closure operator

> For the closure of a subset of a topological space, see at _[[closed subset]]_.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _closure operator_ is a [[monad]] on a [[poset]], typically a [[poset of subobjects]] (of some [[object]]) in some [[category]].  In [[logic]], this is often referred to as a (monadic) _[[modal operator]]_.  The elements of the poset that are fixed by the closure operator are called _closed_ (or perhaps _[[modal type|modal]]_).

Dually, a [[comonad]] on a poset is called a _co-closure operator_ and the elements fixed by it are called _co-closed_.

More generally, in [[type theory]]/[[category theory]], we may think of any [[idempotent monad]] on a [[category]] as being a closure operator, and of any idempotent [[comonad]] as a co-closure operator.

## Definition

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

For $\mathcal{C}$ a [[category]], a **closure operator** $\diamond$ on $\mathcal{C}$ is an [[idempotent monad]] on $\mathcal{C}$, hence an [[endofunctor]] $\diamond \colon \mathcal{C} \to \mathcal{C}$ equipped with [[unit of a monad|unit]] and product [[natural transformations]]

* $\eta_{\diamond} \;\colon\; id_{\mathcal{C}} \to \diamond$

* $\mu_{\diamond} \;\colon\; \diamond \circ \diamond \to \diamond$

such that the [[monad]]-axioms hold and such that the following equivalent conditions hold (idempotency)

* the product map is an [[isomorphism]];

* $\eta_{\diamond(-)}$ is an isomorphism.

=--

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

For $\diamond \colon \mathcal{C} \to \mathcal{C}$ a closure operator, def. \ref{ClosureOperatorAsMonad}, and for $X \in \mathcal{C}$ an object, we say that

* $\diamond X \in \mathcal{C}$ is the **$\diamond$-closure** of $X$;

* $\eta_{X} \colon X \to \diamond X$ is the **closing map** or **map to the closure**;

* $X$ is **$\diamond$-closed** precisely if $\eta_X$ is an [[isomorphism]].

We write

$$
  \mathcal{C}_\diamond \hookrightarrow \mathcal{C}
$$

for the [[full subcategory]] on the $\diamond$-closed objects.

=--

## Examples

* A closure operator on a [[power set]] is also called a _[[Moore closure]]_. See there for more.

* One well-known example of a Moore closure is [[topological closure]], which is precisely a Moore closure that preserves finite joins (unions). Similarly, one can view _[[topological interior]]_ as a co-closure operator on a power set that preserves finite meets, or dually as a closure operator on the *[[opposite poset|opposite]]* of a power set that preserves finite [[joins]] (given by [[intersections]] of subsets).

* A [[Lawvere-Tierney topology]] is a closure operator on the [[subobject classifier]] $\Omega$ of a [[topos]] $E$, viewed as an internal [[meet-semilattice]]. More precisely, it is a just such a closure operator that preserves internal finite meets. Externally, $\hom(-, \Omega) \colon E^{op} \to Set$ provides an example of a [[universal closure operator]].

## Induced closure on slices
 {#InducedClosureOnSlices}

We discuss here how a closure operator on a [[topos]] may induce closure operators on each of its [[slice categories]].

Throughout, our [[topos]] is denoted $\mathcal{C}$.

Given a [[monad]] $\diamond \colon \mathcal{C} \to \mathcal{C}$ we write  $\eta_\diamond \colon id_{\mathcal{C}} \to \diamond$
for its [[unit of a monad|unit]], and write $\mu_\diamond \colon \diamond \circ \diamond \to \diamond$ for its multiplication. As we proceed, we add assumptions on $\diamond$, such as that it preserves certain [[limits]] and/or that it be [[idempotent monad|idempotent]].

For $X \in \mathcal{C}$ any [[object]], we write $\mathcal{C}_{/X}$ for the [[slice topos]] over it. The corresponding [[base change geometric morphism]] ([[dependent sum]] $\dashv$ [[context extension]] $\dashv$ [[dependent product]]) we write

$$
  \left(
    \underset{X}{\sum}
    \dashv
    X^\ast
    \dashv
    \underset{X}{\prod}
  \right)
  \;\colon\;
  \mathcal{C}_{/X}
  \stackrel{\overset{\sum_X}{\to}}{\stackrel{\overset{X^\ast}{\leftarrow}}{\underset{\prod_X}{\to}}}
  \mathcal{C}
  \,.
$$

We denote an object $p \in \mathcal{C}_{/X}$ in the slice also by the corresponding [[morphism]]

$$
  p \coloneqq
  \left(
    \left(\sum_X p\right) \stackrel{p}{\to} X
  \right)
  \coloneqq
  \sum_X \left( p \to \ast_{X} \right)
$$

in $\mathcal{C}$, which is the image under [[dependent sum]] of the unique morphism from $p$ to the [[terminal object]] in $\mathcal{C}_{/X}$. Accordingly, a morphism $\phi \colon p_1 \to p_2$ in the slice we also denote by the corresponding triangular [[commuting diagram]]

$$
  \left(
   \array{
      \sum_X p_1 &&\stackrel{\sum_X \phi}{\to}&& \sum_X p_2
      \\
      & {}_{\mathllap{p}_1}\searrow && \swarrow_{\mathrlap{p_2}}
      \\
      && X
    }
  \right)
$$

in $\mathcal{C}$.

Here we study the following endofunctors on slices induced from a monad on the total topos.

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

For $\diamond \colon \mathcal{C} \to \mathcal{C}$ an [[monad]] on a [[topos]] $\mathcal{C}$, and for $X \in \mathcal{C}$ any [[object]], the **induced operator**

$$
  \diamond_{/X} \colon \mathcal{C}_{/X} \to \mathcal{C}_{/X}
$$

on the [[slice topos]] $\mathcal{C}_{/X}$ is the [[functor]] which sends an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ to $(X \underset{\diamond X}{\times} \diamond E \stackrel{\eta_\diamond(X)^\ast \diamond p}{\to} X)$, hence to the left vertical morphism in the [[pullback]] [[diagram]]

$$
  \array{
    X \underset{\diamond X}{\times} \diamond E
    &\to&
    \diamond E
    \\
    \downarrow^{\mathrlap{\eta_\diamond(X)^\ast \diamond p}} && \downarrow^{\mathrlap{\diamond p}}
    \\
    X &\stackrel{\eta_\diamond(X)}{\to}& \diamond X
  }
  \,,
$$

regarded as an object in $\mathcal{C}_{/X}$, and which sends morphisms to the corresponding universal maps between these pullbacks:

$$
  \diamond_{/X}
  \;\;
  \colon
  \;\;
  \left(
   \array{
     E_1 &&\stackrel{\phi}{\to}&& E_2
     \\
     & \searrow && \swarrow
     \\
     && X
   }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
    \array{
      X \underset{\diamond X}{\times} \diamond E_1
      && \stackrel{X \underset{\diamond X}{\times} \diamond \phi}{\to}
      &&
      X \underset{\diamond X}{\times} \diamond E_2
      \\
      & \searrow && \swarrow
      \\
      && X
    }
  \right)
  \,.
$$

=--

We now want to identify conditions under which $\diamond_{/X}$ is itself a monad. First observe that the [[unit of a monad|unit]]-like map is canonically present.

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

In the situation of def. \ref{InducedOperatorOnSlice}, there is a [[natural transformation]]

$$
  \eta_{\diamond_{/X}}
  \;\colon\;
  id_{\mathcal{C}_{/X}}
  \to
  \diamond_{/X}
$$

from the identity on the slice to the induced operator on the slice, whose component over an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ is the universal morphism into the defining [[pullback]] in def. \ref{InducedOperatorOnSlice} induced from the naturality of the $\diamond$-unit $\eta_{\diamond}$:

$$
  \array{
    \eta_{\diamond}(E) \colon
    & E &\stackrel{\sum_{X} \left(\eta_{\diamond_{/X}}\left(p\right)\right) }{\to}& X \underset{\diamond X}{\times} \diamond E
    &\to&
    \diamond E
    \\
    & &{}_{\mathllap{p}}\searrow& \downarrow^{\mathrlap{\eta_\diamond(X)^\ast \diamond p}} && \downarrow^{\diamond \mathrlap{p}}
    \\
    & && X &\stackrel{\eta_\diamond(X)}{\to}& \diamond X
  }
  \,,
$$

=--

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

We have to show that for all morphisms

$$
  \left(
    \array{
      E_1 &&\stackrel{f}{\to}&& E_2
      \\
      & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
      \\
      && X
    }
  \right)
$$

in $\mathcal{C}_{/X}$ the induced diagram

$$
  \array{
     E_1 &\stackrel{\sum_X (\eta_{\diamond_{/X}}(p_1) )}{\to}&
     X \underset{\diamond X}{\times} \diamond E_1
     \\
     \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{X \underset{\diamond X}{\times} \diamond f }}
     \\
     E_2 &\stackrel{\sum_X (\eta_{\diamond_{/X}}(p_2) )}{\to}&
     X \underset{\diamond X}{\times} \diamond E_2
  }
$$

in $\mathcal{C}$ commutes. Inspection of the defining pullback diagram shows that both composites in this diagram constitute [[cones]] over the pullback diagram that defines the bottom right object. Therefore by the [[universal property]] of the pullback they have to coincide.

=--

Next, to have also a product operation on the induced operator $\diamond_{/X}$ we need that $\diamond$ preserves some pullbacks:

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

Assume that the monad $\diamond \colon \mathcal{C} \to \mathcal{C}$ preserves [[pullbacks]] over objects in its image. Then for each $X \in \mathcal{C}$ the induced endofunctor $\diamond_{/X}$ of def. \ref{InducedOperatorOnSlice} comes with a [[natural transformation]]

$$
  \mu_{\diamond_{/X}} \;\colon\; \diamond_{/X} \circ \diamond_{/X} \to \diamond_{/X}
$$

whose component on an object $(E \stackrel{p}{\to} X) \in \mathcal{C}_{/X}$ is the pullback of the component $\mu_{\diamond} E$ of the product of $\diamond$ itself over the component $\mu_\diamond X$ along the unit components $\eta_{\diamond} X$.

=--

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

First we produce the component map as claimed, then we show that it is indeed the component of a natural transformation.

So for $p \in \mathcal{C}_{/X}$ an object in the slice, consider the defining pullback diagram of $\diamond_{/X} p$ from def. \ref{InducedOperatorOnSlice}

$$
  \array{
    X \underset{\diamond X}{\times} \diamond E
    &\to&
    \diamond E
    \\
    \downarrow^{\mathrlap{\eta_\diamond(X)^\ast \diamond p}} && \downarrow^{\mathrlap{\diamond p}}
    \\
    X &\stackrel{\eta_\diamond(X)}{\to}& \diamond X
  }
  \,.
$$

By the assumption that $\diamond$ preserves pullback diagrams of this form, application of $\diamond$ yields the pullback diagram

$$
  \array{
    \diamond( X \underset{\diamond X}{\times} \diamond E )
    &\to&
    \diamond \diamond E
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\diamond \diamond p}}
    \\
    \diamond X &\stackrel{\diamond(\eta_\diamond(X))}{\to}& \diamond \diamond X
  }
  \,.
$$

Pasting to this the pullback of its left vertical morphism along $\eta_\diamond(X)$ yields

$$
  \array{
    X \underset{\diamond X}{\times} \diamond(X \underset{\diamond X}{\times} \diamond E)
    &\to&
    \diamond( X \underset{\diamond X}{\times} \diamond E )
    &\to&
    \diamond \diamond E
    \\
    \downarrow
    &&
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\diamond \diamond p}}
    \\
    X
    &\stackrel{\eta_\diamond X}{\to}&
    \diamond X &\stackrel{\diamond(\eta_\diamond(X))}{\to}& \diamond \diamond X
  }
  \,,
$$

where the total rectangle is also a pullback, by the [[pasting law]].

We now build a morphism of [[diagrams]] from the underlying [[cospan]] of this diagram to another cospan, such that the induced map on pullbacks is the component of the natural transformation that we are looking for,

To this end, first paste to the above diagram the [[naturality square]] of the monad multiplication map $\mu_\diamond \colon \diamond \circ \diamond \to \diamond$ to obtain

$$
  \array{
    X \underset{\diamond X}{\times} \diamond(X \underset{\diamond X}{\times} \diamond E)
    &\to&
    \diamond( X \underset{\diamond X}{\times} \diamond E )
    &\to&
    \diamond \diamond E
    \\
    \downarrow
    &&
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\diamond \diamond p}}
    & \searrow^{\mathrlap{\mu_{\diamond}(E)}}
    \\
    X
    &\stackrel{\eta_\diamond X}{\to}&
    \diamond X &\stackrel{\diamond(\eta_\diamond(X))}{\to}& \diamond \diamond X
   && \diamond E
   \\
   && && & \searrow^{\mathrlap{\mu_\diamond X} } & \downarrow^{\mathrlap{\diamond p}}
   \\
   && && && \diamond X
  }
  \,,
$$

Then fill in the commuting diagram that exhibits the [[unitality]] axiom of $\diamond$ to obtain

$$
  \array{
    X \underset{\diamond X}{\times} \diamond(X \underset{\diamond X}{\times} \diamond E)
    &\to&
    \diamond( X \underset{\diamond X}{\times} \diamond E )
    &\to&
    \diamond \diamond E
    \\
    \downarrow
    &&
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\diamond \diamond p}}
    & \searrow^{\mathrlap{\mu_{\diamond}(E)}}
    \\
    X
    &\stackrel{\eta_\diamond X}{\to}&
    \diamond X &\stackrel{\diamond(\eta_\diamond(X))}{\to}& \diamond \diamond X
   && \diamond E
   \\
   && &\searrow^{\mathrlap{id_{\diamond X}}} & & \searrow^{\mathrlap{\mu_\diamond X} } & \downarrow^{\mathrlap{\diamond p}}
   \\
   && && \diamond X &\stackrel{id_{\diamond X}}{\to}& \diamond X
  }
  \,.
$$

Finally paste in an identity square, just as to manifestly exhibit a morphism of diagrams

$$
  \array{
    X \underset{\diamond X}{\times} \diamond(X \underset{\diamond X}{\times} \diamond E)
    &\to&
    \diamond( X \underset{\diamond X}{\times} \diamond E )
    &\to&
    \diamond \diamond E
    \\
    \downarrow
    &&
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\diamond \diamond p}}
    & \searrow^{\mathrlap{\mu_{\diamond}(E)}}
    \\
    X
    &\stackrel{\eta_\diamond X}{\to}&
    \diamond X &\stackrel{\diamond(\eta_\diamond(X))}{\to}& \diamond \diamond X
   && \diamond E
   \\
   &\searrow^{\mathrlap{id_X}}& &\searrow^{\mathrlap{id_{\diamond X}}} & & \searrow^{\mathrlap{\mu_\diamond X} } & \downarrow^{\mathrlap{\diamond p}}
   \\
   && X &\stackrel{\eta_\diamond X}{\to}& \diamond X &\stackrel{id_{\diamond X}}{\to}& \diamond X
  }
  \,.
$$

Now observe that the total front [[cospan]] of morphisms is such that the [[limit]] [[cone]] over it is the [[pullback]] that defines $X \underset{\diamond X}{\times} \diamond E$. By functoriality of [[pullbacks]] (by their universal property), this induces a component morphism

$$
  \mu_{\diamond_{/X}}
  \;\colon\;
  X \underset{\diamond X}{\times} \diamond (X \underset{\diamond}{\times} \diamond E)
  \to
  X \underset{\diamond X}{\times} \diamond E
$$

as claimed.

Since this is built just from universal constructions, the fact that this morphism is indeed [[natural transformation|natural]] follows as in prop. \ref{SliceOperatorHasUnit}.

=--

So far we have constructed from a monad that preserves pullbacks over objects in its image an operator on slices which is equipped with a unit-like and a multiplication-like transformation. We now claim that this yields indeed a monad on the slice.

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

For $\diamond \colon \mathcal{C} \to \mathcal{C}$ a [[monad]] which preserves [[pullbacks]] over objects in its image, and for $X \in \mathcal{C}$ any object, the natural transformations

1. $\eta_{\diamond_{/X}} \colon id_{\mathcal{C}_{/X}} \to \diamond_{/X}$ of prop. \ref{SliceOperatorHasUnit}

1. $\mu_{\diamond_{/X}} \colon \diamond_{/X} \circ \diamond_{/X} \to \diamond_{/X}$ from prop. \ref{ProductOnSliceOperator}

constitute the unit and product of a [[monad]] structure $(\diamond_{/X}, \mu_{\diamond_{/X}}, \eta_{\diamond_{/X}})$ on the
slice operator $\diamond_{/X}$ of def. \ref{InducedOperatorOnSlice}.

If moreover $\diamond$ is [[idempotent monad|idemponent]], then so is $\diamond_{/X}$.

=--

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

By forming cospan morphisms and inducing maps between the corresponding pullbacks, this follows from the monad structure $(\diamond, \mu_{\diamond}, \eta_{\diamond})$ by the same arguments as in the proof of prop. \ref{ProductOnSliceOperator}.

=--

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

If $\diamond$ is an [[idempotent monad]], hence a closure operator, then, by the discussion there, the monad unit exhibits an [[equivalence of categories]] between the objects in the image of $\diamond$ and the $\diamond$-closed objects.

Therefore in this case the condition that $\diamond$ preserves pullbacks over objects in its image is equivalently that it preserves pullbacks over $\diamond$-closed objects. In this form we will mostly state this condition in the following.

=--

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

Let $\diamond \colon \mathcal{C} \to \mathcal{C}$ be an [[idempotent monad]] that preserves pullbacks over $\diamond$-closed objects. Then the closed objects $p \in \mathcal{C}_{/X}$, def. \ref{ClosureAndClosedObjects}, of the induced idempotent monad $\diamond_{/X}$ on the slice over any $X \in \mathcal{C}$ are precisely those objects $(E \stackrel{p}{\to} X)$ for which the naturality square of the $\diamond$-unit is a pullback square in $\mathcal{C}$.

=--

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

By def. \ref{ClosureAndClosedObjects} we need to show that for $p \in \mathcal{C}_{/X}$ the corresponding component of the $\diamond_{/X}$-unit $\eta_{\diamond_{/X}}(p)$ is an [[isomorphism]] precisely if

$$
  \array{
    E &\stackrel{\eta_{\diamond} E}{\to}& \diamond E
    \\
    \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{\diamond p}}
    \\
    X &\stackrel{\eta_{\diamond} X}{\to}& \diamond X
  }
$$

is a pullback diagram in $\mathcal{C}$. By prop. \ref{InducedOperatorOnSliceIsMonad} and  prop. \ref{SliceOperatorHasUnit},  the universal map from this diagram, regarded as a [[cone]] over the underlying [[cospan]], to the limiting cone is precisely $\eta_{\diamond_{/X}}(p)$. Hence the claim follows by the universal property of the pullback.

=--

As a special case of def. \ref{ClosureAndClosedObjects} we are therefore now interested in the following.

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

For $\diamond \colon \mathcal{C} \to \mathcal{C}$ an [[idempotent monad]] which preserves pullbacks over $\diamond$-closed objects, write

$$
  (\mathcal{C}_{/X})_{\diamond_{/X}}
  \hookrightarrow
  \mathcal{C}_{/X}
$$

of the [[full subcategory]] of the [[slice topos]] on the $\diamond_{/X}$-closed objects.

=--

## Related concepts

* [[universal closure operator]]

* [[modal logic]]

* [[reflective factorization system]], [[stable factorization system]]

[[!redirects closure]]
[[!redirects closures]]

[[!redirects closure operator]]
[[!redirects closure operators]]
[[!redirects closure operation]]
[[!redirects closure operations]]

[[!redirects coclosure operator]]
[[!redirects coclosure operators]]
[[!redirects coclosure operation]]
[[!redirects coclosure operations]]
[[!redirects co-closure operator]]
[[!redirects co-closure operators]]
[[!redirects co-closure operation]]
[[!redirects co-closure operations]]


Clover May

* [personal page](https://www.math.ucla.edu/~clovermay/)

## Selected writings

On [[commutative ring]]-structure ([[E-infinity ring|$E_\infty$-ring]]-structure) in [[rational equivariant stable homotopy theory]] in general and in [[rational equivariant K-theory]] in particular:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

On [[Kronholm's freeness theorem]] in [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]:

* [[Eric Hogle]], [[Clover May]], *The freeness theorem for equivariant cohomology of $Rep(C_2)$-complexes*, Topology and its Applications **285** 1 (2020) 107413 [[arXiv:2005.07300](https://arxiv.org/abs/2005.07300), [doi:10.1016/j.topol.2020.107413](https://doi.org/10.1016/j.topol.2020.107413)]

* [[Daniel Dugger]], [[Christy Hazel]], [[Clover May]], *Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$*, Journal of Pure and Applied Algebra **228** 3 (2024) 107473 [[arXiv:2203.05287](https://arxiv.org/abs/2203.05287), [doi:10.1016/j.jpaa.2023.107473](https://doi.org/10.1016/j.jpaa.2023.107473)]

category: people


club

#Contents#
* table of contents
{:toc}

## Idea ##

A club is a particular sort of [[doctrine]] or [[monad]] on [[categories]], one which encapsulates the following frequently observed phenomenon: to describe [[free functor|free]] algebras $F(C)$ with respect to the monad, it frequently suffices to describe the free algebra $F(1)$ on the [[terminal category]], and then a certain "[[categorical wreath product]]" gives the free algebra on $C$:

$$F(C) \cong F(1) \wr C$$

Examples of this phenomenon include the monad for [[monoidal categories]], [[symmetric monoidal categories]], [[braided monoidal categories]], categories with finite [[product]]s, [[closed category|closed symmetric monoidal categories]], and many others.

Clubs were introduced by [[Max Kelly]] in a series of papers (see references), and are akin in spirit to [[operad|operads]].  In fact, many (perhaps even all) types of clubs are a special case of [[generalized multicategories|generalized operads]], and also of [[polynomial monads]].

## Clubs over the permutation category

This is the most basic sort of club, introduced by Kelly in [MVFC](#KellyMVFC) and [AAC](#KellyAAC).

### Action by substitution product

Let $\mathbf{P}$ be the category of finite sets $[n] = \{1, \ldots, n\}$ (including the empty set $[0]$) and permutations between them, and let $Cat$ be the category of small categories. We define a "wreath product" action of the category $Cat/\mathbf{P}$ on $Cat$, where $Cat/\mathbf{P}$ is the slice category of $Cat$ over $\mathbf{P}$

$$\circ: Cat/\mathbf{P} \times Cat \to Cat$$

taking a pair $(\Gamma: C \to \mathbf{P}, D)$ to the category whose objects are tuples

$$(c; d_1, \ldots, d_n)$$

with $c \in Ob(C)$, $d_i \in Ob(D)$, and $\Gamma(c) = [n]$. A morphism is a tuple

$$(f; g_1, \ldots, g_n)$$

consisting of a morphism $f: c \to c'$ in $C$ and morphisms $g_i: d_i \to d_{\Gamma(f)(i)}'$ in $D$, composed in the obvious way.

This generalizes the standard notion of wreath product: given a group $G$ and a permutation representation, i.e., a homomorphism $\gamma: G \to Aut([n]) \hookrightarrow \mathbf{P}$, and given a group $H$, the wreath product is defined to be the semidirect product

$$G \int H = G \ltimes H^n$$

with the action of $G$ on $H^n$ induced from the action on $[n]$.

Another, more abstract, way to describe this substitution product is as follows: there is a [[cartesian monad]] $S$ on $Cat$ whose algebras are symmetric strict monoidal categories, and we have $\mathbf{P} = S 1$, where $1$ is the [[terminal category]].  The substitution product $\Gamma \circ D$ can then be described as the following pullback in $Cat$.
$$\array{\Gamma \circ D & \overset{}{\to} & S D\\
  \downarrow && \downarrow^{S !}\\
  C & \underset{\Gamma}{\to} & S 1 & = \mathbf{P}}$$
See below for a general version of this construction.

### Substitution product as monoidal product

We describe how the substitution action $\circ$ lifts to a self-action denoted by the same symbol:

$$\circ: Cat/\mathbf{P} \times Cat/\mathbf{P} \to Cat/\mathbf{P}$$

To start, given a pair $(\Gamma: C \to \mathbf{P}, \Delta: D \to \mathbf{P})$, the domain of $\Gamma \circ \Delta$ is the category $\Gamma \circ D$ described above. The functor

$$\Gamma \circ \Delta: \Gamma \circ D \to \mathbf{P}$$

sends an object $(c; d_1, \ldots, d_n)$ to $\sum_i \Delta(d_i)$.

The effect of $\Gamma \circ \Delta$ on morphisms $(f; g_1, \ldots, g_n)$ may be summarized as "substituting the permutations $\Delta(g_i)$ into the permutation $\Gamma(f)$". To describe this more precisely, we give a little preface. The hom-set $\mathbf{P}([n], [n]) = Aut(n)$ of permutations on $n$ may be identified with the set $Lin(n)$ of total (or linear) orders on $[n]$, if we identify the identity element with the standard order on $\{1, 2, \ldots, n\}$. The sets $Aut(n) = Lin(n)$ are components of an [[operad]] $Lin$ where the operadic multiplication

$$\mu(n; k_1, \ldots, k_n): Lin(n) \times Lin(k_1) \times \ldots \times Lin(k_n) \to Lin(k_1 + \ldots + k_n)$$

takes a linear ordering of $n$ linearly ordered sets and produces the evident linear ordering on the disjoint sum of the sets.

Then, given a morphism $(f; g_1, \ldots, g_n)$ in $\Gamma \circ D$, we define

$$(\Gamma \circ \Delta)(f; g_1, \ldots, g_n) \stackrel{def}{=} \mu(n; k_1, \ldots, k_n)(\Gamma(f), \Delta(g_1), \ldots \Delta(g_n))$$

Given the abstract description above of the substitution product in terms of the cartesian monad $T$, the functor $\Gamma \circ \Delta$ can be described as the composite
$$ \Gamma \circ D \to T D \overset{T \Delta}{\to} T T 1 \overset{\mu_1}{\to} T 1  = \mathbf{P}$$
where $\mu\colon T T \to T$ is the multiplication of the [[monad]] $T$.

The substitution product thus indicated,

$$\circ: Cat/\mathbf{P} \times Cat/\mathbf{P} \to Cat/\mathbf{P},$$

is the product for a [[monoidal category]] structure on $Cat/\mathbf{P}$. The monoidal unit is the functor $I: 1 \to \mathbf{P}$ which names the 1-element set.  Abstractly, we can observe that the cartesian monad $T$ induces a [[pseudomonad]] on the [[bicategory]] $Span(Cat)$ of spans in $Cat$, which has a [[Kleisli category|Kleisli bicategory]] $Span(Cat)_T$ in which a 1-cell $A\to B$ is a span $A \leftarrow X \to T B$ in $Cat$.  We then have
$$ Cat/\mathbf{P} \simeq Cat / (1 \times T 1) \simeq Span(Cat)_T (1, 1) $$
and the monoidal structure above is that induced from the bicategory composition in $Span(Cat)_T$.

Under this monoidal product, the substitution action indicated earlier,

$$\circ: Cat/\mathbf{P} \times Cat \to Cat,$$

carries a structure of [[actegory]] over the monoidal category $Cat/\mathbf{P}$, in the sense that there is a coherent associativity

$$(\Gamma \circ \Delta) \circ E \cong \Gamma \circ (\Delta \circ E)$$

for $\Gamma: C \to \mathbf{P}$, $\Delta: D \to \mathbf{P}$, and a category $E$, and similarly coherent left and right unit isomorphisms.

### Clubs over $\mathbf{P}$

+-- {: .un_defn}
###### Definition
A **club** over $\mathbf{P}$ is a monoid in the monoidal category $(Cat/\mathbf{P}, \circ, I)$.  By the abstract characterization above, this is equivalent to a [[monad]] in the bicategory $Span(Cat)_T$ on the object $1$, or equivalently a $T$-[[generalized multicategory|operad]] in the sense of Leinster.
=--

A club over $\mathbf{P}$ induces (via the actegory structure) a 2-monad on $Cat$, and an algebra over the club is an algebra for this monad. That is, an **algebra** over a club $C$ is a category $D$ together with an action $m: C \circ D \to D$ compatible in the usual way with the monoid structure on $C$.

Given a club structure on $\Gamma: C \to \mathbf{P}$, we think of the objects $c$ as formal operations of arity $n = \Gamma(c)$. An algebra $D$ over $C$, $m: C \circ D \to D$, gives in effect an interpretation of each $c$ of arity $n$ as an actual operation $D^n \to D$.

The free $C$-algebra over a category $D$ is just $C \circ D$. For the free algebra over the terminal category $1$, notice that

$$C \circ 1 \cong C$$

(strictly speaking, the $C$ on the left is a category over $\mathbf{P}$ and the $C$ on the right is just the category). Thus, in a manner of speaking, the free algebra generated by $D$ is obtained by wreathing the free algebra generated by $1$ with $D$, as adumbrated in the idea section above.

### Examples

* The identity $1_{\mathbf{P}}: \mathbf{P} \to \mathbf{P}$ carries a club structure. The multiplication $\mu$ of the club, on the object level, is given by the assignment
$$(n; k_1, \ldots, k_n) \mapsto k_1 + \ldots + k_n$$
and on morphisms, it is given by the operad structure on $Aut(n) = Lin(n)$ discussed above. Algebras over this club are symmetric strict monoidal categories. Algebras in the "pseudo" sense over the induced 2-monad on $Cat$ are [[unbiased]] non-strict [[symmetric monoidal categories]]. Alternatively, if $F(1)$ is the free (biased, non-strict) symmetric monoidal category on one generator, then the symmetric monoidal equivalence $\Gamma: F(1) \to \mathbf{P}$ carries a club structure whose strict algebras are (biased, non-strict) symmetric monoidal categories.

* Let $\mathbf{B}$ be the [[braid category]], equipped with the usual forgetful functor $\Gamma: \mathbf{B} \to \mathbf{P}$. The club multiplication, at the level of morphisms, is "substitution" of $n$ braids into a braid on $n$ elements. Pseudo-algebras over the induced 2-monad on $Cat$ are unbiased braided monoidal categories. Alternatively, as in the previous example, biased non-strict braided monoidal categories are also strict algebras over a club of the form $\Gamma: F(1) \to \mathbf{P}$ where $F(1)$ is the free braided monoidal category on one generator.

* Let $C$ be any (permutative) operad valued in $Set$, with underlying species $\mathbf{P} \to Set$. The [[category of elements]] gives a functor $\Gamma: El(C) \to \mathbf{P}$, and this carries a club structure induced from the operad structure on $C$. In this way, clubs generalize operads.  In fact, operads in $Set$ can be identified with those clubs for which the functor $\Gamma\colon C\to \mathbf{P}$ is a [[discrete fibration]].

## Clubs over finite sets

Essentially the same construction can be given replacing $\mathbf{P}$ by (a skeleton of) the category $FinSet$ of finite sets and functions, or its opposite.  The corresponding cartesian monads are those whose algebras are categories with strictly associative finite coproducts or strictly associative finite products, respectively.  Examples of such clubs include categories with (not necessarily strictly associative) finite coproducts or products.  Clubs of this sort were also defined in [MVFC](#KellyMVFC) and [AAC](#KellyAAC), and studied further in [OCD](#KellyOCD).

## Clubs over cartesian monads

In fact, the foregoing constructions can be generalized to any [[cartesian monad]] $S$ on $Cat$.  Since a monad is a [[monoid]] in the [[monoidal category]] of [[endofunctors]], the [[slice category]] $[Cat,Cat]/S$ inherits a monoidal structure.  We also have an "evaluate at $1$" functor $[Cat,Cat]/S \to Cat/S1$, which has a right adjoint defined by pulling back: given $A\to S1$, we define an endofunctor $\hat{A}$ of $Cat$ by the following pullbacks:

$$ \array{ \hat{A}(X) & \to & A \\
\downarrow && \downarrow\\
S(X) & \to & S(1) }$$

This right adjoint is fully faithful and embeds $Cat/S1$ into $[Cat,Cat]/S$ as the endofunctors equipped with a [[cartesian natural transformation]] to $S$.  Finally, the cartesianness of $S$ implies that $Cat/S1$ is closed under the monoidal structure of $[Cat,Cat]/S$.  This induced monoidal structure on $Cat/S1$ is, in the case when $S$ is the symmetric strict monoidal category monad, exactly the substitution product defined above.  Thus, a club is a monoid for this monoidal structure, and the inclusion of $Cat/S1$ into $[Cat,Cat]/S$ thereby sends it to a monad over $S$.

Note, however, that this construction depends on no special features of $Cat$; it can be performed for any cartesian monad $S$ on any category with finite limits.  This generalization was performed by Kelly in [CDT](#KellyCDT).  More generally, it can be performed for cartesian monads *in* a suitable 2-category; see [[club in a 2-category]].

## Clubs with many objects

To describe structures on a family of categories rather than a single category (such as a monoidal functor between two monoidal categories), we can generalize to $Cat^\Lambda$ for any set $\Lambda$.  In this case the substitution monoidal product lives on $Cat/(\Lambda \times S\Lambda)$.  This generalization can also be found in [MVFC](#KellyMVFC) and [AAC](#KellyAAC); it amounts to nothing more than considering [[generalized multicategories]] for a cartesian monad (with a [[discrete category]] $\Lambda$ of objects) rather than simply generalized operads (those with the [[terminal category]] of objects).

## Clubs over $Cat$

In [CDT](#KellyCDT) Kelly described a further generalization of clubs over $FinSet$, in which $FinSet$ is replaced by $Cat$.  The crucial feature that allows the substitution product to continue working is the [[Grothendieck construction]] for functors into $Cat$.  In fact, Kelly generalized even further to a substitution product that lives not on the ordinary slice category $CAT/Cat$ but on the [[lax slice 2-category]] $CAT\sslash Cat$ (which Kelly denoted $CAT \int^* Cat$).  This has the advantage that the resulting functor $Mon(CAT\sslash Cat,\circ) \to Mnd(Cat)$ from clubs to [[monads]] on $Cat$ is [[fully faithful functor|fully faithful]], which it is not for the previous notions of club.

There is an analogous generalization of the many-object case, though Kelly did not make this explicit.  Similarly, there is a dual notion of club in $CAT\sslash Cat^{op}$ which generalizes clubs over $FinSet^{op}$.

Kelly did not give any interesting examples of clubs in this generality, only remarking that one is "ineluctably led to" the generalization.  But also worth noting is that there could be interesting morphisms between clubs over $FinSet$ that are only visible when they are viewed as clubs in $CAT\sslash Cat$ (or equivalently as monads on $Cat$).

## Clubs as polynomial monads

The cartesian monads $S$ that are used to define clubs over $\mathbf{P}$, $FinSet$, and $FinSet^{op}$ are in fact [[polynomial monads]] (which immediately implies that they are cartesian).  And an endofunctor equipped with a [[cartesian natural transformation]] to a [[polynomial functor]] is itself automatically polynomial.  Thus, the subcategory of $[Cat,Cat]/S$ identified with $Cat/S1$ by the "clubs over cartesian monads" construction is equivalently the category of polynomial endofunctors of $Cat$ cartesianly-over $S$, and the identification with $Cat/S1$ reduces to the identification of cartesian morphisms between polynomial functors with pullback squares of exponentiable morphisms.  Similarly, considering cartesian natural transformations in the [[double category]] of polynomial functors yields a description of clubs with many objects.

This suggests that an arbitrary polynomial monad could be regarded as a "global club", i.e. one not "over any base".  A polynomial endofunctor of $Cat$ is determined by a single [[exponentiable morphism]] $f:B\to A$ in $Cat$, i.e. a [[Conduche functor]], which in turn is determined by a [[pseudofunctor]] $A\to$ [[Prof]].  Composition of polynomial endofunctors thus yields a monoidal structure on the "slice category" whose objects are categories equipped with a pseudofunctor to $Prof$, monoids with respect to which are polynomial monads; thus a polynomial monad could be viewed as a "club over $Prof$".  Clubs in $CAT/Cat$ and $CAT/Cat^{op}$ are the obvious special cases when $Cat$ and $Cat^{op}$ are embedded into $Prof$ by [[representable profunctor|representable]] and corepresentable profunctors and the pseudofunctor is strict; these correspond to polynomial endofunctors for which $f:B\to A$ is a [[split opfibration]] or [[split fibration]] respectively.

Finally, recall that *non-cartesian* natural transformations between polynomial functors determined by $g:D\to C$ and $f:B\to A$ have a similar description as consisting of a map $h:C\to A$ and a map $C\times_A B \to D$ over $C$.  Translated into the classifying maps $A\to Prof$ and $C\to Prof$ this corresponds to a triangle filled by a lax natural transformation whose components are functors.  Thus in the cases of split (op)fibrations we recover Kelly's lax slice 2-categories $CAT\sslash Cat$ and $CAT\sslash Cat^{op}$.

## Clubs of mixed variance

The clubs described in the preceding sections are all examples of _covariant clubs_: the operations on their algebras $D$ are covariant functors $D^n \to D$. However, many doctrines on $Cat$, such as the doctrine of closed categories, involve functors which are covariant in some arguments and contravariant in others, together with [[extranatural transformations]] between them. For example, in the doctrine for closed monoidal categories, there is an extranatural transformation of the form

$$ev_{X, Y}: [X, Y] \otimes X \to Y$$

which is dinatural in $X$ and natural in $Y$.

It turns out that many doctrines of "mixed variance" can also be described by an extension of the club notion. This applies particularly to closed monoidal, closed symmetric monoidal, and $*$-autonomous categories. But there are some subtleties, some of which can be explained by looking at a non-example: the doctrine of compact closed categories.

### Extranaturality graphs

The underlying arity of an operation of mixed variance will be a _signed set_, i.e., a finite set $\{1, 2, \ldots, n\}$ where a sign $+$ or $-$ is assigned to each element $j$. The element $j$ is given the sign $+$ (resp., $-$) if $j^{th}$ argument of the operation appears covariantly (resp., contravariantly). For example, for the operation

$$C^{op} \times C \times C \to C: (a, b, c) \mapsto [a, b] \otimes c$$

the underlying arity is an ordered list of signs $\{-, +, +\}$.

The underlying arity of an extranatural transformation for a given doctrine is an arrow of signed sets $A$, $B$ called a **graph** (or EKM graph, for Eilenberg, Kelly and Mac Lane), which by definition is a partition of the signed elements of $A \cup B$ into mated pairs, such that

* Mated elements, one in $A$ and one in $B$, have the same sign;

* Mated elements, both in $A$ or both in $B$, have opposite signs.

An EKM graph may be identified with a directed graph whose edges are mated pairs, oriented in the direction from an $A$-element to a $B$-element if the mates have the same sign, or from $+$ to $-$ if the mates are $A$-elements, or from $-$ to $+$ if the mates are $B$-elements. Such a graph may also be considered as an oriented 1-cobordism between oriented 0-manifolds without loops (circles).

For example, the arity of the evaluation map in closed monoidal categories,

$$ev_{X, Y}: [X, Y] \otimes X \to Y$$

is an arrow with an oriented edge from the third placeholder of the domain to the first placeholder, and an oriented edge from the second placeholder of the domain to the placeholder of the codomain.

Hence we obtain a directed graph whose vertices are signed sets and whose edges are EKM graphs between the signed sets. An immediate question is how to compose graphs to form a category, and in particular what to do about "loops" or "islands" which may arise in composing such 1-cobordisms. The first answer that may come to mind is simply to ignore them (in other words, regard the pairings as morphisms in a bicategory of co-relations, and compose them as such). An answer more relevant to clubs will emerge in the next section.

### Doctrine of compact closed categories

An example in which loops arise in compositions of extranatural transformations is the doctrine of compact closed (symmetric monoidal) categories. The classic example is the composition

$$1 \stackrel{\eta}{\to} c^* \otimes c \stackrel{\varepsilon}{\to} 1$$

where $1$ denotes a monoidal unit, $\eta$ is a unit for an adjunction $c \dashv c^*$, and $\varepsilon$ is a counit for $c^* \dashv c$ (in a symmetric monoidal category, such adjunctions are inevitably [[ambidextrous adjunction|ambidextrous]]).

This composition obviously does not define an extranatural transformation from a constant functor to itself, precisely because of dependence on $c$ (for example, when interpreted in the compact closed category of finite-dimensional vector spaces, the value is $dim(c)$). In general, the presence of loops in compositions of extranaturality graphs should be seen as reflecting a fatal ill-definedness of composition of the extranatural transformations giving rise to them, and in such situations an alarm should sound: "[Danger, Will Robinson!](http://en.wikipedia.org/wiki/Danger,_Will_Robinson)".

The lesson learned is that the doctrine of compact closed categories is not describable by a club, and that for the purposes of clubs there is no use for compositions of graphs which produce loops. If one insists on assigning a value to composition in such cases, it might as well be a junk value $*$, and therefore it is justifiable to regard signed sets and arrows between them as carrying a structure of category enriched in the symmetric monoidal category of pointed sets (= category of sets and partially defined functions).

### Definition of club of mixed variance

Let $\mathbf{G}$ be the category enriched in pointed sets whose objects are finite signed sets, and whose morphisms are graphs as described above, composed as in the bicategory of cospans between sets unless this results in the creation of loops (in which case the composition is defined to be the basepoint of the hom-set it belongs to). There is no harm in thinking of $\mathbf{G}$ as an ordinary category.

Let $(Cat/\mathbf{G})'$ be the full subcategory of $Cat/\mathbf{G}$ whose objects are functors $\Gamma: C \to \mathbf{G}$ such that for every morphism $f$ of $C$, $\Gamma(f)$ is _not_ the basepoint of the hom-set it belongs to. There is an action

$$\circ: (Cat/\mathbf{G})' \times Cat \to Cat$$

for which the objects of $(\Gamma: C \to \mathbf{G}) \circ D$ are pairs
$(c, \sigma: \Gamma(c) \to Ob(D) \cup Ob(D^{op}))$ where $c$ is an object of $C$ and $\sigma$ is a lift of $sign: \Gamma(c) \to \{+, -\}$ through the function

$$! \cup !: Ob(D) \cup Ob(D^{op}) \to \{+\} \cup \{-\} = \{+, -\}$$

Morphisms of $(\Gamma: C \to \mathbf{G}) \circ D$ are pairs $(f: c \to c', \phi: \Gamma(f) \to U(D))$ where $\phi$ is a morphism of [[quiver|directed graphs]] to the underlying graph of $D$. Again, this action $\circ$ lifts to a monoidal product

$$\circ: (Cat/\mathbf{G})' \times (Cat/\mathbf{G})' \to (Cat/\mathbf{G})'$$

with monoidal unit $I: 1 \to \mathbf{G}$ naming the 1-point set, positively signed. The action of $(Cat/\mathbf{G})'$ on $Cat$ becomes an actegory with respect to the monoidal category structure.

+-- {: .un_defn}
###### Definition
A **club** over $\mathbf{G}$ is a monoid in the monoidal category $((Cat/\mathbf{G})', \circ, I)$.
=--

A club over $\mathbf{G}$ induces (via the actegory structure) a monad on $Cat$, and an **algebra** over the club is an algebra for this monad.  Unlike in the covariant cases, this monad is *not* a 2-monad, but it does give a 2-monad if one restricts to the locally groupoidal 2-category of categories, functors, and natural isomorphisms.

### Examples

As shown above, we cannot expect to describe compact closed categories with a club.  However, we can describe various other structures of mixed variance, such as symmetric monoidal *closed* categories.  Specifically, let $F(1)$ be the free smc category on one generator. There is a "graph functor" $\Gamma: F(1) \to \mathbf{G}$ which takes a morphism of $F(1)$ to its extranaturality graph. The crucial observation is that no basepoint morphism lies in the image of $\Gamma$, i.e., no islands can occur in formal compositions for this doctrine.  This fact is nontrivial; its proof relies on a [[cut-elimination theorem]] (see [KellyCET](#KellyCET)) and works for any structure obtained by adding [[multivariable adjunction|multi-variable right adjoints]] to any of the operations of a club over $\mathbf{P}$.

A morphism in the free symmetric monoidal closed category on a category $C$ is then given by a morphism of $F(1)$ together with a labeling of the oriented edges of its underlying graph (an oriented 1-cobordism) by morphisms of $C$. Then, you compose morphisms of this "[[categorical wreath product]]" in the obvious way, composing the morphism-labels of edges in 1-cobordisms as they get pasted together.

## References

* [[Max Kelly]], *Many-variable functorial calculus. I*. In "Coherence in categories", LNM 281, pp. 66–105
 {#KellyMVFC}

* [[Max Kelly]], *An abstract approach to coherence.*  In "Coherence in categories", LNM 281, pp. 106–147.
 {#KellyAAC}

* [[Max Kelly]], *A cut-elimination theorem.*  In "Coherence in categories", LNM 281, pp. 196–213.
 {#KellyCET}

* [[Max Kelly]], *On clubs and doctrines*. In "Category Seminar (Proc. Sem., Sydney, 1972/1973)", LNM 420, pp. 181–256.
 {#KellyOCD}

* [[Max Kelly]], *On clubs and data-type constructors*, Applications of Categories in Computer Science, Proceedings of the London Mathematical Society Symposium, Durham 1991
 {#KellyCDT}

[[!redirects club]]
[[!redirects clubs]]


club in a 2-category

# Clubs in a 2-category

* table of contents
{: toc}

## Idea

We can generalize the notion of [[club]] over a [[cartesian monad]] on $Cat$ to a cartesian monad $S$ on an object $C$ of any [[2-category]] $K$.

Note that the $Cat$ *on* which the cartesian monad lives does not generalize to the 2-category $K$; it generalizes to the [[object]] $C$.  The 2-category $K$ generalizes the [[large category|larger]] 2-category $CAT$ that contains $Cat$ as an object.

## Definition

### The context

Let $K$ be a 2-category and $C\in K$ an object that *has representable finite limits*, meaning that each [[hom-category]] $K(X,C)$ has [[finite limits]] and the precomposition functors $(-\circ f) : K(X,C) \to K(Y,C)$ [[preserved limit|preserve]] finite limits.

Given morphisms $f,g:X\to C$, a 2-cell $\alpha:f\to g$ is called **cartesian** if for any $h,k:Y\to X$ and 2-cell $\beta:h\to k$, the commutative square
$$ \array{ f h & \overset{f \beta}{\to} & f k\\
  ^{\alpha h}\downarrow && \downarrow^{\alpha k}\\
  g h & \underset{g\beta}{\to} & g k } $$
is a pullback in $K(Y,C)$.  (Note that this is a different notion of "cartesian 2-cell" than the one involved in the notion of [[fibration in a 2-category]].)

A [[monad]] $s$ on $C$ (that is, a 1-morphism $s:C\to C$ equipped with 2-morphisms $\mu : s s\to s$ and $\eta : id_C \to s$ that are associative and unital) is **cartesian** if $\mu$ and $\eta$ are cartesian 2-cells, in the above sense, and $s$ preserves pullbacks in the sense that each functor $(s\circ -):K(X,C) \to K(X,C)$ preserves pullbacks.

### The object of collections

Now suppose that $K$ is [[cartesian closed category|cartesian closed]] --- the weak [[bicategory|bicategorical]] sense suffices, i.e. we have [[equivalences]] $K(X\times Y,Z) \simeq K(X,[Y,Z])$.  Suppose further more that $K$ has a [[terminal object]] and [[comma objects]].  To avoid confusion, we will denote by $\mathbf{1}$ the terminal object of $K$, by $id$  the identity morphisms in $K$, and by $1\in K(X,C)$ the representable terminal object of $C$.

Now any monad $s$ transposes to a morphism $\mathbf{1}\to [C,C]$, and we can form the "slice object" $[C,C]/s$ by the following comma square:
$$ \array{ [C,C]/s & \to & [C,C] \\
  \downarrow & \swArrow & \downarrow \\
  \mathbf{1} & \underset{s}{\to} & [C,C] } $$
This has the representable property that $K(X,[C,C]/s) \simeq K(X\times C,C)/s_X$, where $s_X$ is the composite $X\times C \overset{\pi_2}{\to} C \overset{s}{\to} C$.

Similarly, we have $s 1 : \mathbf{1} \to C$ (meaning the composite $\mathbf{1} \overset{1}{\to} C \overset{s}{\to} C$, where $1:\mathbf{1}\to C$ is the representable terminal object of $C$), and we can form the slice $C/s1$.  Thus has the representable property $K(X,C/s1) \simeq K(X,C)/(s1)_X$, where $(s1)_X$ is the composite $X \overset{!}{\to} \mathbf{1} \overset{s1}{\to} C$.  Now "evaluate at 1" yields a morphism $ev_1:[C,C]/s \to C/s1$, which can be defined representably in a straightforward way.

Given the assumption that $C$ has representable pullbacks as well, we claim that the morphism $ev_1$ has a right adjoint.  This can be constructed representably as follows.  A morphism $X\to C/s1$ is a morphism $a:X\to C$ together with a 2-cell $a \to (s1)_X$, and we want a morphism $b:X\times C\to C$ together with a 2-cell $b \to s_X$.  We define $b$ to be the pullback
$$ \array{ b & \to & a \pi_1\\
  \downarrow && \downarrow\\
  s_X & \to & (s1)_{X\times C} = (s1)_X \pi_1 } $$
It is straightforward to check that this defines a right adjoint whose [[counit]] is fully faithful.  It commutes with precomposition (since the pullbacks in $S$ do), so it induces a right adjoint $C/s1 \to [C,C]/s$ whose counit is also fully faithful, i.e. $C/s1$ is a "reflective subobject" of $[C,C]/s$.

Moreover, by construction, a morphism $X\to [C,C]/s$ factors through $C/s1$ just when for the corresponding map $b:X\times C\to C$, the induced square
$$ \array{ b & \to & b (id\times 1) \pi_1\\
  \downarrow && \downarrow\\
  s_X & \to & (s1)_X \pi_1 } $$
is a pullback.  Now for any $h,k:Y\to X\times C$ and $\beta:h\to k$, consider the rectangle
$$ \array{b h &\to & b k & \to & b (id \times 1) \pi_1 h\\
  \downarrow && \downarrow && \downarrow\\
  s_X h & \to & s_X k & \to & (s1)_Y \pi_1 } $$
The factorization property above says that the right-hand square and the outer rectangle are pullbacks, which by the pasting law for pullbacks implies that the left-hand square is also a pullback, i.e. that $b\to s_X$ is a cartesian 2-cell as defined above.  On the other hand, if $b\to s_X$ is a cartesian 2-cell, then precomposing with the unique 2-cell $id_C \to 1$ yields the above square.  Hence, $C/s1$ as a subobject of $[C,C]/s$ can be said to be the "full subobject consisting of the cartesian transformations into $s$".  We may call it the object of *collections* over $S$.

### The substitution product

Now, $[C,C]$ is always a [[pseudomonoid]] in $K$, and since $s$ is a monoid therein, the comma object $[C,C]/s$ inherits a pseudomonoid structure.  We claim that if $s$ is cartesian, then $C/s1$ is closed under this pseudomonoid structure, i.e. $C/s1$ is a pseudomonoid and the inclusion is a pseudomonoid morphism.

Firstly, the unit of $[C,C]/s$ is the map $\mathbf{1} \to [C,C]/s$ corresponding to $id_C$ equipped with $\eta:id_C \to s$.  Since part of $s$ being cartesian means that $\eta$ is a cartesian 2-cell, this certainly factors through $C/s1$.

Secondly, for the tensor product we again argue representably.  Suppose given two maps $X\to [C,C]/s$, corresponding to $a,b:X\times C\to C$ equipped with cartetsian 2-cells to $s_X$.  Their "product" induced by the pseudomonoid structure on $[C,C]/s$ is the composite
$$ X\times C \overset{\Delta}{\to} X\times X\times C \overset{id\times b}{\to} X\times C \overset{a}{\to} C $$
equipped with the 2-cell
$$ a (id\times b) \Delta \to s_X(id\times b)\Delta \cong s b \to s s \overset{\mu}{\to} s. $$
Then we have the composite rectangle
$$ \array{
  a (id\times b) \Delta & \to &
  s_X(id\times b)\Delta & \cong &
  s b & \to &
  s s_X & \to &
  s_X \\
  \downarrow && \downarrow && \downarrow && \downarrow && \downarrow \\
  a (id\times b) \Delta (id\times 1) \pi_1 & \to &
  s_X(id\times b)\Delta(id\times 1)\pi_1 & \cong &
  s b (id\times 1) \pi_1 & \to &
  s s_X (id \times 1) \pi_1 & \to &
  s 1
} $$
The first square on the left is a pullback since $a\to s_X$ is cartesian.  The second is an isomorphism, hence automatically a pullback.  The third is a pullback since $b\to s_X$ is cartesian and $s$ preserves pullbacks.  And the last is a pullback since $\mu:s s \to s$ is cartesian.  Thus, the composite rectangle is also a pullback, showing that $a(id\times b)\Delta \to s_X$ is also cartesian.

Note that we did not actually need $C$ to have all representable pullbacks nor $s$ to preserve all of them: we only needed the existence of pullbacks of the form
$$ \array{ b & \to & a \pi_1\\
  \downarrow && \downarrow\\
  s_X & \to & (s1)_X \pi_1 } $$
and $s$ to preserve these.

## Examples

### Clubs on Cat

When $K=CAT$, we obtain the notion of club over a cartesian monad described at [[club]].  The usual subcase of interest is when $C=Cat$, and the simple case of clubs over $\mathbf{P}$ (the permutation category) is the further special case when $s$ is the free symmetric (strict) monoidal category monad.

### Double clubs

When $K=DBLCAT$ is the 2-category of (pseudo) [[double categories]], pseudo double functors, and vertical transformations, we obtain essentially the notion of *double club* from [(Garner)](#Garner).  Representable pullbacks in $DBLCAT$ can be constructed "levelwise", by taking pullbacks in the category of objects and vertical arrows, and also in the category of (all) horizontal arrows and cells; except that in general such a pullback may not define a pseudo double functor (it may not preserve identities and composition up to isomorphism).  The condition (hps) required by Garner ensures that the *specific* pullbacks needed above do exist as pseudo double functors.  Cartesianness of monads and transformations can also be tested levelwise.

### Virtual double clubs

When $K=VEQUIP$ is the 2-category of [[virtual equipments]] (or even just [[virtual double categories]] with units) and normal (unit-preserving) functors, we obtain a notion of *virtual double club*.  The units are necessary to have a cartesian closed 2-category (specifically, we need morphisms $\mathbf{1} \to C$, where $\mathbf{1}$ is the terminal object, to correspond to objects of $C$; and without units and normality such a morphism would instead be a monoid in $C$).  However, we can do without requiring composites to exist or be preserved by functors.

This means that it suffices to assume Garner's condition (hps2) on units, but not necessarily his condition (hps1) on composites.  This is nice because in many cases, such as the double category $Cat$ of categories, functors, and profunctors, (hps2) is automatically true for all monads $s$ by [(Garner, Proposition 48)](#Garner).

On the other hand, in this context we only get that the "inclusion" $C/s1 \to [C,C]/s$ is a normal functor of virtual double categories, i.e. lax in the horizontal direction; whereas in Garner's context it is a strong functor.

## References

* [[Richard Garner]], *Double clubs*, Cahiers de Topologie et Geometrie Differentielle Categoriques 47 (2006), no. 4, 261--317; [arXiv](http://arxiv.org/abs/math/0606733)

[[!redirects double club]]
[[!redirects double clubs]]


cluster algebra

#Contents#
* table of contents
{:toc}

## Idea

A matrix $A$ over the field of [[real numbers]] is __[[totally positive matrix|totally positive]]__ (resp. __totally nonnegative__) if every minor (= determinant of any submatrix) is a positive (resp. nonnegative) real number. Total positivity implies a number of remarkable properties; for example all eigenvalues are distinct and positive.

[[George Lusztig]] discovered that total positivity is closely related to some phenomena in the theory of [[Lie groups]] and [[quantum groups]]. Later, [[Sergey Fomin]] and [A. Zelevinsky](http://www.math.neu.edu/zelevinsky/andrei.html) studied the canonical bases for [[quantum groups]] and discovered the combinatorics of simple transformations and defined  associated classical and quantum __cluster algebras__ to such situations.   In particular, Stasheff [[associahedron|associahedra]] are  associated to these cluster algebras. Remarkably, they found an unusual [[algebraic geometry]] related to cluster algebras, possessing new, and at the beginning mysterious, [[Laurent phenomenon]]. Later, the cluster algebras appeared also in the connection to the representations of [[quivers]], tilting theory and the wall crossing phenomenon, with the applications in [[representation theory]] and the study of [[triangulated category|triangulated categories]].

A cluster algebra of rank $n$ comes equipped with some subsets of size $n$ called clusters. Some of these clusters are related by sequences of operations called mutations.

##Definition##

(this is still a stub)

##Links and references##

* Sergey Fomin, Andrei Zelevinsky, _Cluster algebras. I. Foundations_, J. Amer. Math. Soc. 15 (2002), no. 2, 497--529, [math.RT/0104151 ](http://arxiv.org/abs/math/0104151); _Cluster algebras. II. Finite type classifications._ Invent. Math. 154 (2003), no. 1, 63--121 -- classification of cluster algebras of finite type via root systems [math.RA/0208229](http://arxiv.org/abs/math/0208229), _Cluster algebras, IV: Coefficients_, Compos. Math. __143__ (2007), 112–164, [MR2295199](http://www.ams.org/mathscinet-getitem?mr=2295199) [doi](http://dx.doi.org/10.1112/S0010437X06002521)

* Arkady Berenstein, Andrei Zelevinsky, _Quantum cluster algebras_, [math.QA/0404446](http://arxiv.org/abs/math/0404446)

* K. Goodearl, M. Yakimov, _Quantum cluster algebras and quantum nilpotent algebras_
Proc Natl Acad Sci USA 111(27):9696-703  (2014) [doi](https://doi.org/10.1073/pnas.1313071111)

* B. Keller, _Cluster algebras, quiver representations and triangulated categories_, [arXiv:0807.1960](http://arXiv.org/abs/0807.1960) [MR2681708](http://www.ams.org/mathscinet-getitem?mr=2681708)

* Lots of links at [Cluster algebras portal](http://www.math.lsa.umich.edu/~fomin/cluster.html) including to the Fomin's [course slides](http://www.math.lsa.umich.edu/~fomin/nordfjordeid.html).

* description and conference info "Cluster Algebras and Lusztig's Semicanonical Basis", Oregon, June 2011, [html](http://pages.uoregon.edu/njp/cluster.html)

* Maxim Kontsevich, Yan Soibelman, _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_, [arXiv:0811.2435](http://arxiv.org/abs/0811.2435)

* Yuji Kodama, Lauren Williams, _KP solitons, total positivity, and cluster algebras_, Proc. Natl. Acad. Sci. [arxiv/1105.4170](http://arxiv.org/abs/1105.4170), [doi](doi:10.1073/pnas.1102627108)
* [[Bernard Leclerc]], _Cluster algebras and representation theory_, [arxiv/1009.4552](http://arxiv.org/abs/1009.4552)
* Christof Geiss, Bernard Leclerc, Jan Schröer, _Preprojective algebras and cluster algebras_, [arxiv/0804.3168](http://arxiv.org/abs/0804.3168); _Kac-Moody groups and cluster algebras_, [arxiv/1001.3545](http://arxiv.org/abs/1001.3545); _Cluster structures on quantum coordinate rings_, [arxiv/1104.0531](http://arxiv.org/abs/1104.0531)

* Tomoki Nakanishi, _[[dilogarithm|Dilogarithm]] identities for conformal field theories and cluster algebras: Simply laced case_, Nagoya Math. J. __202__ (2011), 23-43, [MR2804544](http://www.ams.org/mathscinet-getitem?mr=2804544) [doi](http://dx.doi.org/10.1215/00277630-1260432)
* Kentaro Nagao, _Donaldson–Thomas theory and cluster algebras_, Duke Math. J. 162(7): 1313-1367 [doi](https://doi.org/10.1215/00127094-2142753)

The connections to the exact WKB method a la Voros are studied in

* Kohei Iwaki, Tomoki Nakanishi, _Exact WKB analysis and cluster algebras_, J. Phys. A 47 (2014) 474009 [arxiv/1401.7094](http://arxiv.org/abs/1401.7094); _Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras_, [arXiv:1409.4641](http://arxiv.org/abs/1409.4641)

Generalized cluster algebras are studied in

* Leonid Chekhov, Michael Shapiro, _Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables_, Int. Math. Res. Notices (2014)  2746-2772 [arxiv/1111.3963](http://arxiv.org/abs/1111.3963) [doi](http://dx.doi.org/10.1093/imrn/rnt016)
* Tomoki Nakanishi, _Structure of seeds in generalized cluster algebras_, [arxiv/1409.5967](http://arxiv.org/abs/1409.5967)

[[!redirects cluster algebras]]
[[!redirects cluster transformation]]
[[!redirects cluster transformations]]


cluster decomposition

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## Idea

In the [[physics]] of [[quantum field theory]], _cluster decomposition_ is a property of [[correlators]]/[[scattering amplitudes]] which expresses the [[local quantum field theory|locality]] property. Roughly it says that interaction between many [[field (physics)|fields]] can be decomposed into those within localized clusters.

## Related concepts

* [[S-matrix]]

## References

* Wikipedia, _[Cluster decomposition theorem](http://en.wikipedia.org/wiki/Cluster_decomposition_theorem)_

[[!redirects cluster decomposition theorem]]


cluster variety

Cluster varieties, their quantization and their duality were gradually introduced in a series of works of Fock and Goncharov including

* V. V. Fock, A. B. Goncharov, _Cluster ensembles, quantization and the dilogarithm_, [arXiv:math.AG/0311245](https://arxiv.org/abs/math/0311245); _Cluster X-varieties, amalgamation and Poisson-Lie groups_, [arXiv:math.RT/0508408](https://arxiv.org/abs/math/0508408)
* V. V. Fock, A. B. Goncharov, _The quantum dilogarithm and representations quantum cluster varieties_,  Invent. math. __175__, 223–286 (2009) [doi](https://doi.org/10.1007/s00222-008-0149-3) [arXiv:0702397](https://arxiv.org/abs/math/0702397)

>  We construct, using the [[quantum dilogarithm]], a series of $\ast$-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces. One of applications is quantization of higher [[Teichmueller space]]s. The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms. We introduce the symplectic/quantum double of cluster varieties and related them to the representations.

A major conjecture has been resolved in

 * [[Mark Gross]], Paul Hacking, Sean Keel, [[Maxim Kontsevich]], _Canonical bases for cluster algebras_, [arxiv:1411.1394](https://arxiv.org/abs/1411.1394)

> In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture. In particular, under suitable hypotheses, for each $Y$ the partial compactification of an affine cluster variety $U$ given by allowing some frozen variables to vanish, we obtain canonical bases for the ring of functions on $Y$ extending to a basis for functions on U. Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space $Y$ we obtain a canonical basis of each irreducible representation of $SL_r$, parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

* Lara Bossinger, _Tropical totally positive cluster varieties_, [arXiv:2208.01723](https://arxiv.org/abs/2208.01723)


clutching construction

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{:toc}

## Idea

The _clutching construction_ is the construction of a $G$-[[principal bundle]] on an [[n-sphere]] from a [[cocycle]] in $G$-[[Cech cohomology]] given by the [[covering]] of the $n$-sphere by two [[hemi-n-spheres]] that overlap a bit at the equator, and one single transition function on that equator $S^{n-1} \to G$.

## Examples

### Basic example

The [[Möbius strip]] is the result of the single non-trivial clutching construction for [[real vector bundles|real]] [[line bundle]] over the [[circle]].

### In physics

In [[physics]], in [[gauge theory]], the clutching construction plays a central role in the discussion of [[Yang-Mills instantons]], and [[monopoles]] ([[Dirac monopole]]). Here the discussion is usually given in terms of [[gauge fields]] on $n$-dimensional [[Minkowski spacetime]] such that they [[vanishing at infinity|vanish at infinity]]. Equivalently this means that one has [[gauge fields]] on the [[one-point compactification]] of Minkowski spacetime, which is the [[n-sphere]]. The transition function of the clutching construction then appears as the asymptotic [[gauge transformation]].

## Related concepts

* [[QCD instanton]], [[Yang-Mills instanton]]

## Literature

Review:

* [[Allen Hatcher]], page 22 of: _Vector bundles and K-theory_ ([web](http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* {#Wirthmuller12} [[Klaus Wirthmüller]], p. 17-28 of: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* Wikipedia, _[Clutching construction](http://en.wikipedia.org/wiki/Clutching_construction)_

[[!redirects clutching constructions]]

[[!redirects clutching function]]
[[!redirects clutching functions]]


CMon-enriched monoidal category

#Contents#
* table of contents
{:toc}

## Idea

A **CMon-enriched monoidal category** is a [[monoidal category]] such that each [[hom-set]] is a [[commutative monoid]] and the [[tensor product]] on [[morphisms]] and the [[composition]] are [[bilinear]].

## Definition

In details, a **CMon-enriched monoidal category** is a monoidal category such that each hom-set $\mathcal{C}[A,B]$ is a commutative monoid (we write $f + g$ for the [[sum]] of two morphisms $f,g \colon A \rightarrow B$ and $0$ for the [[zero]] $A \rightarrow B$), such that for [[every]] $f,g \colon A \rightarrow B$, $h,i \colon C \rightarrow D$ and $j,k \colon B \rightarrow C$:

* $(f+g)\otimes h = f \otimes h + g \otimes h$
* $f \otimes (h+i) = f \otimes h + f \otimes i$
* $0 \otimes f = f \otimes 0 = 0$
* $(f+g);j = f;j + g;j$
* $f;(j+k) = f;j + f;k$
* $0;f = f;0 = 0$

## Properties

\begin{proposition}
If a CMon-enriched monoidal category possesses the [[binary products]], then they are [[biproducts]] and the category is thus [[semiadditive category|semiadditive]]. The same goes if it possesses the [[binary coproducts]].
\end{proposition}

\begin{proposition}
If a CMon-enriched monoidal category possesses a [[terminal object]], then it is a [[zero object]]. The same goes if it possesses an [[initial object]].
\end{proposition}

## Examples

\begin{proposition}
Every [[semiadditive category|semiadditive]] monoidal category is a CMon-enriched monoidal category.
\end{proposition}

## Related entries

* [[CMon-enriched symmetric monoidal category]]

[[!redirects CMon-enriched monoidal categories]]


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* table of contents
{:toc}

## Idea

A **CMon-enriched symmetric monoidal category** is a [[symmetric monoidal category]] such that each [[hom-set]] is a [[commutative monoid]] and both the [[tensor product]] and the [[composition]]-operation on [[morphisms]] are [[bilinear]].

Beware that in discussion of [[differential categories]] and of [[differential linear logic]] such categories are often referred to as "([[pre-additive category|pre-]])[[additive categories]]", while traditionally this terminology refers to [[enriched category|enrichment]] in [[abelian groups]] instead of just [[commutative monoids]]. The term "CMon-enriched symmetric monoidal category" is non-standard but used here to avoid this clash of terminology.

## Definition

In detail, a **CMon-enriched symmetric monoidal category** is a [[symmetric monoidal category]] such that each [[hom-set]] $\mathcal{C}[A,B]$ is a [[commutative monoid]] (we write $f + g$ for the [[sum]] of two morphisms $f,g \colon A \rightarrow B$ and $0$ for the [[zero]] $A \rightarrow B$), such that for [[every]] $f,g \colon A \rightarrow B$, $h,i \colon C \rightarrow D$ and $j,k \colon B \rightarrow C$:

* $(f+g)\otimes h = f \otimes h + g \otimes h$
* $f \otimes (h+i) = f \otimes h + f \otimes i$
* $0 \otimes f = f \otimes 0 = 0$
* $(f+g);j = f;j + g;j$
* $f;(j+k) = f;j + f;k$
* $0;f = f;0 = 0$

[[!redirects CMon-enriched symmetric monoidal categories]]

## Related entries

* [[CMon-enriched monoidal category]]


CMU HoTT Research Group's local activities

## CMU-HoTT Research Group's Local Activities ##

Here are some links to things happening at the CMU-HoTT Research Group.

* CMU-HoTT research group [website](https://cmuhott.wordpress.com)
* Marc Bezem's [[HoTT Mini-Course]], Spring 2016.
* Steve's slides from the 2018 HoTT-MURI meeting: [[awodeyMURI18.pdf:file]]

## See also

* [[homotopy type theory events]]


CNOT gate

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## Idea
 {#Idea}

The *controlled NOT* or *[[reversible computation|reversible]] [[XOR]]-gate* is the [[logic gate]] acting between [[pairs]] of [[bits]] which identically keeps the first bit in the pair and [[negation|negates]] the second IFF the first was set.

This may be understood equivalently as

* the [[controlled quantum gate]]-version of the quantum [[NOT]]-gate

* the [[reversible computation|reversible]] version of the [[XOR]]-gate.

In [[quantum computing]]/[[quantum information theory]], by the *controlled NOT gate* or *CNOT gate* one means the [[quantum logic gate]] which acts on a [[pair]] of [[qbits]] by the [[linear map]] which is given in this way on the canonical [[linear basis]]-elements of a [[pair]] of [[qbits]]:

\begin{imagefromfile}
    "file_name": "CNOTGates-230807.jpg",
    "width": "740",
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Similarly, by the *CCNOT*-gate (or *Toffoli gate*) one means the operation on triples of [[bits]]/[[qbit]] which (always retains the first two and) reverses the third iff the first two are both set.

## Related entries

* [[controlled quantum gate]]

* [[Hadamard gate]]

* [[reversible computation]]

## References

Original discussion:

* [[David E. Deutsch]], Figure 2 in: *Quantum computational networks*, Proceedings of the Royal Society A, **425** 1868 (1989) 73-90 [[doi:10.1098/rspa.1989.0099](https://doi.org/10.1098/rspa.1989.0099), [jstor:2398494](https://www.jstor.org/stable/2398494)]

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], pp. 8 of: *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

Textbook accounts:

* [[Michael A. Nielsen]], [[Isaac L. Chuang]], p. 21 in: *Quantum computation and quantum information*, Cambridge University Press (2000) $[$[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]$]$

See also:

* Wikipedia, *[Controlled NOT gate](https://en.wikipedia.org/wiki/Controlled_NOT_gate)*

* Wikipedia, *[Toffoli gate](https://en.wikipedia.org/wiki/Toffoli_gate)*

[[!redirects CNOT]]
[[!redirects CCNOT]]

[[!redirects Toffoli gate]]

[[!redirects controlled NOT gate]]
[[!redirects controlled-controlled NOT gate]]

[[!redirects quantum CNOT gate]]
[[!redirects quantum CNOT gates]]

[[!redirects controlled quantum NOT gate]]
[[!redirects controlled quantum NOT gates]]


co-associativity

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## Idea

The [[formal dual]] of [[associativity]].

## Definition

Given a [[monoidal category]] $(\mathcal{C}, \otimes)$ and an [[object]] $A$ in $\mathcal{C}$ equipped with morphism ("[[co-multiplication]]") $\Delta \colon A \longrightarrow A \otimes A$, then this is _co-associative_ if the following [[commuting diagram|diagram commutes]]

$$
  \array{
    A &\overset{\Delta}{\longrightarrow}& A \otimes A
    \\
    \downarrow && \downarrow^{\mathrlap{\Delta \otimes id}}
    \\
    A \otimes A
    &\underset{id \otimes \Delta}{\longrightarrow}& A \otimes A \otimes A
  }
  \,.
$$

If in addition there is a [[counit]] on $A$ for which the coproduct satisfies [[co-unitality]], then $A$ is called a  [[co-monoid]] in $(\mathcal{C}, \otimes)$.

[[!redirects coassociativity]]


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* table of contents
{:toc}

## Idea

We discuss (Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} below) how the [[rationalization]] of the [[Whitehead product]] is the co-binary part of the [[Sullivan model|Sullivan differential]] in [[rational homotopy theory]].

## Notation and conventions

We make explicit some notation and normalization conventions that enter the statement.

In the following, for $W$ a $\mathbb{Z}$-[[graded module]], we write

$$
  W \wedge W
  \;\coloneqq\;
  Sym^2(W)
  \;\coloneqq\;
  \big(
    W \otimes W
  \big)
  /
  \big(
   \alpha \otimes \beta
     \sim
   (-1)^{ n_\alpha n_\beta }
   \beta \otimes \alpha
  \big)
  \,,
$$

where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that "$(-)\wedge(-)$" is not to imply here a degree shift of the generators (as it typically does in the usual notation for [[Grassmann algebras]]).

Let $X$ be a [[simply connected topological space]] with [[Sullivan model]]

\[
  \label{SullivanModelX}
  CE( \mathfrak{l} X )
  \;=\;
  \big(
    Sym^\bullet\big(V^\ast\big), d_X
  \big)
\]

for $V^\ast$ the [[graded vector space]] of generators, which is the $\mathbb{Q}$-linear [[dual space|dual]] [[graded vector space]] of the [[graded object|graded]] $\mathbb{Z}$-[[module]] (=[[graded abelian group]]) of [[homotopy groups]] of $X$:

$$
  V^\ast
  \;\coloneqq\;
  Hom_{Ab}\big(
    \pi_\bullet(X),
    \mathbb{Q}
  \big)
  \,.
$$

Declare the [[wedge product]] pairing to be given by

\[
  \label{WedgeProductNormalization}
  \array{
    V^\ast
    \wedge
    V^\ast
    &\overset{\Phi}{\longrightarrow}&
    Hom_{Ab}
    \big(
      \pi_\bullet(X) \wedge \pi_\bullet(X)
      ,
      \mathbb{Q}
    \big)
    \\
    (\alpha, \beta)
    &\mapsto&
    \Big(
      v \wedge w
      \;\mapsto\;
      (-1)^{ n_\alpha \cdot n_\beta }
      \alpha(v)\cdot \beta(w)
      +
      \beta(v)\cdot \alpha(w)
    \Big)
  }
\]

where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.

(Notice that the usual normalization factor of $1/2$ is _not_ included on the right. This normalization follows [Andrews-Arkowitz 78, above Thm. 6.1](#AndrewsArkowitz78).)

Finally, write

\[
  \label{PojectionToBinary}
  [-]_2
  \;\colon\;
  Sym^\bullet\big(V^\ast\big)
  \longrightarrow
  V^\ast \wedge V^\ast
\]

for the linear projection on quadratic polynomials in the graded [[symmetric algebra]].

## Statement

+-- {: .num_prop #CoBinarySullivanDifferentialIsWhiteheadProduct}
###### Proposition
**([[co-binary Sullivan differential is Whitehead product]])**

Let $X$ be a [[simply connected topological space]] of rational [[finite type]], so that it has a [[Sullivan model]] with Sullivan differential $d_X$ (eq:SullivanModelX).

Then the co-binary component (eq:PojectionToBinary) of the Sullivan differential equals the $\mathbb{Q}$-[[linear dual map]] of the [[Whitehead product]] $[-,-]_X$ on the [[homotopy groups]] of $X$:

$$
  [d_X \alpha]_2
  \;=\;
  [-,-]_X^\ast
  \,.
$$

More explicitly, the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    V^\ast
      &\overset{ [-]_2\circ d_X }{\longrightarrow}&
    V^\ast \wedge V^\ast
    \\
    \big\downarrow^{ \mathrlap{=} }
    &&
    \big\downarrow^{ \mathrlap{\Phi} }
    \\
    Hom_{Ab}
    \big(
      \pi_\bullet(X),
      \mathbb{Q}
    \big)
    &
      \underset{
        Hom_{Ab}\big(
          [-,-]_X
          ,
          \;
          \mathbb{Q}
        \big)
      }{
        \longrightarrow
      }
    &
    Hom_{Ab}
    \big(
      \pi_\bullet(X)
        \wedge
      \pi_\bullet(X),
      \;
      \mathbb{Q}
    \big)
  }
  \,,
$$

where the wedge product on the right is normalized as in (eq:WedgeProductNormalization).

=--

([Andrews-Arkowitz 78, Thm. 6.1](#AndrewsArkowitz78), see also [Félix-Halperin-Thomas 00, Prop. 13.16](#FelixHalperinThomas00))

\linebreak

## Examples

### Hopf fibrations
 {#HopfFibrations}

For $X = S^2$ the [[2-sphere]], consider the following two [[elements]] of its [[homotopy groups]] ([[homotopy groups of spheres|of spheres]], as it were):

1. $id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the [[identity function]] $S^2 \to S^2$)

1. $h_{\mathbb{C}} \in \pi_3\big(  S^3 \big)$ (represented by the [[complex Hopf fibration]])

Then the Whitehead product satisfies

\[
  \label{WhiteheadProductOf2Sphere}
  \big[
    id_{S^2},
    \;
    id_{S^2}
  \big]
  \;=\;
  2 \cdot h_{\mathbb{C}}
  \,.
\]

(by [this Example](Whitehead+product#WhiteheadProductCorrespondingToComplexHopfFibration)).

Now let

$$
  vol_{S^2},\; vol_{S^3}
  \;\in\;
  CE\big( \mathfrak{l}S^2 \big)
$$

be the two generators of the [[rational n-sphere|Sullivan model of the 2-sphere]], _normalized_ such that they correspond to the [[volume forms]] of the [[2-sphere]] and (after [[pullback of differential forms|pullback]] along the [[complex Hopf fibration]] $h_{\mathbb{C}}$) of the [[3-sphere]], respectively.

This means that the [[Sullivan model|Sullivan differential]] is

\[
  \label{GenericDifferentialForSullivanModelOf2Sphere}
  d_{S^2} vol_{S^3}
  \;=\;
  c \cdot
  vol_{S^2} \wedge vol_{S^2}
\]

for some [[rational number]] $c \in \mathbb{Q}$.

Notice that with the normalization in (eq:WedgeProductNormalization) we
have

$$
  \begin{aligned}
  \Phi(vol_{S^2}, vol_{S^2})\big(  id_{S^2} \wedge id_{S^2} \big)
  & =
  (-1)^{2 \cdot 2}
  vol_{S^2}\big( id_{S^2}\big)
  \cdot
  vol_{S^2}\big( id_{S^2}\big)
  +
  vol_{S^2}\big( id_{S^2}\big)
  \cdot
  vol_{S^2}\big( id_{S^2}\big)
  \\
  & =
  2
  \end{aligned}
$$

Therefore Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} gives

$$
  \array{
    \big\{
      vol_{S^3}
    \big\}
    &\overset{[-]_2\circ d_{S^2}}{\longrightarrow}&
    c' \cdot ( vol_{S^2} \otimes vol_{S^2} )
    \\
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{\Phi}}
    \\
    \big\{
     h_{\mathbb{H}}
    \big\}
    &\overset{ [-,-]_X^\ast }{\longrightarrow}&
    \big\{
      id_{S^2} \wedge id_{S^2}
      \mapsto
      2
    \big\}
  }
$$

where in the bottom row we used the Whitehead product (eq:WhiteheadProductOf2Sphere).

Hence $c = 1$:

$$
  d_{S^2} vol_3
  \;=\;
  - vol_{S^2} \wedge vol_{S^2}
  \,.
$$

See also [Félix-Halperin-Thomas 00, Example 1 on p. 178](#FelixHalperinThomas00).

## References

Under the general relation between the Sullivan model and the original Quillen model of [[rational homotopy theory]], the statement comes from

* {#Quillen69} [[Daniel Quillen]], section I.5 of _Rational Homotopy Theory_, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](https://www.jstor.org/stable/1970725))

It is made fully explicit in

* {#AndrewsArkowitz78} Peter Andrews, [[Martin Arkowitz]], Theorem 6.1 _Sullivan's Minimal Models and Higher Order Whitehead Products_, Canadian Journal of Mathematics, 30(5), 961-982, 1978 ([doi:10.4153/CJM-1978-083-6](https://doi.org/10.4153/CJM-1978-083-6))

where the result is attributed to

* {#DeligneGriffithsMorganSullivan75} [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], _Real homotopy theory of Kähler manifolds_, Invent Math (1975) 29: 245 ([doi:10.1007/BF01389853](https://doi.org/10.1007/BF01389853))

(which however just touches on it in passing)

and in

* Francisco Belchí, [[Urtzi Buijs]], José M. Moreno-Fernández, Aniceto Murillo, Prop. 3.1 of: _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 ([arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008))

Textbook accounts:

* {#FelixHalperinThomas00} [[Yves Félix]], [[Steve Halperin]], J.C. Thomas, Prop. 13.16 in _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

[[!redirects the co-binary Sullivan differential is the dual Whitehead product]]
[[!redirects the co-binary Sullivan differential is the dual of the Whitehead product]]


co-concrete object

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### Context
#### Discrete and concrete objects
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=--
#### Category Theory
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[[!include category theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

For $\Gamma : \mathcal{E} \to \mathcal{B}$ a [[functor]] we say that it _has [[discrete objects]]_ if it has a  [[full and faithful functor|full and faithful]] [[left adjoint]] $Disc : \mathcal{B} \hookrightarrow \mathcal{E}$.

An object in the [[essential image]] of $Disc$ is called a **discrete object.

This is for instance the case for the [[global section]] [[geometric morphism]] of a [[connected topos]] $ (Disc \dashv \Gamma ) : \mathcal{E} \to \mathcal{B}$.

In this situation, we say that a **co-concrete object** $X \in \mathcal{E}$ is one for which the $(Disc\dashv \Gamma)$-[[counit of an adjunction]] is an [[epimorphism]].

The dual concept is the of a _[[concrete object]]_.

## References

* [[Mike Shulman]], _Discreteness, Concreteness, Fibrations, and Scones_ ([blog post](http://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html))
 {#Shulman}

[[!redirects co-concrete objects]]

[[!redirects coconcrete object]]
[[!redirects coconcrete objects]]


co-H-space

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of __co-H-space__ is the [[Eckmann-Hilton duality|Eckmann-Hilton dual]] of that of [[H-spaces]].

They are [[co-H-object]]s in the [[category]] of [[pointed object|pointed]] [[topological spaces]]. Thus a co-H-space $(X, \phi)$ is a pointed space, $X$, together with a map $\phi: X \to X \vee X$ (the [[wedge sum]]), such that $p_i \circ \phi$ is homotopic to $1_X$, where $p_i, i = 1, 2$, are the projections $X \vee X \to X$. Alternatively, $(X, \phi)$ is a co-H-space if and only if $j \circ \phi$ is homotopic to $\Delta$, where $j: X \vee X \to X \times X$ is the inclusion and $\Delta: X \to X \times X$ is the [[diagonal map]].

The importance of the notion is that $X$ is a co-H-space if and only if for every space $Y$, $[X, Y]$ has a binary operation with unit. Further properties of $\phi$ are of interest, in particular being (co)associative and having right and left (co)inverses. In this case $X$ is a [[cogroup]]. The [[suspension]] of a topological space is a cogroup.

Every co-H-space is [[path-connected space|path-connected]], and its [[fundamental group]] is [[free group|free]].

##Reference##

* [[Martin Arkowitz]], _Co-H-spaces_, chapter 23 of [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_

[[!redirects co-H-spaces]]


co-Heyting algebra

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### Context
#### Category theory
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#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
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=--

# Contents
* table of contents
{:toc}

##Idea

When the [[lattice of open subsets]] of a [[topological space]] is the primordial example of a [[Heyting algebra]] then its dual _lattice of [[closed subsets]]_ is the primordial example of a **co-Heyting algebra**.

In general, co-Heyting algebras are [[duality|dual]] to Heyting algebras and like them come equipped with non-Boolean logical operators that make them interesting players in [[modal logic|modal]], [[paraconsistent logic|paraconsistent]], and co-intuitionistic logic, [[linguistics]], [[topos theory]], [[continuum physics]], [[quantum theory]] and in [[mereology]].

##Definition

A **co-Heyting algebra** is a bounded [[distributive lattice]] $L$ equipped with a binary _[[subtraction]]_ operation $ \backslash :L\times L\to L$ such that $x\backslash y\leq z$ iff $x\leq y\vee z$.[^Dist]

[^Dist]: Existence of $\backslash$ amounts to an [[adjunction]] \_$\backslash y\dashv y\vee$\_ and the existence of a left adjoint implies that $y\vee$\_ preserves limits $\wedge$ hence the assumption of distributivity in the definition is redundant and has been put in for emphasis only.

A **bi-Heyting algebra** is a bounded distributive lattice $L$ that carries a Heyting algebra structure with implication $\Rightarrow$ and a co-Heyting algebra structure with subtraction $\backslash$.

## Remarks

* Co-Heyting algebras were initially called _Brouwerian algebras_ . Bi-Heyting algebras were introduced and studied in a series of papers by Cecylia Rauszer in the 1970s who called them _semi-Boolean algebras_ which suggests to view them as a generalization of [[Boolean algebra|Boolean algebras]].

* A [[topos]] $\mathcal{E}$ such that the lattice $sub(X)$ of subobjects is a bi-Heyting algebra for every object $X\in\mathcal{E}$ is called a [[bi-Heyting topos]].

## Examples

* The lattice of closed subsets of a topological space is a co-Heyting algebra with $X\backslash Y=\overline{X\cap Y^c}\quad$.

* A Boolean algebra provides a (degenerate) example of a bi-Heyting algebra by setting $x\Rightarrow y:=\neg x\vee y$ and $x\backslash y:=x\wedge\neg y$.

* An [[irreflexive]] [[comparison]], such as an [[apartness relation]] or a [[linear order]], is an [[enriched poset]] on the co-Heyting algebra $\Omega^\op$, where $\Omega$ is the Heyting algebra of [[truth values]].

##Properties

* $a\backslash b=0$ iff $a\backslash b\leq 0$ iff $a\leq b\vee 0$ iff $a\leq b$. In particular, $a\backslash a=0$.

* As \_$\backslash x$ has a right adjoint it preserves colimits hence: $(a\vee b)\backslash x =(a\backslash x)\vee (b\backslash x)\quad$.

* $a\backslash 0\leq a\backslash 0$ iff $a\leq 0\vee (a\backslash 0)$ iff $a\leq a\backslash 0\quad$. On the other hand, $a\leq 0\vee a$ and the adjunction yield $a\backslash 0\leq a\quad$, hence $a\backslash 0 = a\quad$.

* Suppose $a\leq b\vee x $ then $a\backslash b\leq x$. As from $a\backslash b\leq a\backslash b$ follows $a\leq b\vee (a\backslash b)$, hence $a\backslash b =\Wedge\{x|a\leq b\vee x\}\quad$.

* The subtraction operation permits to define the [[co-Heyting negation]] $\sim: L\to L$, called _non a_ in Lawvere (1991), by setting $\sim a:=1\backslash a$.

* $\sim$ in turn can then be used to define the [[co-Heyting boundary|co-Heyting boundary operator]] $\partial :L\to L$ by $\partial a:=a\wedge\sim a$. That $\partial a$ is not necessary trivial is dual to the non-validity of the _tertium non datur_ for general Heyting algebras and points to the utility of co-Heyting algebras for [[paraconsistent logic]].

* In [[bi-Heyting toposes]] like e.g. [[essential subtoposes]] of presheaf toposes ([Lawvere](#Law91a), [Reyes](#Reyes)), the co-Heyting algebra operations are generally not preserved by [[inverse image functor|inverse image functors]], so that the co-Heyting logical operators are subject to _[[de re and de dicto]]_ effects. The parallel between this and the [[commutator]] in quantum mechanics has been suggested by Lawvere thereby somewhat anticipating the view of Döring (2013).

##Related entries

* [[co-Heyting negation]]
* [[co-Heyting boundary]]
* [[Heyting algebra]]
* [[Heyting category]]
* [[distributive category]]
* [[mereology]]
* [[bi-Heyting topos]]
* [[bitopological space]]
* [[coframe]]
* [[cotopos]]

##References

* G. Bellin, _Categorical Proof Theory of Co-Intuitionistic Linear Logic_ , arXiv:1407.341 (2014). ([pdf](http://arxiv.org/pdf/1407.3416v1))

* G. Bezhanishvili, N. Beshanishvili, D. Gelaia, A. Kurz, _Bitopological duality for distributive lattices and Heyting algebras_ , Math. Struc. Comp. Sci. **20** no.3 (2010) pp.359-393. ([preprint](http://www.cs.le.ac.uk/people/nb118/Publications/PairwiseStone.pdf))

* [[Andreas Döring|A. Döring]], _Topos-based Logic for Quantum Systems and Bi-Heyting Algebras_ , arXiv:1202.2750 (2013). ([pdf](http://arxiv.org/pdf/1202.2750.pdf))

* {#RRZ94} M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _The non-Boolean logic of natural language negation_ , Phil. Math. **2** no.1 (1994) pp.45-68.

* M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Models for non-Boolean negation in natural languages based on aspect analysis_ , pp.241-260 in Gabbay, Wansing (eds.), _What is Negation?_, Kluwer Dordrecht 1999.

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* {#Law86} [[William Lawvere|F. W. Lawvere]], _Introduction_ , pp.1-16 in Lawvere, Schanuel (eds.), _Categories in Continuum Physics_ , Springer LNM **1174** 1986.

* {#Law91a} [[William Lawvere|F. W. Lawvere]], _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in A. Carboni, M. Pedicchio, G. Rosolini (eds.) ,  _[[Como|Category Theory - Proceedings of the International Conference held in Como 1990]]_ , LNM **1488** Springer Heidelberg 1991.

* T. Mormann, _Heyting Mereology as a Framework for Spatial Reasoning_ , Axiomathes **23** no.1 (2013) pp.237-264. ([draft](http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf))

* {#Reyes}[[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari,  _Bi-Heyting Algebras, Toposes and Modalities_ , J. Phi. Logic **25** (1996) pp.25-43.

* C. Rauszer, _Semi-Boolean algebras and their applications to intuitionistic logic with dual operations_ , Fund. Math. **83** no.3 (1974) pp.219-249. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83120.pdf))

* J.G. Stell, M.F. Worboys, _The algebraic structure of sets of regions_ , pp.163-174 in Hirtle, Frank (eds.), _Spatial Information Theory_, Springer LNCS **1329** (1997).

* [[Marshall Stone|M. H. Stone]], _Topological representation of distributive lattices and Brouwerian logics_ , Cas. Mat. Fys. **67** (1937) pp.1-25. ([pdf](http://dml.cz/bitstream/handle/10338.dmlcz/124080/CasPestMatFys_067-1938-1_1.pdf))

* F. Wolter, _On logics with coimplication_ , J. Phil. Logic **27** (1998) pp.353-387. ([draft](http://lips.informatik.uni-leipzig.de/files/1998-24.pdf))

[[!redirects coHeyting algebra]]
[[!redirects Co-Heyting algebra]]
[[!redirects CoHeyting algebra]]
[[!redirects bi-Heyting algebra]]
[[!redirects biHeyting agebra]]
[[!redirects Brouwerian algebra]]


co-Heyting boundary

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

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

[[!redirects coHeyting boundary]]
[[!redirects Co-Heyting boundary]]
[[!redirects boundary operator]]

## Idea

In a [[co-Heyting algebra]] it is possible to define an equivalent to the [[boundary]] operation in [[topology]]. In applications this affords an intrinsic view of [[mereology]] without consideration of the embedding of bodies into an ambient space.

## Definition

Let $a$ be an element of a [[co-Heyting algebra]] $L$ with subtraction $\backslash$ and [[co-Heyting negation]] $\sim$. The **(co-Heyting) boundary** of $a$ is defined as $\partial a :=a\wedge\sim a$.

## Properties

>

* $\partial (a\wedge b) = (\partial a \wedge b)\vee (a\wedge\partial b)\quad$. (**Leibniz rule**)

* $\partial (a\vee b)\vee\partial (a\wedge b) =\partial a\vee\partial b\quad$.

* The boundaries $x=\partial a$ can be characterized as those $x$ with $\partial x = x$ (in particular, $\partial^2=\partial$), or, alternatively, those $x$ with $\sim x=1$, showing that boundary parts are precisely the (intuitively) **thin parts**.

* Every part is the sum of its _regular core_ and its boundary: $a=\sim\sim a\vee\partial a$. This suggests to view $\partial a$ as the _irregular_ part of $a$.

## Remark

As the lattice of [[subtopos|subtoposes]] of a given [[topos]] $\mathcal{E}$ comes naturally with a co-Heyting structure it becomes (in principle) possible to define the boundary $\partial\mathcal{A}$ of a subtopos $\mathcal{A}$ in this lattice and then in turn the boundary $\partial T'$ of the [[geometric theory]] $T'$ that $\mathcal{A}$ [[classifying topos|classifies]] which is an extension of the theory $T$ classified by $\mathcal{E}$ ([Lawvere 1991](#Law91a), [Caramello 2009](#Cara09)).[^Law]

[^Law]: The boundary operation is one in Lawvere's arsenal of (mereological) tools for the study of logical theories in the context of a topos and its subtoposes (besides e.g. the [[Aufhebung|Aufhebungs relation]]). For a more complete picture of the toolbox see [Lawvere (2002)](#Law02).

When a subtopos $\mathcal{A}$ is complemented in the lattice of subtoposes, as occurs e.g. for [[open subtopos|open]], [[closed subtopos|closed]] or [[locally closed subtopos|locally closed]] subtoposes, its boundary vanishes: $\partial\mathcal{A}=0$ since in a co-Heyting algebra the complement $\neg a$ necessarily coincides with $\sim a$ (cf. at [[co-Heyting negation]]).

## Related entries

* [[boundary]]
* [[co-Heyting negation]]
* [[co-Heyting algebra]]
* [[Heyting algebra]]
* [[Heyting category]]
* [[mereology]]
* [[subtopos]]
* [[Aufhebung]]

## References

Closed boundaries of subtoposes are defined in an exercise of [SGA4](#SGA4) (cf. at [[open subtopos]] for some of the details). The concept for co-Heyting algebras seems to stem from Lawvere ([1976](#Law76),[1986](#Law86), [1991](#Law91a)) although the 1927 [article of M. Zarycki](#Zar27) already studies properties and axiomatic potential of the boundary operator in topology. [La Palme Reyes, Reyes&Zolfaghari (2004)](#RRZ04) has an introductory exposition in the context of bi-Heyting algebras. For boundaries of geometric theories [Caramello (2009)](#Cara09) is essential reading although they don't appear there explicitly. For mereological applications of the concept see [Lawvere (1986)](#Law86), [Stell&Worboys (1997)](#SW97), [Pagliani (2009)](#Pagl09) and [Mormann (2013)](#Mor13).

* {#SGA4} [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462) {#SGA4}

* {#Cara09} [[Olivia Caramello|O. Caramello]], _Lattices of theories_ , arXiv:0905.0299v1 (2009). ([pdf](http://arxiv.org/pdf/0905.0299v1))

* {#KL89}[[G. M. Kelly]], F. W. Lawvere, _On the Complete Lattice of Essential Localizations_ , Bull.Soc.Math. de Belgique **XLI** (1989) 261-299 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* {#KRRZ10} C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, _Levels in the toposes of simplicial sets and cubical sets_ , JPAA **215** no.5 (2011) pp.949-961. ([preprint](http://arxiv.org/abs/1003.5944))

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

* {#Law76} [[William Lawvere]], _Variable Quantities and Variable Structures in Topoi_ , pp.101-131 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg_ , Academic Press New York 1976.

* {#Law86} [[William Lawvere|F. W. Lawvere]], _Introduction_ , pp.1-16 in _Categories in Continuum Physics_ , Springer LNM **1174** 1986.

* {#Law91a} F. W. Lawvere, _Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes_ , pp.279-281 in Springer LNM **1488** (1991).

* {#Law94b} F. W. Lawvere, _Tools for the Advancement of Objective Logic: Closed Categories and Toposes_, pp.43-56 in: J. Macnamara, G. E. Reyes (eds.), _The Logical Foundations of Cognition_ , Oxford UP 1994.

* {#Law02} F. W. Lawvere, _Linearization of graphic toposes via Coxeter groups_ , JPAA **168** (2002) pp.425-436.

* [[Matías Menni|M. Menni]], C. Smith, _Modes of Adjointness_ , J. Philos. Logic **43** no.3-4 (2014) pp.365-391.

* {#Mor13} T. Mormann, _Heyting Mereology as a Framework for Spatial Reasoning_ , Axiomathes **23** no.1 (2013) pp.237-264. ([draft](http://philpapers.org/go.pl?id=MORCMA-3&proxyId=&u=http%3A%2F%2Fphilpapers.org%2Farchive%2FMORCMA-3.pdf))

* {#Pagl09} P. Pagliani, _Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis_ , pp.123-130 in
Polkowski, Skowron (eds.), _RSCTC 1998_ , Springer LNCS **1424** (2009).

* M. van Lambalgen, _Logical constructions suggested by vision_ , Proceedings of ITALLC98, CLSI 2000. ([draft](http://staff.science.uva.nl/~michiell/docs/ITALLCCSLI.pdf))

* C. Rauszer, _Semi-Boolean algebras and their applications to intuitionistic logic with dual operations_, Fund. Math. **83** no.3 (1974) pp.219-249. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm83/fm83120.pdf))

* [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari,  _Bi-Heyting Algebras, Toposes and Modalities_ , J. Phi. Logic **25** (1996) pp.25-43.

* {#SW97} J.G. Stell, M.F. Worboys, _The algebraic structure of sets of regions_ , pp.163-174 in Hirtle, Frank (eds.), _Spatial Information Theory_, Springer LNCS **1329** (1997).

* {#Zar27} M. Zarycki, _Quelque notions fondamentales de l'Analysis Situs au point de vue de l'Algèbre de la Logique_ , Fund. Math. **IX** (1927) pp.3-15. ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm9/fm912.pdf))

* H. Zolfaghari, _Topos et Modalités_ , Thèse de doctorat Université de Montréal 1991.