2 \subseteq R_1$).

The three classes of map $(E,F,M)$ are then defined by $E=L_1$, $F = L_2\cap R_1$, and $M=R_2$.  This is justified by:

+--{: .un_prop}
###### Proposition
Given a ternary factorization system as above, any morphism $f:A\to B$ factors as
$$ A \overset{L_1}{\to} im_2(f) \overset{L_2 \cap R_1}{\to} im_1(f) \overset{R_2}{\to} B $$
in an essentially unique way.
=--
+--{: .proof}
###### Proof
Consider the two ternary factorizations of $f$ obtained by

1. First factoring $f$ into an $L_1$-map followed by an $R_1$-map, then factoring the $R_1$-part into an $L_2$-map followed by an $R_2$-map; and
1. First factoring $f$ into an $L_2$-map followed by an $R_2$-map, then factoring the $L_2$-part into an $L_1$-map followed by an $R_1$-map.

Note that both start with an $L_1$ map and end with an $R_2$ map.  By a straightforward exercise in orthogonality, we can get comparison maps in both directions between these two factorizations which make them isomorphic.  Therefore, since the first produces a middle map which is in $L_2$ and the second produces a middle map which is in $R_1$, this middle map must in fact be in $L_2\cap R_1$.  Finally, any other such ternary factorization of $f$ induces an $(L_1,R_1)$ and $(L_2,R_2)$ factorization by composing pairwise, and uniqueness of these two implies uniqueness of the ternary factorization.

More explicitly, we factor $f$ as

\begin{center}\begin{tikzcd}
& C_1 \ar[r,"{\lambda_2}"] \ar[drr,"{r_1}"'] & C_2 \ar[dr,"{\rho_2}"] \\
A \ar[ur,"{\ell_1}"] \ar[dr,"{\lambda_1}"'] \ar[drr,"{\ell_2}"] &&& B\\
& D_1 \ar[r,"{\rho_1}"'] & D_2 \ar[ur,"{r_2}"']
\end{tikzcd}\end{center}

with $\lambda_i, \ell_i\in L_i, \rho_j,r_j\in R_j$.  Then since $R_2\subseteq R_1$, we have $r_2 \rho_1 \in R_1$, so that $(\ell_1,r_1)$ and $(\lambda_1,r_2\rho_1)$ are both $(L_1,R_1)$-factorizations of $f$ and thus we have a unique compatible isomorphism $C_1\cong D_1$.  Similarly, $(\lambda_2 \ell_1, \rho_2)$ and $(\ell_2,r_2)$ are both $(L_2,R_2)$-factorizations, so we have a unique compatible isomorphism $C_2\cong D_2$.  This gives a diagram

\begin{center}\begin{tikzcd}
& C_1 \ar[r,"{\lambda_2}"] \ar[dd,"\cong"] & C_2 \ar[dr,"{\rho_2}"] \ar[dd,"\cong"'] \\
A \ar[ur,"{\ell_1}"] \ar[dr,"{\lambda_1}"'] &&& B\\
& D_1 \ar[r,"{\rho_1}"'] & D_2 \ar[ur,"{r_2}"']
\end{tikzcd}\end{center}

with two commutative triangles, and the middle square also commutes since both sides are lifts in a lifting problem of $\ell_1$ against $r_2$.  Finally, since $\lambda_2\in L_2$ is isomorphic to $\rho_1\in R_1$ in the arrow category, both are in fact in $L_2\cap R_1$.

=--

Conversely, just as for a binary factorization system, the extra requirement of orthogonality can be deduced from uniqueness of the factorizations, a unique and *functorial* ternary factorization implies that it "splits" into a pair of binary factorization systems, i.e. a ternary factorization system as defined here.  This is remarked on [here](http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html#c034162).

One can also characterize the notion in terms of a ternary factorization with a "ternary orthogonality" property; see the paper of Pultr and Tholen referenced below.

## The sixth class of maps

In addition to $L_1$, $R_1$, $L_2$, $R_2$, and $L_2\cap R_1$, a ternary factorization system also determines a sixth important class of morphisms, namely those whose $(L_2\cap R_1)$-part is an isomorphism, or equivalently those that can be factored as an $L_1$-map followed by an $R_2$-map.  We therefore call this class $R_2 L_1$.

+--{: .un_prop}
###### Proposition
In a ternary factorization system, $L_1 = L_2 \cap R_2L_1$ and $R_2 = R_1 \cap R_2L_1$.
=--
+--{: .proof}
###### Proof
In both cases $\subseteq$ is obvious.  Conversely, if $f \in L_2 \cap R_2 L_1$, say $f = m e$ for $m\in R_2$ and $e\in L_1$, then orthogonality in the square
$$\array{a & \overset{e}{\to} & c\\
  ^f \downarrow && \downarrow ^m\\
  b & \underset{id}{\to} & b}$$
exhibits $f$ as a retract of $e$ in $Arr(C)$, whence $f\in L_1$ since $L_1$ is closed under retracts.
=--

In many examples, the class $R_2 L_1$ is closed under composition, but this is not always the case.

+--{: .num_prop #R2L1Comp}
###### Proposition
In a ternary factorization system, the class $R_2L_1$ is closed under composition if and only if $L_1R_2 \subseteq R_2L_1$.
=--
+--{: .proof}
###### Proof
This is Proposition 3.6 of [Pultr and Tholen](#PultrTholen).

"Only if" is obvious, since $L_1R_2 \subseteq (R_2L_1)(R_2L_1)$ which is contained in $R_2L_1$ if it is closed under composition.  Conversely, if $L_1R_2 \subseteq R_2L_1$ then $(R_2L_1)(R_2L_1) = R_2 (L_1 R_2) L_1 \subseteq R_2 (R_2 L_1) L_1 \subseteq R_2 L_1$.
=--

+--{: .num_cor #R2L1NotComp}
###### Corollary
In a ternary factorization system, if every morphism factors as an $R_2$-map followed by an $L_1$-map, then $R_2L_1$ is closed under composition if and only if it consists of all morphisms, and if and only if $L_2 \cap R_1$ consists of only the isomorphisms.
=--
+--{: .proof}
###### Proof
This is remarked after Examples 3.12 in [Pultr and Tholen](#PultrTholen).

By the Proposition \ref{R2L1Comp}, if $L_1R_2$ is all morphisms and $R_2L_1$ is closed under composition, then it is also all morphisms, and by uniqueness of ternary factorizations this is equivalent to $L_2 \cap R_1$ being the isomorphisms.
=--

Concrete examples are below.

## Relation to iterated distributive laws

Since [[strict factorization systems]] can be identified with [[distributive laws]] in the [[bicategory]] of [[spans]], it is natural to think that "strict" ternary factorization systems can be identified with [[iterated distributive laws]] in [[Span]].  However, this is not true in general, due to the lack of closure of $R_2L_1$ under composition.

In a triple distributive law there are three monads $A,B,C$ and distributive laws $B A \to A B$, $C A \to A C$, and $C B\to B C$ satisfying the [[Yang-Baxter equation]].  Thus, in particular there are three composite monads $A B$, $A C$, and $B C$, and the Yang-Baxter equation ensures we get distributive laws $(B C)A\to A(B C)$ and $C(A B) \to (A B)C$ and a unique induced monad structure on $A B C$.

In a ternary factorization system, we would naturally take $A = R_2$, $B = L_2 \cap R_1$, and $C = L_1$.  We then get distributive laws $B A \to A B$ and $C B \to B C$, since $A B = R_1$ and $B C = L_2$ are closed under composition.  However, if $A C = R_2L_1$ is not closed under composition, we do not get a distributive law $C A \to A C$.

It is now natural to conjecture that this is the only obstacle, though: strict ternary factorization systems for which $R_2L_1$ is closed under composition should be equivalent to triple distributive laws in $Span$.  Similarly, the non-strict case should correspond to triple distributive laws over the groupoid of isomorphisms in [[Prof]], as for ordinary [[orthogonal factorization systems]] and ordinary distributive laws.

## Examples

* In [[Top]], let $L_1=$ quotient maps, $R_1=$ injective continuous maps, $L_2=$ surjective continuous functions, and $R_2=$ subspace embeddings.  Here $L_2\cap R_1=$ bijective continuous maps, and the two intermediate objects in the ternary factorization of a continuous map are obtained by imposing the coarsest and the finest compatible topologies on its set-theoretic image.

* More generally, if a category has both ([[epimorphism|epi]], [[strong monomorphism|strong mono]]) and ([[strong epimorphism|strong epi]], [[monomorphism|mono]]) factorizations, then since strong epis are epi, we have a ternary factorization.  Here $L_2\cap R_1$ is the class of monic epics. The maps in $R_2 L_1$ are sometimes called [[strict morphisms]].

  By Corollary \ref{R2L1NotComp}, if every morphism factors as a strong mono followed by a strong epi, the strict morphisms are only closed under composition if they are the isomorphisms.  For instance, this is true if the category has coproducts whose injections are strong mono, since we can then factor $A\to B$ as $A \to A+B \to B$ where the second factor is even [[split epi]].  This is the case in [[Top]], so that is a concrete example where $R_2L_1$ is not closed under composition.

* On [[Cat]] there is a 2-categorical version of a ternary factorization system, determined by the [[factorization system on a 2-category|2-categorical factorization systems]] ([[essentially surjective functor|eso]]+[[full functor|full]], [[faithful functor|faithful]]) and (eso, [[fully faithful functor|full and faithful]]).  Here $L_2\cap R_1$ is the class of eso+faithful functors, while $R_2 L_1$ is the class of full functors.  This factorization system plays an important role in the study of [[stuff, structure, property]].

  Restricted to [[groupoids]] this is the [[1-image]]-[[2-image]] factorization, the 3-stage [[Postnikov system]] of groupoids.

* On [[Topos]] there is also a 2-categorical ternary factorization system composed of the binary 2-categorical factorization systems ([[hyperconnected geometric morphism|hyperconnected]], [[localic geometric morphism|localic]]) and ([[geometric surjection|surjection]], [[geometric embedding|inclusion]]). Here the maps in $L_2\cap R_1$ have no name other than "localic surjections," and those in $R_2 L_1$ have no established name (although they are briefly mentioned in A4.6.10 of the [[Elephant]]).

* Suppose that $C$ has a binary factorization system $(E,M)$ and that $p\colon A\to C$ is an [[ambifibration]] relative to $(E,M)$: i.e. every arrow in $E$ has an opcartesian lift and every arrow in $M$ has a [[cartesian morphism|cartesian]] lift.  (In particular, $p$ could be a [[bifibration]].)  Then there is a ternary factorization system on $A$ for which $L_1$ is the class of opcartesian arrows over $E$, $R_2$ is the class of cartesian arrows over $M$, and $L_2\cap R_1$ is the class of vertical arrows (those lying over identities).  See [this comment](http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html#c034162).

  For instance, the above factorization system on $Top$ is induced in this way via the [[forgetful functor]] $Top\to Set$ from the (epi,mono) factorization system on [[Set]].

* A similar example is given by a [[span]] $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ of categories where $p$ is a [[fibration]] whose cartesian morphisms are $q$-vertical and $q$ is an [[opfibration]] whose opcartesian morphisms are $p$-vertical (that is, the span $(p,q)$ is both a left and a right fibration in the sense of Street).  Then the two factorization systems on $E$ given by the $q$-opcartesian and $q$-vertical morphisms on the one hand, and the $p$-vertical and $p$-cartesian morphisms on the other, satisfy the $L_1 \subseteq L_2$ condition above, so that every morphism in $E$ factors as a $q$-opcartesian morphism followed by a morphism that is both $p$- and $q$-vertical, followed by a $p$-cartesian morphism.

  Such a span is a [[two-sided fibration]] if $L_1R_2 \subseteq R_2L_1$, that is if the three-way factorization of the composite of a $p$-cartesian morphism followed by a $q$-opcartesian one has its middle term an isomorphism.

## Related concepts

* The notion of [[model category]] involves a pair of [[weak factorization systems]] called (acyclic cofibration, fibration) and (cofibration, acyclic fibration) which are compatible in the same sense as above.  However, non-uniqueness of these factorizations means that the resulting "ternary factorization" of a morphism is not unique.  The class corresponding to $R_2 L_1$ is important, however: it is precisely the class of [[weak equivalences]].

* The notion of [[k-ary factorization system]] is a generalization to factorizations into $k$ morphisms.

## References

Ternary factorisation systems are introduced as *double factorisation systems* in:

* {#PultrTholen} A. Pultr and W. Tholen, *Free Quillen Factorization Systems*.  Georgian Math. J.9 (2002), No. 4, 807-820

* [Cafe discussion](http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html)

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ternary frame

# Ternary frames
* table of contents
{: toc}

## Warning

*The term 'frame' is used in a different sense here than in [[geometric logic]]; see [[frame]].  The usage here is analogous to [[Kripke frames]] in [[modal logic]].*

## Idea

A **ternary frame** is a way of presenting a model for a [[substructural logic]] (such as [[linear logic]] and [[relevant logic]]) in terms of a set of "worlds" or "states of information" and a ternary [[relation]].

## Definition

A **ternary frame** is a set $A$ together with a ternary relation $R$ on $A$; we write $R x y z$ when $R$ holds of three elements $x,y,z\in A$.

We may additionally ask that $A$ have a [[partial ordering]]; in this case we demand the compatibility condition that if $R x y z$ and $x'\le x$, $y'\le y$, and $z\le z'$, then also $R x' y' z'$.

## Modeling substructural logic

We can model [[logic]] using a ternary frame with a "forcing" or "satisfaction" relation between points of $A$ and formulas.  We begin by assigning to each [[atomic formula]] a set of points of $A$ which satisfy it.  If $A$ has a partial order, as above, then we ask each of these sets to be up-closed.

The [[logical connectives]] can then be defined inductively by clauses such as the following:

* $x \Vdash P \& Q$ (the negative [[conjunction]]) if and only if $x\Vdash P$ and $x\Vdash Q$.
* $x \Vdash P \oplus Q$ (the positive [[disjunction]]) if and only if $x \Vdash P$ or $x\Vdash Q$.
* $x \Vdash \mathbf{0}$ (the positive [[falsity]]) never.
* $x \Vdash \top$ (the negative [[truth]]) always.
* $x \Vdash P \multimap Q$ (the one-sided linear implication) if and only if for all $y,z$, if $R x y z$ and $y\Vdash P$, then $z\Vdash Q$.
* $x \Vdash P \otimes Q$ (the positive conjunction) if and only if there exist $y,z$ such that $R y z x$ and $y\Vdash P$ and $z\Vdash Q$.

The logic obtained thereby will generally be [[substructural logic|substructural]]: it need not satisfy the structural rules like [[weakening rule|weakening]], [[contraction rule|contraction]], or even [[exchange rule|exchange]].  On this page, we have used the notation for substructural connectives from [[linear logic]].

## Additional structure

We can impose properties or structure on the ternary frame to affect the logic.  For instance, if $R x y z$ implies $R y x z$, then the logic we obtain will satisfy the exchange rule.

We also need additional structure in order to model positive truth, negative falsity, negative disjunction, and negation.

### Truth sets

A **truth set** in an ordered ternary frame is a subset $T\subseteq A$ such that $x\le y$ if and only if there exists a $t\in T$ with $R t x y$, and if and only if there exists an $s\in T$ with $R x s y$.  (If $R$ is commutative, as above, then the two conditions are equivalent.)  Alternatively, given an unordered ternary frame $A$ and a subset $T$, we could *define* $x\le y$ in this way, and then require as a property of $T$ that the resulting relation is a [[partial order]].

If $T$ is a truth set, then it makes sense to define

* $x \Vdash \mathbf{1}$ (the positive truth) if and only if $x\in T$.

### Compatibility relations

One way to model negation (and thereby obtain negative falsity $\bot$ and negative disjunction $\parr$ by duality from positive truth $\mathbf{1}$ and positive conjunction $\otimes$) is with a **compatibility relation**, which is just a binary relation $C$.  If $A$ has a partial order, we demand additionally that if $x C y$ and $x'\le x$ and $y\le y'$, then $x' C y'$.

Given such a $C$, we define

* $x \Vdash \neg P$ if and only if for all $y$, if $x C y$ then not $y\Vdash P$.

### Falsity sets

Negation can alternatively be modeled using a *false set*.  Suppose given a subset $F\subseteq A$, to be the interpretation of the negative falsity $\bot$:

* $x\Vdash \bot$ if and only if $x\in F$.

We can then, if we wish, interpret negation and negative disjunction by defining $\neg P \coloneqq (P\multimap \bot)$ and $P\parr Q \coloneqq \neg (\neg P \otimes \neg Q)$.

The latter is most sensible if negation is involutive, which it need not be in general --- that is, if $x \Vdash \neg \neg P$ we need not have $x\Vdash P$.  One solution to this (if we want negation to be involutive) is to close up $\Vdash$ under double-negation.  This entails replacing the clauses defining the interpretation of the positive connectives $\otimes$, $\oplus$, $\mathbf{1}$, and $\mathbf{0}$ with their double-negation closure.  This is commonly done in the [[phase semantics]] for linear logic (see below).

## From ternary frames to quantales

Since the models above associate to formulas *subsets* of $A$, it seems natural to describe them in a purely algebraic way using structure on the [[powerset]] of $A$.  In the case when $A$ is a poset, instead of the powerset of $A$ we must use the set of up-closed subsets of $A$.  Since this subsumes the unordered case (use the discrete ordering), we henceforth assume $A$ to be a poset, with $R$ having the assumed compatibility relation.

In fact, this axiom (which we repeat here for the reader's convenience):

* if $R x y z$ and $x'\le x$, $y'\le y$, and $z\le z'$, then also $R x' y' z'$.

says precisely that $R$ is a $\mathbf{2}$-[[enriched category|enriched]] [[profunctor]] $A^{op} \times A^{op} ⇸ A^{op}$.  (Here we are identifying posets with $\mathbf{2}$-enriched categories, where $\mathbf{2}$ is the [[interval category]].)

Therefore, by [[Day convolution]], $R$ induces a binary tensor product on the $\mathbf{2}$-enriched [[presheaf category]] $\mathbf{2}^A$, which is precisely the poset of up-closed sets in $A$.  This tensor product is precisely the above interpretation of $\otimes$.  By the usual Day convolution arguments, this tensor product functor has both left and right adjoints, which are precisely the interpretation of $\multimap$ and its dual.

Of course, the interpretations of $\&$ and $\oplus$ are just the categorical product and coproduct in $\mathbf{2}^A$.  Similarly, that of $\mathbf{0}$ and $\top$ are the initial and terminal objects.

A truth set $T$ corresponds to a profunctor $1 ⇸ A^{op}$ which is a unit for the pro-multiplication $R$.  Therefore, in this case the tensor product on $\mathbf{2}^A$ has a unit object, which is precisely $T$, the interpretation of the positive truth $\mathbf{1}$.

If we were to additionally add the assumption that $R$ is *associative*, in the sense that

* there exists a $z$ with $R x y z$ and $R z u v$ if and only if there exists a $w$ with $R x w v$ and $R y u w$

then $A^{op}$ would become a [[promonoidal category|promonoidal]] poset, and hence $\mathbf{2}^A$ would be a complete and cocomplete closed monoidal poset, i.e. a [[quantale]].

A false set $F$ is of course just an arbitrary object of this quantale.  Closing up under double-negation means restricting to the sub-poset of elements that are equal to their "double dual" $x = (x \multimap F) \multimap F$.  Since the self-adjunction $(-\multimap F)$ is [[idempotent adjunction|idempotent]], this sub-poset is itself a quantale, and indeed a [[star-autonomous category|*-autonomous]] one.

The quantale-theoretic content of a compatibility relation is somewhat trickier: as defined it is a profunctor from $A$ to itself, whereas the negation of a pro-$*$-autonomous poset would be a profunctor from $A^{op}$ to $A$.  (This would induce a functor $(\mathbf{2}^A)^{op} \to \mathbf{2}^{A}$ due to the special property that $\mathbf{2}\cong \mathbf{2}^{op}$.)  Moreover, the definition of $x \Vdash \neg P$ for a compatibility relation is also the (metatheoretic) *negation* of the map on $\mathbf{2}^A$ that would be induced profunctorially.  If we put these together, we can see that negation ought to be the the composite of the map $\mathbf{2}^A \to \mathbf{2}^A$ induced by the profunctor $C$ with the isomorphism $\mathbf{2}^A \cong (\mathbf{2}^{A^{op}})^{op}$ (using again that $\mathbf{2}\cong \mathbf{2}^{op}$).  At least if $A$ is discrete, so that $A\cong A^{op}$, then this has the correct domain and codomain, so we should be able to assert an axiom ensuring that it makes $\mathbf{2}^A$ $*$-autononmous.

+--{: .query}
To be completed...
=--

### Generalizations

The quantale-theoretic viewpoint suggests a generalization replacing $\mathbf{2}$ by any other quantale.  That is, for any quantale $Q$, we can define a notion of "$Q$-valued ternary frame" that generates a new quantale by Day convolution.  Everything goes through without significant change, except that compatibility relations seem to require $Q$ itself to be $*$-autonomous.

## Examples

### Phase spaces

If $A$ is a [[magma]] with multiplication $\cdot$, then we can make it a ternary frame by defining $R x y z$ to mean $x \cdot y = z$.  More generally, if $A$ is a poset equipped with a binary multiplication $\cdot$, we can make it an ordered ternary frame by defining $R x y z$ to mean $x \cdot y \le z$.

If $\cdot$ has a unit object $t$, then $T = \{t\}$ (in the unordered case) or $T = \{x | t \le x \}$ (in the ordered case) is a truth set.

Categorically, this corresponds to the usual way of regarding a [[monoidal category]] as a [[promonoidal category]].

In the special case when $A$ is a commutative [[monoid]] equipped with a "false set" $F$ as above (usually written $\bot$) in this context, this semantics for linear logic is called [[phase semantics]] (see there for more).  It is usually expressed in terms of the quantale $\mathbf{2}^A$ obtained by Day convolution, but after passing to fixed points of the double-negation monad (in order to obtain an involutive negation).  In this context, fixed points of $\neg\neg$ are referred to as *facts*.

It is also possible to interpret the exponential modalities $!$ and $?$ of linear logic using phase space semantics.  For instance, we can define

* $x \Vdash !P$ if and only if $x$ belongs to the $\neg\neg$-closure of the set of all idempotents $y$ such that $y\Vdash P \& \mathbf{1}$.

and obtain $?P$ by duality.

Phase space semantics is *complete* for [[linear logic]], in the sense that a formula is provable if and only if in any phase space semantics we have $1\vDash P$, where $1$ is the unit element of the monoid $A$.

### PCAs

If $A$ is a [[partial combinatory algebra]] (PCA), we can make it a ternary frame by defining $R x y z$ to mean that $x \cdot y$ is defined and equals $z$.  (Similarly, if $A$ is an ordered PCA we can make it an ordered ternary frame.)  The resulting interpretation of $\multimap$ almost coincides with the usual interpretation of implication in [[realizability]] over $A$, and the combinators $k,s$ have the property that $k \Vdash P \multimap (Q \multimap P)$ and $s \Vdash (P \multimap Q \multimap R) \multimap (P \multimap Q) \multimap P \multimap R$ for any $P,Q,R$, as would be expected from typed [[combinatory logic]].

## References

* R. Routley and R.K. Meyer, *The Semantics of Entailment I*, Truth, Syntax and Modality, ed. H. Leblanc, North-Holland Publishing Company (Amsterdam), pp. 199-243.  (1973)

* R. Routley and R.K. Meyer, *The Semantics of Entailment, II-III*. Journal of Philosophical Logic, 1, 53-73 and 192-208.  (1972)

* Natasha Kurtonina, *Frames and Labels - A modal analysis of categorial inference*.  Ph.D. Thesis, Institute of Logic, Language and Information (ILLC), Amsterdam, 1994

* J. M. Dunn and R. K. Meyer, *Combinators and Structurally Free Logic*.  Logic journal of the IGPL, 5(4):505-538, July 1997

* Greg Restall, *An introduction to substructural logic*.  Routledge, 2000.

[[!redirects ternary frames]]


ternary relation

A __ternary relation__ (or __triadic relation__) is a [[relation]] in three variables (instead of the more common two of a [[binary relation]]).

The elementary theory of [[category|categories]] (in the [[strict category|strict]] sense) may be defined with a single ternary relation (and several axioms) in untyped [[predicate logic]] with equality; the relation holds of [[morphisms]] $f$, $g$, and $h$ if and only if $f$ is the [[composite]] (in a fixed order) of $g$ and $h$.

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Terry Farrelly

* [Institute page](https://smp.uq.edu.au/profile/7450/terry-farrelly)

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

## Selected writings

On [[holographic tensor network]] [[quantum error correcting codes]]:

* [[Terry Farrelly]], [[Robert J. Harris]], [[Nathan A. McMahon]], [[Thomas M. Stace]], *Tensor-network codes* ([arXiv:2009.10329](https://arxiv.org/abs/2009.10329))

category: people


Terry Gannon

* [webpage](http://www.math.ualberta.ca/~tgannon/)

## Selected writings

On the [[Verlinde formula]] for [[logarithmic CFTs]]:

* [[Thomas Creutzig]], [[Terry Gannon]], _Logarithmic conformal field theory, log-modular tensor categories and modular forms_, J. Phys. A __50__, 404004 (2017) ([doi:10.1088/1751-8121/aa8538](https://doi.org/10.1088/1751-8121/aa8538), [arXiv:1605.04630](https://arxiv.org/abs/1605.04630))

category: people


tertiary radical

Third or tertiary radical of a (possibly noncommutative) unital ring has been studied in

* L. Lesieur, _Théorie noethérienne des anneaux non commutatifs : une propriété caractéristique des idéaux tertiaires_, Séminaire Dubreil. Algèbre et théorie des nombres, 11 no. 2 (1957-1958), Exp. No. 14, 7 p. [numdam](http://www.numdam.org/numdam-bin/item?id=SD_1957-1958__11_2_A1_0)
* L. Lesieur, R. Croisot, _Sur les anneaux premiers noethériens à gauche_, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 76 no. 3 (1959), p. 161-183, [numdam](http://www.numdam.org/numdam-bin/item?id=ASENS_1959_3_76_3_161_0);  _Algèbre nœthérienne non commutative_, Mémorial des sciences mathématiques, 154 (1963), p. 1-119 [numdam](http://www.numdam.org/numdam-bin/item?id=MSM_1963__154__1_0)

where the tertiary decomposition theory has been introduced.

See also

* Gerhard Michler, _Goldman's primary decomposition and the tertiary decomposition_, J. Algebra __16__, 1970 129–137 [MR268218](http://www.ams.org/mathscinet-getitem?mr=268218) doi
A generalization of tertiary decomposition theory to a class of [[Grothendieck categories]] is in

* Nicolae Popescu, _An introduction to Abelian categories with applications to rings and modules_, Acad. Press 1973

category: algebra
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Teruji Thomas

* [webpage](http://users.ox.ac.uk/~mert2060/home/index.shtml)

## related $n$Lab entries

* [[CPT theorem]]

category: people

[[!redirects Teru Thomas]]


tessarines

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# Contents
* table of contents
{:toc}

## Definition

The **tessarines** are an [[algebra]] over the [[real numbers]]. Every tessarine $t$ may be represented as
$$
t = a_0 + a_1 i + a_2 j + a_3k
$$
where the basis elements satisfy the following products:

| $\times$ | i | j | k |
|--|--|--|--|
| i | -1 | k | -j |
| j | k | +1 | i |
| k | -j | i | -1 |

and conjugation $t^* = a_0 - a_1 i - a_2 j - a_3k$.

These are closely related to the [[quaternions]], as the generators satisfy similarly-looking relations.

The tessarines are also referred to as **bicomplex numbers** since they may be obtained using the [generalization of the Cayley-Dickson construction](https://ncatlab.org/nlab/show/Cayley-Dickson+construction#cayleydicksonalbert_construction) applied on the [[complex number|complex numbers]].

## Related concepts

* [[quaternions]]

* [[split-quaternion]]

* [[split-complex numbers]]

## References
* Elena Luna-Elizarrarás, Michael Shapiro, Daniele Struppa (2013) *Bicomplex Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers*. Birkhauser ISBN 978-3-319-24868-4

* Fidelis Zanetti de Castro, Marcos Eduardo Valle. *A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks*. Neural Networks 122, February 2020, Pages 54-67. ([doi](https://doi.org/10.1016/j.neunet.2019.09.040))

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tesselation

#Contents#
* table of contents
{:toc}

## Idea

A *regular tesselation* or *tiling* is a [[covering]] of a [[plane]] by [[regular polygons]] (the tiles), so that polygons overlap only on their [[boundary]] and such the same number of tiles meet at each [[vertex]].

A tesselation by regular $n$-gons such that $k$ of them meet at each [[vertex]] is denoted by the [[Schläfli symbol]] $\{n,k\}$.

### Euclidean tesselation

For example $\{3,6\}$, $\{4,4\}$ and $\{6,3\}$ denote the familiar tilings of the [[Euclidean plane]] by [[triangles]], [[squares]] and [[hexagons]], respectively. In fact these are all tesselations of the Euclidean plane which exist.

### Hyperbolic tesselations

Similarly a *hyperbologic tesselation* is such a tiling of the [[hyperbolic plane]] by [[regular polygons]]. In contrast to the Euclidean case, there are infinitely many different such tesselations.

For example the hyperbolic tesselation $\{5,4\}$ by [[pentagons]]:

In discussion of [[holographic entanglement entropy]] and [[quantum error correcting codes]] this controls the [[HaPPY code]], see also [JGPE 19](Majorana+dimer+code#JGPE19).

Or the hyperbolic tesselation $\{4,6\}$ by [[squares]]:

## Related concepts

* [[wallpaper group]]

* [[holographic entanglement entropy]]

## References

### General

Introduction and survey:

* [[David Joyce]], *[Hyperbolic Tesselations](https://mathcs.clarku.edu/~djoyce/poincare/index.html)*

* Damon Motz-Storey, *Constructing Hyperbolic Polygon Tessellations*, 2016 ([pdf](https://damonmotzstorey.files.wordpress.com/2016/11/thesiswithrevisions_damonmotzstorey.pdf), [[MotzStoreyTesselations.pdf:file]])

See also:

* Wikipedia, *[Tesselation](https://en.wikipedia.org/wiki/Tessellation)*

* Wikipedia, *[Uniform tilings in hyperbolic plane](https://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane)*

* Wikipedia, *[Hyperbolic Geometry -- Hyperbolic Tesselations](https://mathstat.slu.edu/escher/index.php
/Hyperbolic_Geometry#Hyperbolic_Tessellations)*

Claim of a hyperbolic tessalation approximated by a [[photonic crystal]]:

* Alicia J. Kollár, Mattias Fitzpatrick, Andrew A. Houck, *Hyperbolic Lattices in Circuit Quantum Electrodynamics*, Nature **571**  (2019) 45–50 [[arXiv:1802.09549](https://arxiv.org/abs/1802.09549), [doi:10.1038/s41586-019-1348-3](https://doi.org/10.1038/s41586-019-1348-3)]

### Via inflation rules

On iterative constructions of hyperbolic tesselations from "inflation rules":

* Latham Boyle, Madeline Dickens, Felix Flicker, Section IV of: _Conformal Quasicrystals and Holography_, Phys. Rev. X 10, 011009 (2020) ([arXiv:1805.02665](https://arxiv.org/abs/1805.02665))

following:

* Latham Boyle, Paul J. Steinhardt, *Self-Similar One-Dimensional Quasilattices* ([arXiv:1608.08220](https://arxiv.org/abs/1608.08220))

See also:

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], around Fig. 6 in: _Central charges of aperiodic holographic tensor network models_, Phys. Rev. A 102, 042407 ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

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Test


test > history

< [[test]]


test category

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{:toc}

## Idea

The notion of a  _test category_ ([Grothendieck 83](#Grothendieck83))
is meant to axiomatize common features of [[categories]] of [[geometric shapes for higher structures|shapes]] used to model [[homotopy types]] in [[homotopy theory]], such as the categories of [[simplicial sets]], [[cubical sets]] or [[cellular sets]].

## Definition

Given any [[small category]] $\mathcal{C}$, one considers $\mathcal{C}$-sets, hence [[presheaves]] on $\mathcal{C}$, hence [[contravariant functors]] from $\mathcal{C}$ to [[Set]].

Given an [[object]] $c\in C$, one considers the [[representable functor]] $Hom_{\mathcal{C}}(-,c)=:\Delta^c$. If $X \colon \mathcal{C}^{op} \to Set$ is a $\mathcal{C}$-set, the [[elements]] of $X(c)$ are called the _$c$-cells_. By the [[Yoneda lemma]], they correspond to the [[natural transformations]] $\Delta^c\to X$.

Let the **cell category** of $X$, denoted $i_{\mathcal{C}} X$, be the [[full subcategory]] of the [[overcategory]] $\mathcal{C}Set/X$ whose [[objects]] are the transformations of the form $\Delta^c\to X$. (This is another name for the [[category of elements]] of $X$.)

The correspondence $X\mapsto i_{\mathcal{C}}X$ extends to a [[functor]] $i_{\mathcal{C}} \colon\mathcal{C}Set
\to $ [[Cat]], which has a [[right adjoint]] $i_{\mathcal{C}}^* \colon Cat\to\mathcal{C}Set$ whose object part is given by the formula

$$ i_{\mathcal{C}}^*(D)(c)
  \coloneqq
Hom_{Cat}(\mathcal{C}/c,D)
\,.
$$

Denote the [[counit of the adjunction]] $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$.

A [[morphism]] of $\mathcal{C}$-sets $f\colon X\to Y$ is a **weak equivalence** is the induced map $i_{\mathcal{C}}(f)\colon i_{\mathcal{C}} X\to i_{\mathcal{C}} Y$ of their cell categories, i.e., the induced map of [[nerves]] ("[[classifying spaces]]") $B(i_{\mathcal{C}} X)\to B(i_{\mathcal{C}} Y)$ is a [[weak equivalence of simplicial sets]].
Two $\mathcal{C}$-sets $X$ and $Y$ are called **weakly equivalent** if there is a weak equivalence $f \colon X\to Y$. The functor
$$i_{\mathcal{C}}:\mathcal{C}Set\to Cat$$
induces a functor
$$i_{\mathcal{C}*}:Ho(\mathcal{C}Set)\to Ho(Cat)$$
of the [[homotopy categories]].

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

A $\mathcal{C}$-set $X$ is called **aspherical** if the category $i_{\mathcal{C}}(X)$ is weakly contractible, i.e. the nerve $B(i_{\mathcal{C}}(X))$ is a weakly contractible simplicial set. Note that if $\mathcal{C}$ is a weakly contractible category, then this is equivalent to the condition that the map $X \to 1$ to the terminal presheaf is a weak equivalence of $\mathcal{C}$-sets.

A **weak test category** is a small category $\mathcal{C}$ such that,
for any category $D$ in $Cat$ which has a [[terminal object]], the $\mathcal{C}$-set $i_{\mathcal{C}}^\ast(D)$ is aspherical.

A **test category** is any small category $\mathcal{A}$ such that

* ($\mathcal{A}$ is aspherical) its ([[geometric realization]] of the) [[nerve]] ("[[classifying space]]") $\vert \mathcal{A}\vert $ is [[contractible space|contractible]]

* ($\mathcal{A}$ is a "local test category") for every [[object]] $a$ in $\mathcal{A}$ require the [[overcategory]] $\mathcal{A}/a$ to be a weak test category. Thus for each $a \in \mathcal{A}$ and any category $D$ with a terminal object, we require that $B(i_{\mathcal{A}/a}(i_{\mathcal{A}/a}^\ast(D)))$ be a weakly contractible simplicial set.

A **strict test category** is a test category $\mathcal{A}$ such that

* $i_{\mathcal{A}} : \mathcal{A}Set \to Cat$ preserves [[finite products]] up to weak equivalence,

or equivalently, such that

* the induced functor $i_{\mathcal{A}*}:Ho(\mathcal{A}Set)\to Ho(Cat)$ preserves finite products.

=--

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

One can show that a small category $\mathcal{C}$ is a weak test category if and only if both the unit $\epsilon : i_{\mathcal{C}}i_{\mathcal{C}}^*\to Id_{Cat}$ and the co-unit $\eta : Id_{\mathcal{C}Set}\to i^*_{\mathcal{C}}i_{\mathcal{C}}$ are weak equivalence; see Lemme 1.3.8 and Proposition 1.3.9 in ([Maltsiniotis](#Maltsin05)). In particular, one then has an equivalence of categories
$$Ho(\mathcal{C}Set)\cong Ho(Cat)\, .$$

An example of weak test category which is not a test category is the category $\Delta_+$ (so that a presheaf on $\Delta_+$ is a [[semi-simplicial set]]). This comes from a more general family of examples studied in ([Cisinski, Maltsiniotis](#CisMal11)).

=--

## Properties

### Homotopy category

The [[homotopy category]] of a [[category of presheaves]] over a (weak) test category, as a [[category with weak equivalences]] is equivalent to the standard homotopy category of [[homotopy theory]]: that of the category of [[simplicial sets]]/[[topological spaces]] with weak equivalences being [[weak homotopy equivalences]].

In other words, presheaves over a (weak) test category are models for [[homotopy types]] of [[∞-groupoids]]. There is a relative analogue of this situation for local test categories (see below).

### Model category structure

The [[presheaf category]] over a test category with the above weak equivalences admits a [[model category]] structure: the [[model structure on presheaves over a test category]]. This is due to ([Cisinski](#Cisinski)) with further developments due to ([Jardine](#Jardine)). In fact the possibility of such model structures characterizes **local** test categories:

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

A small category $\mathcal{A}$ is a local test category if and only if the category of presheaves of sets on $\mathcal{A}$ is equipped with a [[model category]] structure whose cofibrations are the monomorphisms with the weak equivalences defined as above. Such a model category structure is always [[proper model category | proper]]. Furthermore, if $\mathcal{A}$ is a local test category, the assignment $X\mapsto (N(i_{\mathcal{A}}(X))\to N(\mathcal{A}))$ is a [[Quillen equivalence]]
$$\mathcal{A}Set\to SSet/N(\mathcal{A})$$
(where $SSet/\mathcal{C}$ is equipped with the sliced [[classical model structure on simplicial sets | Kan-Quillen model structure]]),
and thus induces equivalences of homotopy categories
$$Ho(\mathcal{A}Set)\cong Ho(Cat/\mathcal{A})\cong Ho(SSet/N(\mathcal{A}))\, .$$

=--

+-- {: .proof}
###### Proof
See Théorème 1.4.3 and Corollaire 4.2.18 in ([Cisinski](#Cisinski)) for the first and second assertion. The Quillen equivalence is a particular case of Proposition 4.4.28 in loc. cit. The properness of the model structure is a particular case of Théorème 4.4.30 in loc. cit.
=--

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

If $\mathcal{A}$ is a local test category, one can describe the model structure on $\mathcal{A}$-sets independently from the functor $i_\mathcal{A}$ as follows: it is the minimal [[Cisinski model structure]] such that any morphism between representable presheaves is a weak equivalence and such that any presheaf on $\mathcal{A}$ is canonically an homotopy colimit of representable presheaves. This is Proposition 6.4.26 in ([Cisinski](#Cisinski)).

=--

## Examples

Apart from the archetypical example of the [[simplex category]] we have the following

* The [[cube category]] is a test category ([Grothendieck](#Grothendieck), [Cisinski](#Cisinski)), however not a strict one ([Kan](#Kan)). (The corresponding [[model category]] is discussed at [[model structure on cubical sets]].)

  The category of [[cubes]] equipped with [[connection on a cubical set | connections]] is even a strict test category ([Maltsiniotis, 2008](#Maltsiniotis)).

* The [[Theta category]] and its [[groupoid]]-analog $\tilde \Theta$ are strict test categories ([Cisinski, Maltsiniotis](#CisMal11) and [Ara](#Ara)).

* The [[tree category]] $\Omega$ is a test category. This was proven in an unpublished note of Cisinski, and later appeared as [Ara, Cisinski, Moerdijk, 2019](#AraCisinskiMoerdijk).

* The [[cycle category]] of Connes is a local test category. The corresponding model structure models homotopy types over $K(\mathbb{Z},2)$; see paragraph 8.5.14 and Proposition 8.5.19 in ([Cisinski](#Cisinski)).

* Given a local test category $\mathcal{A}$ and a small category $\mathcal{C}$, the [[product]] $\mathcal{A}\times\mathcal{C}$ is a local test category. It is a test category if both $\mathcal{A}$ and $\mathcal{C}$ have weakly contractible nerves. This is a special case of a more general fact: if $\mathcal{A}$ is a local test category, for any a [[Grothendieck fibration]] $p:\mathcal{X}\to\mathcal{A}$ (with small fibers), the category $\mathcal{X}$ is a local test category. This follows from Corollaire 1.7.15, Exemple 3.2.2 and the dual version of Proposition 3.2.9 in ([Maltsiniotis, 2005](#Maltsin05)). In particular, given any presheaf of groups $G$ on a local test category $\mathcal{A}$, one gets a nice homotopy theory of representations of $G$; see Scholie 7.2.15 in [Cisinski](#Cisinski)).

Any small category $\mathcal{A}$ which is closed under finite products and which contains an interval is a strict test category (where an interval is an object $I$ equipped with two morphisms from the terminal object $d^e:*\to I$, $e=0,1$, such that there is no maps from an object of $\mathcal{A}$ which factors through both $d^0$ and $d^1$); see Corollaire 1.5.7 in ([Maltsiniotis](#Maltsin05)). Particular cases include the following examples.

* The category of non-empty finite sets is a strict test category (hence [[symmetric simplicial sets]] are a model of homotopy types of CW-complexes).

* The full subcategory of the category of topological spaces, whose objects are closed balls in euclidian spaces, is a strict test category.

* The category of open balls in euclidian spaces, with $C^\infty$-maps as morphisms, is a strict test category.

* The category of contractible [[Stein manifold | Stein manifolds]], with holomorphic maps as morphisms, is a strict test category.

## Related concepts

* [[direct category]]

* [[basic localizer]]

* [[test topos]]

## References

The notion of test category was introduced in

* {#Grothendieck83} [[Alexandre Grothendieck]], _[[Pursuing Stacks]]_, 1983 ([djvu](http://people.math.jussieu.fr/~maltsin/groth/ps/Pursuing_Stacks.djvu))

Various conjectures made there are proven in

* {#Cisinski} [[Denis-Charles Cisinski]], _Les préfaisceaux comme modèles des types d'homotopie_, Asterisque __308__, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf).

which moreover develops the main toolset and establishes the [[model structure on presheaves over a test category]].

General surveys include

* {#Maltsin05} [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck_, Astérisque, 301, pp. 1-140, (2005) (see
[html](http://people.math.jussieu.fr/~maltsin/textes.html))

* [[J. F. Jardine]], _Categorical homotopy theory_, Homot. Homol. Appl. __8__ (1), 2006, pp.71–144, ([HHA](http://www.intlpress.com/HHA/v8/n1/a3/),  [pdf](http://www.intlpress.com/HHA/v8/n1/a3/v8n1a3.pdf))

That the [[cube category]] is a test category is asserted without proof in  ([Grothendieck](#Grothendieck)). A proof is spelled out in ([Cisinski](#Cisinski))

That it is not a strict test category is implicitly already in

* {#Kan} [[Dan Kan]], _Abstract homotopy. I_ , Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092–1096. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528202/pdf/pnas00727-0082.pdf))

and led to the preference for [[simplicial sets]] over [[cubical sets]].

That the category of [[cubes]] _equipped with [[connection on a cubical set]]_ forms a _strict_ test category is shown in

* [[Georges Maltsiniotis]], _La catégorie cubique avec connexions est une catégorie test stricte_ . (French. English summary) Homology, Homotopy Appl. 11 (2009), no. 2, 309–326. ([web](http://www.intlpress.com/HHA/v11/n2/a15/))
 {#Maltsiniotis}

The test category nature of the groupoidal [[Theta category]] is discussed in

* {#Ara} [[Dimitri Ara]], _The groupoidal analogue $\tilde \Theta$ to Joyal's category $\Theta$ is a test category_ ([arXiv:1012.4319](http://arxiv.org/abs/1012.4319))

and the [[Theta category]] itself is discussed in

* [[Denis-Charles Cisinski]], [[Georges Maltsiniotis]], _La catégorie $\Theta$ de Joyal est une catégorie test_ , JPAA **215** no.5 (2011) pp.962-982. ([pdf](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/theta.pdf), in French)
{#CisMal11}

That fact that the [[tree category]] is a test category was proved in

* {#AraCisinskiMoerdijk} [[Dimitri Ara]], [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _The dendroidal category is a test category_, Mathematical Proceedings of the Cambridge Philosophical Society, 167(1), 107-121, 2019, ([doi:10.1017/S030500411800021X](https://doi.org/10.1017/S030500411800021X), [arXiv:1703.07098](http://arxiv.org/abs/1703.07098)).

A short introduction can be found in

* [[Chris Kapulkin]], _Introduction to Test Categories_ [PDF](http://www.math.uwo.ca/faculty/kapulkin/notes/test_categories.pdf).

[[!redirects test categories]]
[[!redirects strict test category]]
[[!redirects strict test categories]]
[[!redirects weak test category]]
[[!redirects weak test categories]]
[[!redirects Grothendieck homotopy theory]]


test topos

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

#Contents#
* table of contents
{:toc}

## Idea

We would like a general procedure to determine when a [[topos]] can be a category of models of homotopy types of CW-complexes, the same way simplicial sets, cubical sets, or any other category of presheaves on a [[test category]] do. In particular, given a sheaf $X$ on a site $C$, we would like that the [[shape theory | shape]] of the topos $Sh(C)_{/X}$ to be strongly related with the (weak) homotopy type of $X$. Typical examples coming from geometry should be sheaves of sets on suitable categories of manifolds. Grothendieck never mentioned the concept of test topoi themselves, but the idea of `test topologies' on test categories is expressed from time to time in Pursuing Stacks. As in the case of test categories, the natural (i.e most robust) notion is in fact local: it is worth looking at models of homotopy types over a given homotopy type.

In other terms, given a topos $X$, a natural question will be to formulate how the objects of $X$ are models of homotopy types over the homotopy type of $X$ (defined through shape theory). The following is a summary of part of ([Cisinski03](#Cisinski03])).

## Artin-Mazur weak equivalences

A morphism of Grothendieck topoi $f\colon X\to Y$ is an **Artin-Mazur weak equivalence** if, for any integer $n\geq 0$ and any locally constant sheaf $F$ on $Y$, the induced map
\[
f^*\colon H^n(Y,F)\to H^n(X,F)
\]

is bijective, with $n=0$ if $F$ is a sheaf of sets, $n=1$ if $F$ is a sheaf of groups, and $n$ is arbitrary if $F$ is a sheaf of abelian groups. These are simply called **weak homotopy equivalences** of topoi in ([Moerdijk95](#Moerdijk95)).
In particular, any morphism inducing an equivalence of [[shape theory | shapes]] of the associated hypercomplete $\infty$-topoi is
Artin-Mazur weak equivalence; see ([Hoyois18](#Hoyois18)).

Given a [[Grothendieck topos]] $T$, a morphism $\colon x\to y$ in $T$ is an **Artin-Mazur weak equivalence** if the induced morphism of topoi $T_{/x}\to T_{/y}$ (whose pull-back functor is defined via the assignment $(t\to y)\mapsto (t\times_y x\to x)$) is an Artin-Mazur weak equivalence.

+-- {: .num_remark}
###### Remark
If $f\colon C\to D$ is a functor between small categories, it induces a morphism of topoi $f\colon PSh(C)\to PSh(D)$ whose pull-back functor is defined by precomposition with $u$. The nerve of the functor $f$ is a weak homotopy equivalence of simplicial sets if and only if the corresponding morphism of topoi is an Artin-Mazur weak equivalence. This is a reformulation of a well known theorem of Whitehead asserting that one can detect weak homotopy equivalences through cohomology with coefficients in local systems.

=--

There is an analogue of (a relative version of) Quillen's Theorem A which is a direct application of (hyper)[[descent]] (e.g. see Thm. 3.4.25 in ([Cisinski03](#Cisinski03))):

+-- {: .num_prop #theorem A for topoi}
###### Proposition
_Let_
\[
\array{ X & & \overset{f}{\to} & & Y\\ & _p \searrow & & \swarrow_q \\ & & S }
\]
_be a commutative triangle of Grothendieck topoi.
Let $(S_i)_{i\in I}$ be a small family of maps in $S$ which is a covering, i.e. so that the induced map from $\coprod_i S_i$ to the terminal object of $S$ is an epimorphism. If, for each index $i$ the induced map $X_{/p^*(S_i)}\to Y_{/q^*(S_i)}$ is an Artin-Mazur weak equivalence, then the map $f:X\to Y$ is
an Artin-Mazur weak equivalence as well._
=--

A particular case is the following.

+-- {: .num_prop #theorem A in topoi}
###### Corollary
_Let_
\[
\array{ X & & \overset{f}{\to} & & Y\\ & _p \searrow & & \swarrow_q \\ & & S }
\]
_be a commutative triangle in a Grothendieck topos $T$.
Let $(S_i\to S)_{i\in I}$ be a small family of maps in $T$ which is a covering, i.e. so that the induced map $\coprod_i S_i\to S$ is an epimorphism.
If, for each index $i$ the induced map $S_i\times_S X\to S_i\times_SY$ is an Artin-Mazur weak equivalence, then the map $f:X\to Y$ is
an Artin-Mazur weak equivalence as well._
=--

+-- {: .num_prop #saturation}
###### Proposition
_Let $T$ be a Grothendieck topos. The class of monomorphisms which are Artin-Mazur weak equivalences is [[saturated class of maps | saturated]] and stable under small filtered colimits._
=--

+-- {: .proof}
###### Proof
This follows right away from Prop. 3.4.22 in ([Cisinski03](#Cisinski03)).
=--

+-- {: .num_defn #asphtopos}
###### Definition
A Grothendieck topos $T$ is **aspherical** if the map $T\to pt$ to the terminal topos is an Artin-Mazur weak equivalence.

An object $x$ of a topos $T$ is **aspherical** if the corresponding topos $T_{/x}$ is aspherical.

A Grothendieck topos $T$ is **locally aspherical** there is a generating family of $T$ which consists of aspherical objects.
=--

+-- {: .num_example #ex1}
###### Example
If $C$ is a small category, the corresponding topos of presheaves of sets $PSh(C)$ is locally aspherical.
Indeed, representable presheaves generate $PSh(C)$, and, for any presheaf $x$ on $C$ represented by an object $c$ of $C$, we have a canonical equivalence
$$PSh(C)_{/x}\cong PSh(C_{/c})$$
with $PSh(C_{/c})$ aspherical because the slice category $C_{/c}$ has a terminal object.
=--

+-- {: .num_example #ex3}
###### Example
A topos $T$ is locally aspherical if and only if $T\cong Sh(C)$ where $C$ is a small site such that, for any representable presheaf $U$ on $C$, with associated sheaf $U^{\wedge}$,
the topos $T_{/U^{\wedge}}$ is aspherical. Equivalently (using the toposic analogue of Quillen's Theorem A above), for any presheaf $F$ on $C$ with associated sheaf $F^{\wedge}$, the induced morphism of topoi
$T_{/F^{\wedge}}\to PSh(C)_{/F}$ is an Artin-Mazur weak equivalence.
=--

## Characterization and examples

+-- {: .num_defn #interval}
###### Definition
An **interval** in a topos $T$ is an object $I$ with the following properties:

* it has two disjoint global sections: there is a monomorphism from the disjoint union of two copies of the terminal object to $I$;

* for any object $X$ in $T$, the projection $I\times X\to X$ is an Artin-Mazur weak equivalence (equivalently, there exists
a small family $(S_i)_{i\in I}$ which covers $T$ such that the projection $I\times S_i\to S_i$ is an Artin-Mazur weak equivalence).
=--

+-- {: .num_prop #characterization of local pretest topoi}
###### Proposition

Let $T$ be a Grothendieck topos. The following conditions are equivalent:

* Any morphism of $T$ with the right ligting property with respect to monomorphisms is an Artin-Mazur weak equivalence;

* There exists an interval in $T$;

=--

+-- {: .proof}
###### Proof
Since the class of Artin-Mazur weak equivalences has the 2-out-of-3 property and is saturated, this follows from Lemma 3.3 in ([Cisinski02](#Cisinski02)).
=--

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

A **local test topos** is a topos which is locally aspherical and which has an interval.

A **test topos** is a local test topos which is aspherical.

=--

+-- {: .num_example #ex4}
###### Example

A small category $A$ is a (local) [[test category]] if and only if the associated topos $PSh(A)=Fun(A^{op},Set)$ is a (local) test topos:
this follows from the proposition right above and from the characterization of local test categories provided by Thm. 1.5.6 in ([Maltsiniotis05](#Maltsin05)).

=--

+-- {: .num_example #ex5}
###### Example

Let $T$ be a topos and $C$ a small site such that $T\cong Sh(C)$. We assume that $C$ is a local [[test category]] and that,
for any representable presheaf $F$ on $C$ with associated sheaf $F^\wedge$, the topos $T_{/U^{\wedge}}$ is aspherical.
Then $T$ is a local test topos: any interval of $PSh(C)$ is then an interval of $T$, and $T$ is clearly
locally aspherical; see Thm. 4.2.8 in ([Cisinski03](#Cisinski03)).

=--

+-- {: .num_theorem #toposic local test model structure}
###### Theorem

Let $T$ be a local test topos. Then there is a proper [[combinatorial model category]] structure on $T$ whose
cofibrations are the monomorphisms, and whose weak equivalences are the Artin-Mazur weak equivalences.
Futhermore, this model structure provides models for homotopy types over the homotopy type of $T$.
More precisely, if $C$ is any small site whose underlying category is a local test category, such that
$T\cong Sh(C)$, and such that, for any representable presheaf $F$ on $C$ with associated sheaf $F^\wedge$,
the topos $T_{/F^{\wedge}}$ is aspherical, then the sheafification functor is a left Quillen equivalence
$PSh(C)\to T$, and there is a left Quillen equivalence $PSh(C)\to SSet_{/N(C)}$ which sends any presheaf
$F$ to the nerve of its category of elements $C_{/F}$.

=--

+-- {: .proof}
###### Proof
The first assertion on the existence of a combinatorial proper model structure comes from Cor. 4.2.12 in ([Cisinski03](#Cisinski03))
and is an example of a [[Cisinski model structure]] provided by Thm. 3.9 in ([Cisinski02](#Cisinski02)).
For the second part, it is obvious that the sheafification functor is a left Quillen equivalence from $PSh(C)$ to $T$
and an Artin-Mazur weak equivalence. Therefore, it is sufficient to prove the case where $T=PSh(C)$. This is then a particular
case of Proposition 4.4.28 in ([Cisinski06](#Cisinski06)).

=--

+-- {: .num_prop #toposic test model structure}
###### Corollary

Let $T$ be a test topos. Then there is a proper combinatorial model category structure on $T$ whose
cofibrations are the monomorphisms, and whose weak equivalences are the Artin-Mazur weak equivalences which
models homotopy types of CW-complexes.

=--

+-- {: .num_example #ex6}
###### Example

Let $T$ be a local test topos and $C$ an internal category of $T$. On can then consider the topos $PSh(C)$ of internal
presheaves on $C$ (which only depends on the stack associated to $C$). Then, by virtue of Thm. 6.2.7
in ([Cisinski03](#Cisinski03])), $PSh(C)$ is a local test topos as well.
In particular, if $G$ is a sheaf of groups on $T$, then $Rep(G)=PSh(BG)$ is the category of representations of $G$ (i.e.
the objects of $T$ equipped with an action of $G$). If the Artin-Mazur weak equivalences are stable under finite products in $T$,
then one can show that the Artin-Mazur weak equivalences of $Rep(G)$
simply are the $G$-equivariant maps $X\to Y$ which are Artin-Mazur weak equivalences in $T$ when we forget the action
of $G$; to see this, one reduces easily to the case where $T$ is a category of presheaves on a local test category,
and this follows then from Prop. 7.3.8 in ([Cisinski02](#Cisinski02])).

=--

+-- {: .num_example #ex7}
###### Example

Let us consider a full subcatogry $C$ of the category of locally ringed spaces with the following properties:

* for any space $X$ in $C$, there is a basis of contractible open subsets of $X$ which are $C$;

* for any locally ringed space $X$, if each point of $X$ admits an open neighborhood in $C$, then $X$ belong to $X$;

* the forgetful functor from $C$ to the category of topological spaces commutes with products (up to weak equivalence);

* there exists a contractible space with two distinct closed points in $C$.

We consider $C$ as site with the Grothendieck topology induced by open coverings.
Then, for any $X$ in $C$, the topos $Sh(C)_{/X}$ is aspherical because there is an adjunction between $Sh(X)$ and $Sh(C)_{/X}$ in the $2$-category of topoi.
Therefore, since $C$ is a [[strict test category]], we see that $Sh(C)$
is a test topos in which the Artin-Mazur weak equivalences are stable under finite products; see Thm. 6.1.8 in ([Cisinski03](#Cisinski03])).
If $X$ is a locally ringed space which has a covering by spaces in $C$, then the sheaf on $C$ represented by $X$ corresponds
in the test model structure on $Sh(C)$ to the classical weak homotopy type of $X$ (this follows by hyperdescent in shape theory). In practice, all objects of $C$
admit CW-structures, so that we even get the classical homotopy type of $X$ characterized by the sheaf represented by $X$ on $C$.
Such geometric models of weak homotopy types of manifolds are used for instance by Madsen and Weiss ([Madsen-Weiss07](#Madsen-Weiss07)) and Kupers ([Kupers19](#Kupers19)).

Examples of such 'test sites' $C$ are given by: euclidian spaces (seen as $C^\infty$-manifolds), or contractible Stein complex manifolds.
If $G$ is a real (or complex) Lie group, then representations of $G$ in the topos of sheaves on differentiable (or complex analytic) manifolds
is a local test topos. One may also replace $G$ by any orbifold, presented as an internal groupoid in the category of manifolds.

=--

## Related concepts

* [[test category]]

## References

* {#Artin-Mazur69} [[Michael Artin]] and [[Barry Mazur]], _Etale homotopy_, Lecture Notes in Mathematics 100, Springer (1969).

* {#Cisinski02} [[Denis-Charles Cisinski]], _Théories homotopiques dans les topos_, J. Pure Appl. Algebra 174, No. 1 (2002). ([doi](https://www.sciencedirect.com/science/article/pii/S0022404901001761))

* {#Cisinski03} [[Denis-Charles Cisinski]], _Faisceaux localement asphériques_, preprint (2003) [[pdf](http://www.mathematik.uni-regensburg.de/cisinski/mtest2.pdf), [[Cisinski-FLA.pdf:file]]]

* {#Hoyois18} [[Marc Hoyois]], _Higher Galois theory_, J. Pure Appl. Algebra 222, no. 7 (2018), ([arXiv](http://arxiv.org/abs/1506.07155))

* {#Kupers19} Alexander Kupers, _Three applications of delooping to h-principles_, Geom. Dedicata 202 (2019), pp. 103–151. ([journal](https://link.springer.com/article/10.1007/s10711-018-0405-7), [arXiv](https://arxiv.org/abs/1701.06788))

* {#Madsen-Weiss07} [[Ib Madsen]] and [[Michael Weiss]], _The stable moduli space of Riemann surfaces: Mumford’s conjecture_, Ann. of Math. (2)165(2007), no. 3, 843–941. ([journal](http://doi.org/10.4007/annals.2007.165.843), [arXiv](https://arxiv.org/abs/math/0212321))

* {#Maltsin05} [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck_, Astérisque 301 (2005) (see
[html](http://people.math.jussieu.fr/~maltsin/textes.html))

* {#Moerdijk95} [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_, Lecture Notes in Mathematics 1616, Springer (1995).

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TestPageB

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[[!redirects TestPageF]]

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Nothing again. x x


TestPageD

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A monad is a monoid in an endofunctor category


tetracategory

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

#Contents#
* table of contents
{:toc}

## Idea

Tetracategories are the [[algebraic definition of higher category]] for the most general (i.e. weak) [[n-category|4-categories]].

## Properties

By the discussion at [[k-tuply monoidal (n,r)-category]] a tetracategory with a single object may be regarded as the [[delooping]] of a monoidal [[tricategory]]. This is discussed in ([Hoffnung](#Hoffnung)).

## Related concepts

* [[category]]

* [[bicategory]]

* [[tricategory]]

* **tetracategory**

## References

The structure of a tetracategories was given by [[Todd Trimble]].

* Todd Trimble [Notes on Tetracategories](http://math.ucr.edu/home/baez/trimble/tetracategories.html)

  * [[TetracategoricalData.pdf:file]]

  * [[K6associativity.pdf:file]]

  * [[U52unitcondition.pdf:file]]

  * [[U53unitconditionA.pdf:file]]

  * [[U54unitcondition.pdf:file]]

A polished writeup of the definition appears in section 3.2 of

* [[Alex Hoffnung]], _Spans in 2-Categories: A monoidal tricategory_ ([arXiv:1112.0560](http://arxiv.org/abs/1112.0560))
 {#Hoffnung}

[[!redirects tetracategories]]

[[!redirects 4-category]]
[[!redirects 4-categories]]


tetrahedral group

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

#Contents#
* table of contents
{:toc}

## Idea

The _tetrahedral group_ is the [[finite group|finite]] [[symmetry group]] of a [[tetrahedron]].

As a symmetry group of one of the [[Platonic solids]], the tetrahedral group participates in the _ADE pattern_:

[[!include ADE -- table]]

More in detail, there are variants of the tetrahedral group corresponding to the stages of the [Whitehead tower of O(3)](Whitehead+tower#OfTheOrthogonalGroup):

* the full tetrahedral group is the [[subgroup]] of [[orthogonal group|O(3)]]

  $$
    T_d \hookrightarrow O(3)
  $$

  which is the [[stabilizer subgroup|stabilizer]] of the standard embedding of the [[tetrahedron]] into [[Cartesian space]] $\mathbb{R}^3$;

* the _rotational tetrahedral group_ $T \hookrightarrow SO(3)$ is the restriction to [[orientation]]-preserving symmetries, hence to [[special orthogonal group|SO(3)]]; this is [[isomorphism|isomorphic]] to the [[alternating group]] $A_4$;

* next the _binary tetrahedral group_ $2T$ is the [[double cover]], hence the lift of $T$ to [[spin group|Spin(3)]]$\simeq$ [[special unitary group|SU(2)]], this is equivalently the [[special linear group]] over the [[prime field]] $\mathbb{F}_3$

  $$
    2T \simeq SL(2,\mathbb{F}_3)
  $$

* then there is a [[string 2-group]] lift $String_{2T} \hookrightarrow String_{SU(2)}$ of the tetrahedral group to a _[[Platonic 2-group]]_ ([Epa 10](#Epa10), [Epa-Ganter 16](#EpaGanter16))

$$
  \array{
     String_{2T} &\hookrightarrow& String_{SU(2)}
     \\
     \downarrow && \downarrow
     \\
     2 T &\hookrightarrow & Spin(3) \simeq SU(2)
     \\
     \downarrow && \downarrow
     \\
     T \simeq A_4 &\hookrightarrow&  SO(3)
     \\
     \downarrow && \downarrow
     \\
     T_d \simeq S_5 &\hookrightarrow & O(3)
  }
$$

## Definition

The binary tetrahedral group $2T$ is, up to [[isomorphism]], the [[subgroup]] of the group $S(\mathbb{H}) \simeq$ [[special unitary group|SU(2)]] $\simeq Spin(3)$ of unit [[quaternions]] consisting of  the 8 unit [[quaternions]] $\pm 1$, $\pm i$, $\pm j$, $\pm k$ and the 16 unit quaternions given by
$\frac1{2}(\varepsilon_0 1 + \varepsilon_1 i + \varepsilon_2 j + \varepsilon_3 k)$ where $(\varepsilon_0, \ldots, \varepsilon_3) \in \{-1, 1\}^4$.

(These are also the [[vertices]] that span the [[24-cell]] as a  convex [[regular polytope]] in $\mathbb{R}^4$).

## Properties

### Basic properties

The full tetrahedral group is [[isomorphism|isomorphic]] to the [[symmetric group]] $S_4$ of [[permutations]] of four elements (see _[Full tetrahedral group is isomorphic to S4](http://groupprops.subwiki.org/wiki/Full_tetrahedral_group_is_isomorphic_to_S4)_).

The [[subgroup]] of [[orientation]]-preserving symmetries is isomorphic to the [[alternating group]] $A_4$.

The [[order of a group|group order]] is:

$\vert T_d\vert = 24$

$\vert T\vert = 12$

$\vert 2T\vert = 24$

+-- {: .num_prop #QuaternionSubgroup}
###### Proposition
**([[quaternion group]] inside [[binary tetrahedral group]])**

The binary tetrahedral group contains the [[quaternion group]] of [[order]] 8, hence the [[binary dihedral group]] of [[order of a group|order]] 8, as a [[subgroup]], in fact as a [[normal subgroup]]:

$$
  2 D_4 =Q_8 \subset 2 T
  \,.
$$

In fact the only [[finite subgroups of SU(2)]] which contain $2 D_4 =Q_8$ as a proper subgroup are the exceptional ones, hence the [[binary tetrahedral group]], the [[binary octahedral group]] and the [[binary icosahedral group]].

=--

See [this Prop](quaternion+group#InclusionInLargerFininteSubgroupsOfSU2) at _[[quaternion group]]_.

### As part of the ADE pattern

[[!include ADE -- table]]

### Character table

[[!include character table of 2T]]

### Group cohomology

The [[group cohomology]] of the tetrahedral group is discussed in [Groupprops](#Groupprops), [Tomoda & Zvengrowski 08, Sec. 4.1](#TomodaZvengrowski08) [Kirdar 13](#Kirdar13), [Epa & Ganter 16, p. 12](#EpaGanter16).

## Related concepts

* [[GL(2,3)]]

## References

Discussion in the context of [[classification of finite rotation groups]] goes back to:

* {#Klein1884} [[Felix Klein]], chapter I.4 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

Exposition:

* Tony Phillips, _[The Geometry of the Binary Tetrahedral Group in SU(2)](http://www.math.stonybrook.edu/~tony/bintet/)_

See also:

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

Discussion of the [[group cohomology]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

* {#Kirdar13} Mehmet Kirdar, _On The K-Ring of the Classifying Space of the Symmetric Group on Four Letters_ ([arXiv:1309.4238](http://arxiv.org/abs/1309.4238))

* {#Groupprops} [[Groupprops]], _[Group cohomology of symmetric group:S4](http://groupprops.subwiki.org/wiki/Group_cohomology_of_symmetric_group:S4)_

Discussion of [[higher central extension]] to [[Platonic 2-groups]] is in

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

[[!redirects tetrahedral groups]]

[[!redirects binary tetrahedral group]]
[[!redirects binary tetrahedral groups]]

[[!redirects 2T]]
[[!redirects SL(2,3)]]


tetrahedron

#Contents#
* table of contents
{:toc}

## Idea

A *tetrahedron* is the convex [[regular polyhedron]] with four faces, as such one of the [[platonic solids]].

Also known as the 3-dimensional *[[simplex]]*.

\begin{tikzpicture}[scale=1.5]
  \coordinate (a) at (.4,-.3);
  \coordinate (b) at (1.7,.4);
  \coordinate (c) at (.6,2);
  \coordinate (d) at (-.6,.6);

\draw (a) -- (d);

\draw[dashed]
  (d) -- (b);

\draw[
    line width=1.1,
]
    (d) -- (a) -- (c) -- cycle;

\draw[
    line width=1.1,
    fill=black!20,
    fill opacity=.6
]
  (a) -- (b) -- (c) -- cycle;

\draw[fill=black] (a) circle (.07);
\draw[fill=black] (b) circle (.07);
\draw[fill=black] (c) circle (.07);
\draw[fill=black] (d) circle (.07);

\end{tikzpicture}

## Related concepts

[[!include ADE -- table]]

## Related concepts

* [[circle]]

* [[triangle]]

* [[square]]

## References

* {#Klein1884} [[Felix Klein]], chapter I.6 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

See also:

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

[[!redirects tetrahedra]]

.


tetraquark

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _tetraquark_ is an [[exotic meson]] which is a [[bound state]] of four [[quarks]], or rather: of a [[diquark]] and an [[antiparticle|anti]]-[[diquark]].

## Related concepts

* [[hadron supersymmetry]]

* [[XYZ meson]]

## References
 {#References}

### General

The idea of tetraquarks is mentioned already in the original articles on [[quarks]].

The original proposal for tetraquarks with two [[heavy quark]] [[constitutent quark|constitutents]]:

* J.P. Ader,  J.M. Richard, P. Taxil, *Do narrow heavy multiquark states exist?*, Phys. Rev. D 25 (1982) 2370 ([doi:10.1103/PhysRevD.25.2370](https://doi.org/10.1103/PhysRevD.25.2370), [spire:166170](https://inspirehep.net/literature/166170))

and fully heavy tetraquarks:

* L. Heller and J. A. Tjon, *Bound states of heavy *Q^2 Q^{-2}* systems*, Phys. Rev. D **32** (1985) 755 ([doi:10.1103/PhysRevD.32.755](https://doi.org/10.1103/PhysRevD.32.755))

See also:

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

### Via hadron supersymmetry

Discussion in view of [[hadron supersymmetry]] and [[diquark]]-models:

in [[holographic light front QCD]]:

* [[Hans Günter Dosch]], [[Stanley Brodsky]], [[Guy de Teramond]], [[Marina Nielsen]], [[Liping Zou]], _Exotic States in a Holographic Theory_, [QCD 20)](https://inspirehep.net/conferences/1778269) ([arXiv:2012.02496](https://arxiv.org/abs/2012.02496))

in regards to experimental results:

* M. A. Bedolla, J. Ferretti, C. D. Roberts, E. Santopinto, *Spectrum of fully-heavy tetraquarks from a diquark+antidiquark perspective* ([arXiv:1911.00960](https://arxiv.org/abs/1911.00960), [doi:10.1140/epjc/s10052-020-08579-3](https://doi.org/10.1140/epjc/s10052-020-08579-3))

### Experimental detection
 {#ExperimentalDetection}

#### Fully charmed tetraquark

[[experiment|Experimental]] detection of a tetraquark resonance $X(6900)$ made entirely of four [[charm quarks]] and argued to be the first observed "true" tetraquark state (instead of just a "[[diquark]] molecule"):

* [[LHCb collaboration]], *Observation of structure in the $J/\psi$-pair mass spectrum*, Science Bulletin 65 (2020) 1983 ([arXiv:2006.16957](https://arxiv.org/abs/2006.16957), [doi:10.1016/j.scib.2020.08.032](https://doi.org/10.1016/j.scib.2020.08.032), [CERN news](https://phys.org/news/2020-07-cern-physicists-discovery-unique-particle.html))

Popular exposition:

* Davide Castelvecchi, *Exotic four-quark particle spotted at Large Hadron Collider*, Nature News 10 Aug 2021 ([doi:10.1038/d41586-021-02174-6](https://doi.org/10.1038/d41586-021-02174-6))

Theoretical explanation:

* Ming-Sheng liu, Feng-Xiao Liu, Xian-Hui Zhong, Qiang Zhao, *Full-heavy tetraquark states and their evidences in the LHCb di-J/ψ spectrum* ([arXiv:2006.11952](https://arxiv.org/abs/2006.11952))

* Qi-Fang Lü, Dian-Yong Chen, Yu-Bing Dong, *Masses of fully heavy tetraquarks $Q Q \overline{Q} \overline{Q}$ in an extended relativized quark model*,  Eur. Phys. J. C 80, 871 (2020) ([arXiv:2006.14445](https://arxiv.org/abs/2006.14445), [doi:10.1140/epjc/s10052-020-08454-1](https://doi.org/10.1140/epjc/s10052-020-08454-1))

reviewed in:

* [[Jean-Marc Richard]], *About the $J/\psi$ $J/\psi$ peak of LHCb: fully-charmed tetraquark?*, Science Bulletin Volume 65, Issue 23, 15 December 2020, Pages 1954-1955 ([arXiv:2008.01962](https://arxiv.org/abs/2008.01962), [doi:10.1016/j.scib.2020.08.020](https://doi.org/10.1016/j.scib.2020.08.020))

* [[Jean-Marc Richard]], *Fully-heavy tetraquarks and other heavy multiquarks*, Nuclear and Particle Physics Proceedings 00 (2021) 1–5 ([arXiv:2105.02503](https://arxiv.org/abs/2105.02503))

#### $T^+_{c c}$ tetraquark

Indication of a tetraquark $c c \bar u \bar d$ with two [[charm quarks]], and [[anti-particle|anti]]-[[up quark]] and an [[anti-particle|anti]]-[[down quark]]:

* [[LHCb collaboration]], *Observation of an exotic narrow doubly charmed tetraquark* ([arXiv:2109.01038](https://arxiv.org/abs/2109.01038))

* [[LHCb collaboration]], *Study of the doubly charmed tetraquark $T^+_{c c}$* ([arXiv:2109.01056](https://arxiv.org/abs/2109.01056))

Further discussion:

* Sean Fleming, Reed Hodges, Thomas Mehen, *$T_{c c}^+$ Decays: Differential Spectra and Two-Body Final States* ([arXiv:2109.02188](https://arxiv.org/abs/2109.02188))

* Huimin Ren, Fan Wu, Ruilin Zhu, *Hadronic molecule interpretation of $T_{c c}^+$ and its beauty-partners* ([arXiv:2109.02531](https://arxiv.org/abs/2109.02531))

[[!redirects tetraquarks]]


Tetsuji Kimura

* [personal page](http://www2.yukawa.kyoto-u.ac.jp/~tetsuji.kimura/index.html)

* [inspire page](https://inspirehep.net/authors/1020358)

## Selected writings

On [[RR-field tadpole cancellation]] on [[orbifolds]]:

* [[Tetsuji Kimura]], Mitsuhisa Ohta, Kei-Jiro Takahashi, _Type IIA orientifolds and orbifolds on non-factorizable tori_, Nucl. Phys. B **798** (2008) 89-123 [[arXiv:0712.2281](https://arxiv.org/abs/0712.2281), [doi:10.1016/j.nuclphysb.2008.01.030](https://doi.org/10.1016/j.nuclphysb.2008.01.030)]

On [[defect brane|defect]] [[(p,q)5-branes]]:

* [[Tetsuji Kimura]], _Defect $(p,q)$ Five-branes_, Nucl.Phys. B **893** (2015) 1-20 [[arXiv:1410.8403](https://arxiv.org/abs/1410.8403), [doi:10.1016/j.nuclphysb.2015.01.023](https://doi.org/10.1016/j.nuclphysb.2015.01.023)]

* [[Tetsuji Kimura]], Shin Sasaki, Kenta Shiozawa, *On Geometries and Monodromies for Branes of Codimension Two* [[arXiv:2312.03358](https://arxiv.org/abs/2312.03358)]

On [[exotic branes]] in [[string theory]]/[[M-theory]]:

* [[Jose J. Fernandez-Melgarejo]], [[Tetsuji Kimura]], [[Yuho Sakatani]], *Weaving the Exotic Web*, JHEP **2018** 72 (2018) [doi:10.1007/JHEP09(2018)072]

category: people


Tetsuji Miwa

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

## Selected writings

On [[holonomic quantum fields]]:

On [[holonomic quantum fields]]:

* [[Tetsuji Miwa]], [[Michio Jimbo]], *Introduction to holonomic quantum fields*, pp. 28--36 in: _The Riemann problem, complete integrability and arithmetic applications_, Lec. Notes in Math. **925**, Springer (1982) $[$[doi:10.1007/BFb0093497](http://dx.doi.org/10.1007/BFb0093497)$]$

* {#JMSM05} [[Michio Jimbo]], [[Tetsuji Miwa]], [[Mikio Sato]], Yasuko Môri, *Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields —*,  In: [[Konrad Osterwalder|K. Osterwalder]] (ed.), *Mathematical Problems in Theoretical Physics*, Lecture Notes in Physics, **116** Springer (2005) $[$[doi:10.1007/3-540-09964-6_310](https://doi.org/10.1007/3-540-09964-6_310)$]$

On the [hypergeometric construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Etsuro Date]], [[Michio Jimbo]], [[Atsushi Matsuo]], [[Tetsuji Miwa]], *Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation*,  International Journal of Modern Physics B **04** 05 (1990) 1049-1057   $[$[doi:10.1142/S0217979290000528](https://doi.org/10.1142/S0217979290000528)$]$

On [[solitons]]:

* [[Michio Jimbo]], [[Tetsuji Miwa]], [[Etsuro Date]], (transl. from Japanese by [[Miles Reid|M. Reid]]), *Solitons: differential equations, symmetries and infinite dimensional algebras*,  Cambridge Tracts in Mathematics **135** (2012) [[ISBN:9781107404199](https://www.cambridge.org/us/universitypress/subjects/mathematics/geometry-and-topology/solitons-differential-equations-symmetries-and-infinite-dimensional-algebras)]

category: people


Tetsuo Gotō

## Selected writings

Introducing discussion of what came to be called the *[[Nambu-Goto action]]* for [[strings]] (then motivated as explanation for the "dual resonance model" for [[hadrons]], cf. *[[Polyakov gauge-string duality]]*):

* {#Gotō71} [[Tetsuo Gotō]], *Relativistic Quantum Mechanics of One-Dimensional Mechanical Continuum and Subsidiary Condition of Dual Resonance Model*, Progress of Theoretical Physics **46** 5 (1971) 1560–1569 [[doi:10.1143/PTP.46.1560](https://doi.org/10.1143/PTP.46.1560)]

category: people

[[!redirects Tetsuo Goto]]


TeV

A [[unit]] of [[energy]]: tera-[[electronvolt]]

## Related entries

* [[GeV]]

* [[MeV]]

## References

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


Tevian Dray

* [webpage](http://people.oregonstate.edu/~drayt/)

## selected writings

On [[octonions]], [[quaternions]], [[normed division algebras]] etc.:

* [[Tevian Dray]], [[Corinne Manogue]], _The Geomety of Octonions_, World Scientific 2015 ([doi:10.1142/8456](https://doi.org/10.1142/8456))

On [[supersymmetry and division algebras]]:

* [[Tevian Dray]], J. Janesky, [[Corinne Manogue]], _Octonionic hermitian matrices with non-real eigenvalues_, Adv. Appl. Clifford Algebras 10 (2000), 193–216 ([arXiv:math/0006069](https://arxiv.org/abs/math/0006069))

On [[E7]] via [[division algebra]]:

* [[Tevian Dray]], [[Corinne Manogue]], [[Robert Wilson]], *A New Division Algebra Representation of $E_7$* [[arxiv:2401.10534](https://arxiv.org/abs/2401.10534)]

## Related  entries

* [[octonions]]

* [[exceptional Jordan algebra]]

* [[E6]]

category: people


Thanu Padmanabhan

Thanu Padmanabhan (1957-2021)

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

* [GoogleScholar entry](https://scholar.google.com/citations?user=ZgZXr20AAAAJ&hl=en)

## Selected writinga

On [[Bekenstein-Hawking entropy]]:

* [[Thanu Padmanabhan]], _Thermodynamical Aspects of Gravity: New insights_, Rep. Prog. Phys. **73** (2010) 046901 [[arXiv:0911.5004](http://arxiv.org/abs/0911.5004)]

On [[gravity]] in relation to [[thermodynamics]]:

* [[Thanu Padmanabhan]], *Gravity and/is Thermodynamics*,  Current Science, **109** (2015) 2236-2242 [[doi:10.18520/v109/i12/2236-2242](https://doi.org/10.18520/v109/i12/2236-2242)]

* [[Thanu Padmanabhan]], *Exploring the Nature of Gravity*, talk notes [[arXiv:1602.01474](https://arxiv.org/abs/1602.01474)]

On the [[worldline formalism]] for [[quantum propagators]]:

* [[Thanu Padmanabhan]], *World-line Path integral for the Propagator expressed as an ordinary integral: Concept and Applications*, Found Phys. **51** 35 (2021) [[arXiv:2104.07041](https://arxiv.org/abs/2104.07041), [doi:10.1007/s10701-021-00447-8](https://doi.org/10.1007/s10701-021-00447-8)]

category: people


the Barsotti-Tate group of an abelian variety

[[!redirects the p-divisible group A(p)]]
Recall that a [[p-divisible group]] $G$ has the defining properties that $p \,id_G:G\to G$ is an epimorphism with finite kernel satisfying $G=\cup_j ker\, p^j \, id_G$.

Now let $A$ be any commutative algebraic $k$-group such that $p\, id_A:A\to A$ is an epimorphism. Then

$$A(p):=\cup_j ker \,p^j \,id_A$$

is a $p$-divisible group.


The Blind Spot

[[!redirects Lectures on Logic]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

This page collects links related to

* [[Jean-Yves Girard]]

  \linebreak

  **The Blind Spot: Lectures on Logic**

  \linebreak

  European Mathematical Society

  2011

  [doi:10.4171/088](https://doi.org/10.4171/088)

on [[formal logic]] and its [[categorical semantics]] with an emphasis on [[linear logic]] and in fact on some kind of [[quantum logic]] (section 17), but emphatically not in the original sense of [Birkhoff & von Neumann (1936)](BirkhoffvonNeumann36).

A kind of survey is in:

* [[Jean-Yves Girard]], *The Blind Spot*, in Jocelyn Benoist, Jean-Yves Girard, Yoshinori Ogawa, Mitsuhiro Okada (eds.) *Towards New Logic and Semantics* (2006) [[ISBN:978-4-7664-1331-1 C3310](https://www.keio-up.co.jp/kup/eng/philo/13310.html), talk slides: [pdf](https://girard.perso.math.cnrs.fr/keio2006.pdf), [[Girard_Keio2006.pdf:file]]]

## Foreword

from p. xii:

> Among the magisterial mistakes of logic, one will first mention [[quantum logic]], whose ridiculousness can only be ascribed to a feeling of superiority of the language – and ideas, even bad, as soon as they take a written form – over the physical world. Quantum logic is indeed a sort of punishment inflicted on nature, guilty of not yielding to the prejudices of logicians... just like Xerxes had the Hellespont – which had destroyed a boat bridge – whipped.

from p. xiii:

> The blind spot is what one does not see and what one is not even conscious of not seeing. The most trivial blind spot is the cheap modal logic justified by an even cheaper Kripke semantics and vice versa; but one finds similar blindings in the most elaborated interpretations. The good news of these lectures is that the procedural standpoint seems to be capable of dislodging the unsaid, the unseen. Simply, while the absence of Hauptsatz is enough to show that logic S5 is nonsense, one has to work much more to imagine what could be wrong in the principles justifying – say – the function $2^n$.

## Part I The basics

## Part II Around Curry-Howard

## Part III Linear logic

## Part IV Polarised interpretations

## Part V Iconoclasm

## Part VI Geometry of Interaction

## Envoi. The phantom of transparency

category: reference