Chu construction

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# The Chu construction
* table of contents
{: toc}

## Idea

In [[category theory]], the _Chu construction_ [[Chu 1979](#Chu79); [Barr 1991](#Barr91), [1996](#Barr96)] is a method for constructing a [[star-autonomous category]] from a [[closed monoidal category|closed symmetric monoidal category]].

Early discussion of application of the Chu construction as [[categorical semantics]] for [[linear logic]] is due to [Barr 1991](#Barr91) and, motivated [[Pratt 1992](quantum+logic#Pratt92)] by [[categorical semantics]] of [[quantum logic]] in [[star-autonomous categories]], due to [Pratt 1993](Pratt93), [1999](#Pratt99).

In outline, given a [[closed symmetric monoidal category]] $C$ with [[pullbacks]] and an [[object]] $d$ of $C$, there is a [[star-autonomous category]] $Chu(C, d)$ and a [[strong monoidal functor|strong]] [[symmetric monoidal functor]]

$$
  i
  \;\colon\;
  C
    \longrightarrow
  Chu(C, d)
$$

which realizes $C$ as a [[coreflective subcategory]] of $Chu(C, d)$. Being star-autonomous, $Chu(C, d)$ is self-dual, in that its [[opposite category]] $C^{op}$ also embeds as a full subcategory of $Chu(C, d)$, this time [[reflective subcategory|reflectively]].

Many [[concrete dualities]] in mathematics can be seen as embedded in a larger ambient self-duality on a Chu construction. This applies in particular to the category of _Chu spaces_, $Chu(Set, 2)$ (see below).

Th Chu construction may be [[categorified]] to what might be called a **2-Chu construction**, producing for example $Chu(Cat, Set)$ (see [Shulman17](#Shulman17)).  It also generalizes to a [[double Chu construction]] and to operations on [[multicategories]] and [[polycategories]].

## Definition ##

### As a category with self-duality

Let $C$ be a [[closed symmetric monoidal category]] and $d\in C$ an [[object]].  The objects of $Chu(C, d)$ are [[triples]] $(a, b; r \,\colon\, a \otimes b \to d)$ (called _$d$-valued pairings_ between $a$ and $b$), where $a$ and $b$ are objects of $C$ and $r$ is a [[morphism]] of $C$. The special triple $(d, I; \rho: d \otimes I \cong d)$, where $\rho$ is an instance of the canonical isomorphism (the right [[unitor]]) for the [[monoidal unit]] $I$, will play the role of [[dualizing object]] in $Chu(C, d)$.

The morphisms of $Chu(C, d)$,

$$
  (a, b; r \,\colon\, a \otimes b \to d)
   \longrightarrow
  (x, y; s: x \otimes y \to d)
  \,,
$$

are pairs of morphisms $f \colon a \to x$, $g \colon y \to b$ which are adjoints with respect to the pairings, that is, making the following [[commuting diagram|diagram commute]]:

$$\array{
  & a \otimes y &
  \overset{1_a \otimes g}{\longrightarrow}
  & a \otimes b
  \\
  &
    {{}^{\mathllap{f \otimes 1_y}}}
    \Big\downarrow
      & &
    \Big\downarrow{\mathrlap{{}^r}}
  \\
  &
    x \otimes y & \underset{\;\; s \;\;}{\longrightarrow}
  &
  d
  \mathrlap{\,,}
}
$$

There is an evident self-duality [[functor]]

$$Chu(C, d)^{op} \to Chu(C, d)$$

which takes an object $(a, b; r \,\colon\, a \otimes b \longrightarrow d)$ to

$$
  \big(
    b, a; r^\dagger = [b \otimes a \overset{\sigma}{\cong} a \otimes b \overset{r}{\to} d]
  \big)
  \,,
$$

where $\sigma$ is an instance of the symmetry isomorphism, so that $r^\dagger$ is the evident transpose. On morphisms, it takes a pair $(f, g)$ to $(g, f)$; note well that the directions of the arrows make the functor contravariant on $Chu(C, d)$.

### As a $\ast$-autonomous category

Armed with just this much knowledge, and knowledge of how star-autonomous categories behave (as categorified versions of [[linear logic]]), the star-autonomous structure on $Chu(C, d)$ can pretty much be deduced (or strongly guessed) by the diligent reader, and this is actually a very good exercise. One could sketch this as follows:

* The monoidal unit of $Chu(C, d)$ should be the dual of the dualizer, and so is $(I, d; \lambda: I \otimes d \cong d)$ where $\lambda$, the transpose of $\rho$, is a canonical isomorphism for the unit $I$.

* The internal hom $A \multimap X$ in $Chu(C, d)$ should internalize the external hom, i.e., the set of maps from the monoidal unit to $A \multimap X$ should be in natural bijection with the set of maps $A \to X$ in $Chu(C, d)$.

This suggests the first component $(A \multimap X)_0$ of the triple

$$A \multimap X = ((A \multimap X)_0, (A \multimap X)_1; pairing)$$

should be the object of adjoint pairs of maps: given

$$A = (a, b; r: a \otimes b \to d), \qquad X = (x, y; s: x \otimes y \to d)$$

define the first component as [[pullback]]:

$$
  \array{
    (A \multimap X)_0
      & \longrightarrow &
    x^a
    \\
    \Big\downarrow
      & &
    \Big\downarrow\mathrlap{{}^{\tilde{s}}}
    \\
    b^y
      & \underset{\tilde{r}}{\longrightarrow} &
   d^{a \otimes y}
   \mathrlap{\,,}
}
$$

where exponentials are used to denote internal homs in $C$, $\tilde{r}$ is the result of [[currying]] $r$ to $b \to d^a$ and exponentiating, and similarly for $\tilde{s}$.

The pullback is paired with $a \otimes y$, i.e., there is a map

$$(A \multimap X)_0 \otimes (a \otimes y) \to d$$

obtained by decurrying either leg of the pullback, so one defines (with fingers crossed) $(A \multimap X)_1$ to be $a \otimes y$, and the pairing to be this map into $d$.

* To form the tensor product $A \otimes X$, use the formula $ A \otimes X \cong (A \multimap X^*)^* $ which holds true in any star-autonomous category (as a categorified analogue of the Boolean algebra equation $a \wedge x = \neg (a \Rightarrow \neg x)$).

With notation as above, this works out to

$$(a \otimes x, y^a \times_{d^{a \otimes x}} b^x)$$

where the second component is a pullback, and the pairing omitted but obvious. The main thing to check is the presence of a canonical isomorphism

$$Chu(C, d)(A \otimes X, Y) \cong Chu(C, d)(X, A \multimap Y)$$

but this is left as an exercise.

Also one should check that if $D$ is the dualizing object, that $ A^* \cong A \multimap D $, but this is straightforward.

* There is a strong symmetric monoidal functor
$$i: C \to Chu(C, d)$$
taking $c$ to $(c, d^c; eval: c \otimes d^c \to d)$. (This does **not** take $d$ to the dualizing object in $Chu(C, d)$, unless of course the canonical map $I \to d^d$ is an isomorphism.) This embedding admits a right adjoint
$$p: Chu(C, d) \to C$$
given by the obvious projection, that is also strong symmetric monoidal. The unit of the adjunction is an isomorphism, hence $C$ is a coreflective (full) subcategory of $Chu(C, d)$.

* If $C$ is complete and cocomplete, then so is $Chu(C, d)$. The formula for colimits is the obvious expected one:
$$colim_i (a_i, b_i; r_i: a_i \otimes b_i \to d) = (colim_i a_i, lim_i b_i; r)$$
where $r$ is the decurrying of
$$lim_i (b_i \to d^{a_i}) \cong lim_i b_i \to d^{colim_i a_i}$$
and the formula for limits is obtained by dualizing the formula for colimits in $Chu(C, d)$.

### As a polycategory

We can also deduce the $\ast$-autonomous structure of $Chu(C,d)$ by constructing it directly as a [[star-polycategory]] and then observing that it is representable.  This has the additional benefit of giving a universal property to the tensor product, as well as applying to more general inputs $C$ (and giving a result that may not be representable), and giving a convenient way to phrase the universal property of $Chu(C,d)$ itself (see below).

Let $C$ be a (symmetric) [[polycategory]] in which every morphism has codomain arity 0 or 1 (a *co-subunary polycategory*).  For instance, if $C$ is any [[symmetric multicategory]] and $d\in C$ an object, we can obtain a co-subunary polycategory by defining $C(x_1,\dots,x_n; ) = C(x_1,\dots,x_n; d)$.  (In fact the construction can be generalized even further; see [Shulman 18](#Shulman18).)

The objects of $Chu(C)$ are now pairings $(a, b; r : (a,b) \to ())$ in the appropriate polycategorical sense.  A morphism $(a,b,r) \to (x,y,s)$ is a pair of morphisms $f:a\to x$ and $g:y\to b$ that are "adjoint" with respect to the pairings, in the sense that the following square commutes:
\begin{tikzcd}
  (a,y) \ar[r,"{(f,1)}"] \ar[d,"{(1,g)}"'] & (x,y) \ar[d,"s"] \\ (a,b) \ar[r,"r"'] & ().
\end{tikzcd}
More generally, we can directly make $Chu(C)$ into a polycategory by taking the morphisms to be a suitable kind of [[multivariable adjunctions]].  For instance, a morphism $((a,b,r),(x,y,s)) \to (u,v,t)$ consists of three morphisms
$$ f : (a,x) \to u \qquad g : (a,v) \to y \qquad h : (x,v) \to b $$
such that the following three composites are equal modulo symmetries:
$$ (a,x,v) \xrightarrow{(f,1)} (u,v) \xrightarrow{t} () $$
$$ (x,a,v) \xrightarrow{(1,g)} (x,y) \xrightarrow{s} () $$
$$ (a,x,v) \xrightarrow{(1,h)} (a,b) \xrightarrow{r} (). $$
Similarly, a morphism $(a,b,r) \to ((x,y,s),(u,v,t))$ consists of three morphisms
$$ f : (a,y) \to u \qquad g : (x,b) \to u \qquad h : (y,v) \to b $$
such that three composite morphisms $(a,y,v) \to ()$ are equal.  With a suitable extension to $m$-$n$-ary morphisms we obtain a polycategory, and indeed a $\ast$-polycategory: the dual of $(a,b,r)$ is $(b,a,r^\dagger)$.

It is then a straightforward exercise to check that if $C$ is closed, representable, has pullbacks, and a "counit" in the sense that $C(\Gamma;) \cong C(\Gamma;d)$, then $Chu(C)$ is representable on both sides and hence a $\ast$-autonomous category, where all the structure coincides with the previous definition.

## Chu spaces (General case)##

While the Chu construction is worthy of exploration for many types of symmetric monoidal categories $C$, a great deal of attention has been focused just on the particular case $Chu(Set, 2)$ (or $Chu(Set,TV)$, where $TV$ is the set of [[truth value]]s, to be [[constructive mathematics|constructive]]), called the category of **Chu spaces**, and on relatives like $Chu(E, \Omega)$ where $E$ is a [[topos]] and $\Omega$ its [[subobject classifier]]. The reason is that a great many concrete categories of interest are fully embedded in Chu spaces. Moreover, the 2-element set $TV$ carries a panoply of [[ambimorphic object|ambimorphic]] (formerly, schizophrenic) object structures which induce concrete dualities between these categories, and all of these dualities are embedded in (i.e., are restrictions of) the one overarching duality that obtains on Chu spaces.

The way this works is invariably the same: if $U: C \to Set$ is a [[concrete category]] and $\mathbf{2}$ is an object of $C$ over the 2-element set $TV$, then there is a functor

$$i: C \to Chu(Set, 2): c \mapsto (U c, hom(c, \mathbf{2}); U c \times hom(c, \mathbf{2}) \to U\mathbf{2} = 2)$$

which is faithful by the notion of concrete category.

What is striking is that this functor $i$ is also *full* in many cases of interest. This is because the adjointness condition for a pair $(f, g): i c \to i d$ to be a Chu space morphism, together with faithfulness of $U$, forces

$$g: hom(d, \mathbf{2}) \overset{monic}{\to} hom(U d, 2) \overset{f^*}{\to} hom(U c, 2)$$

to be a restriction of the [[preimage]] function $f^*$ -- and then the mere additional fact that $f^*(\phi) = \phi \circ f \in hom(c, \mathbf{2})$ whenever $\phi \in hom(d, \mathbf{2})$ is often enough to force $f$ to be (the underlying function of) a morphism of $C$. All that is required is that there be sufficiently many morphisms $d \to \mathbf{2}$ to detect $C$-structure on $d$. Some examples follow:

* As explained above, $Set$ fully embeds in $Chu(Set, 2)$ by $X \mapsto (X, 2^X; eval: X \times 2^X \to 2)$.

* For $C = Top$, taking $\mathbf{2}$ to be [[Sierpinski space]], we have for each [[topological space]] $c$ an identification $Open(c) = hom(c, \mathbf{2})$. Then the adjointness condition on a morphism $(f, g): i c \to i d$ between the corresponding Chu spaces expresses precisely the continuity condition that the [[preimage]] $f^*$ takes opens of $d$ to opens of $c$. Hence the functor $Top \to Chu(Set, 2)$ is full.

* For $C = Pos$, the category of [[poset]]s, take $\mathbf{2}$ to be the partially ordered set of truth values. Here we have that for a partial order $c$, $hom(c, \mathbf{2})$ is the set of [[upper set]]s (upward-closed subsets) of $c$. Given a function $f: U c \to U d$ between posets, the condition that the preimage $f^*(v)$ of an upper set of $d$ is an upper set of $c$ is enough to force $f$ to be a poset map (consider $v = \{q \in d: f p \leq q\}$). It follows that the functor $Pos \to Chu(Set, 2)$ is full.

* For $C = Sup$, the category of [[sup-lattice]]s (whose morphisms are those functors between the underlying posets that are left adjoints), take $\mathbf{2}$ to be the partially ordered set of truth values, but this time as the _opposite_ of the poset $TV$. For a sup-lattice $c$, $hom(c, \mathbf{2})$ may be identified with the set of representable functors $c(-, x): c^{op} \to TV$. The Chu condition then is that $d(-, y) \circ f = d(f-, y)$ is representable for every representable $d(-, y)$. But this condition is equivalent $f$'s being a left adjoint. Therefore the functor $Sup \to Chu(Set, 2)$ is full.

For other examples of concrete categories, the presence of enough elements in $\hom_C(c, \mathbf{2})$ to detect the $C$-structure of $c$ often requires some form of choice principle, such as the [[axiom of choice]] or [[ultrafilter theorem]]:

* For $C = Vect_{\mathbb{F}_2}$, the category of vector spaces over the 2-element field, any object $d$ has a [[basis theorem|basis]], let us say $S$. For each $s \in S$, the characteristic or indicator $\chi_{s}: S \to \mathbf{2}$ extends uniquely to a linear function $\phi_s: d \to \mathbf{2}$. If $f: U c \to U d$ is a function such that $\phi_s \circ f: c \to \mathbf{2}$ is linear for every basis element $s \in S$, then
$$f(a x + b y) = \sum_{s \in S} \phi_s(f(a x + b y))s = \sum_s (a\phi_s(f(x)) + b\phi_s(f(y)))s = a f(x) + b f(y)$$
It follows that the functor $Vect_{\mathbb{F}_2} \to Chu(Set, 2)$ is full.

Similar considerations apply to Boolean algebras, Stone Boolean algebras, algebraic lattices, and so on.

In all of these cases, the fullness of these embeddings entitles one to identify a topological space, a Boolean algebra, a vector space over $\mathbb{F}_2$, etc. with its corresponding Chu space, and the same consideration applies to the duals (opposites) of these categories.

Now many of these formal categorical duals are themselves concrete categories, as in the famous example of classical [[Stone duality]] between Boolean algebras and Stone spaces, i.e., [[compact space|compact]] [[Hausdorff space|Hausdorff]] [[connected space|totally disconnected]] [[topological space]]s. In many such Stone duality situations, and certainly wherever Stone duality applies to the categories listed above, a contravariant equivalence between a concrete category $C$ and an algebraic category $D$ (i.e., where $U: D \to C$ is monadic),

$$\array{
 & C^{op} & \overset{\sim}{\to} & D & \\
hom(-, \mathbf{2}) & \searrow & & \swarrow & U\\
 & & Set, & &
}$$

is effected by lifting the object $\mathbf{2}$ of $C$ to a $D$-algebra structure in $C$ (making $\mathbf{2}$ an [[ambimorphic object]], carrying $C$- and $D$-structures compatible with one another); equivalently, seeing $hom(-, \mathbf{2}): C^{op} \to Set$ as an algebra over the monad $U F$ for which $U: D \to Set$ is monadic. For example, classical Stone duality is the case where $C$ is the category of Stone spaces, $D$ is the category of Boolean algebras, and the Boolean operations on $\mathbf{2}$ are continuous with respect to its Stone space structure, making $\mathbf{2}$ a Boolean algebra object in the category of Stone spaces. (For much more on this, see Johnstone's classic treatise _[[Stone Spaces]]_, especially the chapter on general concrete dualities.)

The point is that in each of these situations, a Stone duality is a restriction of the more global duality on Chu spaces, in that the diagram

$$
  \array{
    C^{op}
      & \overset{hom(-, \mathbf{2})}{\longrightarrow} &
    D
    \\
    \mathllap{{}^{i^{op}}}\Big\downarrow
      & &
    \Big\downarrow\mathrlap{{}^{i}}
    \\
    Chu(Set, 2)^{op}
      & \underset{(-)^*}{\longrightarrow} &
    Chu(Set, 2)
}$$

(where the vertical arrows are full embeddings as described above) commutes up to canonical isomorphism.

The same principle extends to other situations. For example, [[Pontryagin duality]] is fully embedded in the larger duality which obtains on $Chu(TopAb, S^1)$, where $TopAb$ is the category of abelian groups internal to a [[nice category of spaces]].  ([Barr](#BarrTobAb) has also shown that Pontryagin duality is embedded in a sub-$\ast$-autonomous category of $Chu(Ab,S^1)$, and one might conceivably hope to find it in $Chu(Top,S^1)$ as well by analogy to the above appearance of algebras inside $Chu(Set,2)$.)

Similarly, the 2-Chu construction, $Chu(Cat, Set)$, exhibits dualities, such as [[Gabriel-Ulmer duality]].

## Chu spaces (Simple examples)##

One of the simplest occurrences of Chu space constructions, and the one explored in [Pratt 1999](#Pratt99), leads to examples that although extremely simple have a well developed theory with connections to areas of logic and to [[formal concept analysis]]. This will be explored in a separate entry, [[Chu spaces, simple examples]].

## Properties ##

* Chu spaces give fairly easy to construct examples of closed monoidal categories in which [[coproduct]] injections are not necessarily monic; see [this MO answer](http://mathoverflow.net/a/208643/49).

### Universal property

Recall from the above that $Chu(C)$ is a $\ast$-polycategory whenever $C$ is a co-subunary polycategory.  This construction is functorial, and we claim that it defines a right adjoint to the forgetful functor $U$ from $\ast$-polycategories to co-subunary polycategories.

To see this, consider first the $\ast$-polycategory $F 1$ freely generated by one object.  It has two objects $x$ and $x^*$, and in addition to identity morphisms it has a [[dual pair]] $(x^*,x)\to ()$ and  $() \to (x,x^*)$.  Of these, all but the last are co-subunary.  Thus, a morphism $U F 1 \to C$ consists of two objects $a$ and $b$ (plus their identity morphisms) and a morphism $(a,b)\to ()$ --- in other words, an object of $Chu(C)$.

Now consider the $\ast$-polycategory $F (1,1)$ freely generated by two objects $x$ and $y$ and a morphism $f:x\to y$.  It contains two copies of $F 1$ generated by $x$ and $y$, together with $f:x\to y$, its dual $f^*: y^* \to x^*$, and two other copies of it $f^\dagger : (x,y^*) \to ()$ and ${}^\dagger f : () \to (x^*,y)$.  Of these all but the last are co-subunary, so a morphism $U F 1 \to C$ is determined by two objects of $Chu(C)$ together with three more morphisms (and some composition laws relating them to the duality morphisms).  This yields the two morphisms $a\to x$ and $y\to b$ of a morphism in $Chu(C)$, along with their common composite $(a,y) \to ()$.

Similarly, one can characterize all the morphisms in $Chu(C)$ by mapping out of $U F (m,n)$, where $F(m,n)$ is the $\ast$-polycategory freely generated by a $m$-ary co-$n$-ary morphism.

As a special case of this universal property, if all the $\ast$-polycategories are representable and the subunary polycategories are representable, closed, with a counit, and have pullbacks, we obtain an adjunction between a certain category of $\ast$-autonomous categories and a certain category of closed symmetric monoidal categories with chosen objects.  This is the way it was phrased in [Pavlovic 97](#Pavlovic97).

## Related pages

* [[double Chu construction]]
* [[2-Chu construction]]
* [[Dialectica construction]]
* [[double gluing]]
* [[poset-valued set]]
* [[ternary frame]]
* [[multivariable adjunction]]

## References ##

The construction originates with

* {#Chu79} [[Po-Hsiang Chu]], *Constructing $\ast$-autonomous categories*, appendix to: [[Michael Barr]], _$\ast$-Autonomous Categories_, Lecture Notes in Mathematics **752**, Springer (1979) 103-138 [[doi:10.1007/BFb0064579](https://doi.org/10.1007/BFb0064579)]

* {#Barr91} [[Michael Barr]], *$\ast$-Autonomous categories and linear logic*, Math. Structures Comp. Sci. **1** 2 (1991) 159–178 [[doi:10.1017/S0960129500001274](https://doi.org/10.1017/S0960129500001274), [pdf](https://www.math.mcgill.ca/barr/papers/scatll.pdf), [[Barr-AutomomousCatAndLinLogic.pdf:file]]]

* {#Barr96} [[Michael Barr]], *The Chu construction*, Theory Appl. Categories **2** 2 (1996) 17–35 [[tac:2-02](http://www.tac.mta.ca/tac/volumes/1996/n2/2-02abs.html)]

Further early discussion:

* [[Michael Barr]], *Separability of tensor in Chu categories of vector spaces*, Math. Structures Comp. Sci. **6** (1996) 213–217 [[doi:10.1017/S0960129500000955](https://doi.org/10.1017/S0960129500000955), [pdf](https://www.math.mcgill.ca/barr/papers/sepvec.pdf), [[Barr-ChuOfVect.pdf:file]]]

* {#Pavlovic97} [[Duško Pavlović]], *Chu I: cofree equivalences, dualities and $\ast$-autonomous categories*, Mathematical Structures in Computer Science **7** 1 (1997) 49-73 [[doi:10.1017/S0960129596002046](https://doi.org/10.1017/S0960129596002046)]

* [[Michael Barr]], *The separated extensional Chu category* Theory Appl. Categories **4** 6 (1998) 127–137 [[tac:4-06](http://www.tac.mta.ca/tac/volumes/1998/n6/4-06abs.html)]

* {#Pratt93} [[Vaughan Pratt]], *The second calculus of binary relations*, Mathematical Foundations of Computer Science 1993. MFCS 1993, Lecture Notes in Computer Science **711**, Springer (1993) [[doi:10.1007/3-540-57182-5_9](https://doi.org/10.1007/3-540-57182-5_9)]

* {#Pratt99} [[Vaughan Pratt]], *Chu Spaces* (1999) [[pdf](http://boole.stanford.edu/pub/coimbra.pdf), [[Pratt-ChuSpaces.pdf:file]], [webpage](http://chu.stanford.edu/)]

* {#BarrTobAb} [[Michael Barr]], *On duality of topological abelian groups*, [[pdf](http://www.math.mcgill.ca/barr/ftp/pdffiles/abgp.pdf)]

* [[Vaughan Pratt]], *Linear process algebra*, in: *Distributed Computing and Internet Technology ICDCIT 2011*,  Lecture Notes in Computer Science **6536**, Springer (2011) [[doi:10.1007/978-3-642-19056-8_6](https://doi.org/10.1007/978-3-642-19056-8_6), [pdf](http://boole.stanford.edu/pub/bhub.pdf)]

  > (uses $Chu(Set,K)$ where $K$ is a 4-element set to model concurrency)

On the history of the notion:

* [[Michael Barr]], *The Chu construction: history of an idea*, TAC **17** 1 (2006) 10-16 [[tac:17-01](http://www.tac.mta.ca/tac/volumes/17/1/17-01abs.html), [pdf](http://www.math.mcgill.ca/barr/papers/chu-hist.pdf)]

Further resources:

* [[Vaughan Pratt]], *Chu Spaces* [[webpage](http://chu.stanford.edu/)]

  * [Guide to Papers on Chu spaces](http://chu.stanford.edu/guide.html)

  * *Algebras and Bialgebras via categories with distinguished objects* [[pdf](http://boole.stanford.edu/pub/PrattCDOOct2016.pdf)]

See also:

* Wikipedia, *[Chu space](https://en.wikipedia.org/wiki/Chu_space)*

* [[Todd Trimble]] on the <a href="http://golem.ph.utexas.edu/category/2007/09/category_theory_in_machine_lea.html#c012536">n-Cafe</a>.

On [[categorifications]] and generalizations:

* {#Shulman17} [[Mike Shulman]], _The 2-Chu Construction_, ([blog post](https://golem.ph.utexas.edu/category/2017/11/the_2chu_construction.html))

* {#Shulman18} [[Mike Shulman]], _The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions_, 2018 [arxiv](https://arxiv.org/abs/1806.06082)

category: category

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Chu spaces, simple examples

# Simple examples of Chu spaces
* table of contents
{: toc}

## Idea

The simplest cases of [[Chu spaces]] can be thought of simply  as matrices over a set $\Sigma$, that is, a rectangular array whose
entries are drawn from $\Sigma$. The case most usually considered is $\Sigma = \mathbf{2} \coloneqq \{0,1\}$, and special cases of this then correspond to many relational structures. In fact, such a 'dyadic' Chu space is just another way of representing a relation from the set of labels for the rows, to that of the labels of columns of the matrix. The role of $\mathbf{2}$ can be replaced by an arbitrary set with suitable modifications of the resulting theory.

## Definitions

The definition we will give here is just an ultra-special case of that given in [[Chu construction]].

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

A _(dyadic or two valued) Chu space,_ $\mathcal{P}$, is a triple, $(P_o, \models_P, P_a)$, where $P_o$ is a set of _objects_, and $P_a$ is a set of _attributes_.  The _satisfaction_ relation $\models_P$ is a subset of $P_o\times P_a$.
=--

The terminology used here is motivated by the link with [[formal concept analysis]].  Alternative terminologies include (from Pratt's [Coimbra notes](http://boole.stanford.edu/pub/coimbra.pdf)) $P_o$ is a set of  *points* constituting the *carrier*, whilst $P_a$ is the set of *states*, which  constitutes the *cocarrier* of the Chu space.

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

A _morphism_ or _Chu transform_ from a Chu space $(P_o, \models_P, P_a)$ to a Chu space $(Q_o, \models_Q, Q_a)$ is a pair of functions $(f_a,f_o)$ with $f_o : P_o\to Q_o$ and $f_a : Q_a \to P_a$ such that, for any $x\in P_o $ and $y \in Q_a$,

$$f_o(x)\models_Q y  \iff  x \models_P f_a(y).$$
=--

This looks very much like some form of adjointness condition, and in particular cases, of course, it is.

In the above, the Chu space was thought of as 'relating' $P_o$ to $P_a$, but, equally well, such a relation relates $P_a$ to $P_o$, i.e. given any dyadic Chu space, there is a dual one:

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

If $\mathcal{P} = (P_o, \models_P, P_a)$ is a dyadic Chu space, then $\mathcal{P}^\perp = (P_a, \models_P^{op}, P_o)$ is the _dual Chu space_ of $\mathcal{P}$. (It just reverses the roles of objects and attributes.)
=--

## Related concepts

* [[formal concept analysis]]

* [[topological system]] (as in the book of [[Steve Vickers]] 'Topology via Logic').

* [[Stone gamut]]

## References

* [[Vaughan Pratt]], [Chu Spaces](http://boole.stanford.edu/pub/coimbra.pdf)

The links with [[formal concept analysis]] are in:

* [[Guo-Qiang Zhang]],   _[Chu spaces, concept lattices, and domains](http://www.entcs.org/files/mfps19/83018.pdf)_ in Brookes, S., Panangaden, P., eds.: Electronic Notes in Theoretical Computer Science. Volume 83., (2004)

* [[P. Hitzler]], [[Guo-Qiang Zhang]], _A cartesian closed category of approximating concepts_
In: Proceedings of the 12th International Conference on Conceptual Structures,
ICCS 2004, Huntsville, Al, July 2004. Volume 3127 of Lecture Notes in Artificial
Intelligence., Springer-Verlag (2004) 170–185.

* Guo-Qiang Zhang, Shen, G.: [Approximable Concepts, Chu spaces, and information systems](http://www.tac.mta.ca/tac/volumes/17/5/17-05.pdf), Theory and Applications of Categories 17, 2006, no.7

General applications of Chu spaces are in:

* [Special TAC Volume on Chu spaces and Applications](http://www.tac.mta.ca/tac/volumes/17/toc17.pdf)

[[!redirects Chu spaces, simple examples]]
[[!redirects simple examples of Chu spaces]]
[[!redirects simple example of Chu spaces]]


Chuangjie Xu

[website](https://cj-xu.github.io)

## Selected writings
 {#SelectedWritings}

On [[ordinals]] in [[type theory]] and [[marked extensional well-founded orders]]:

* {#deJongEtAl23} [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], [[Chuangjie Xu]], *Set-Theoretic and Type-Theoretic Ordinals Coincide* [[arXiv:2301.10696](https://arxiv.org/abs/2301.10696)]

category: people


Chung Tao Yang

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

## Selected writings

Introducing the [[slice theorem]] for [[topological G-spaces]]:

* [[Deane Montgomery]], [[Chung Tao Yang]], _The existence of a slice_, Ann. of Math. 65 (1957), 108-116 ([jstor:1969667](https://www.jstor.org/stable/1969667), [doi:10.2307/1969667](https://doi.org/10.2307/1969667))

On [[Hopf fibrations]] and more general [[spherical fibrations]] over [[spheres]]:

* [[Herman Gluck]], [[Frank Warner]], [[Chung Tao Yang]], _Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics_, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 ([euclid:dmj/1077303489](https://projecteuclid.org/euclid.dmj/1077303489))

category: people


Chung-Kil Hur

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

## Selected writings

On [[higher-order abstract syntax]]:

* [[Marcelo Fiore]], [[Chung-Kil Hur]]. _Second-order equational logic_, International Workshop on Computer Science Logic. Springer, Lecture Notes in Computer Science **6247** (2010) [[doi:10.1007/978-3-642-15205-4_26](https://doi.org/10.1007/978-3-642-15205-4_26), [pdf](https://www.cl.cam.ac.uk/~mpf23/papers/Types/soeqlog.pdf)]

category: people


Church monoid

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

#Contents#
* table of contents
{:toc}

## Idea

A *Church monoid* is a particular type of [[algebra|algebraic]] [[mathematical structure]] that provides [[semantics]] for a flavour of [[relevance logic]], a weak form of [[substructural logic]].

## Definition

A Church monoid is a [[partially ordered set|partially ordered]] [[commutative monoid]] $(M,\circ, 1,\leq)$ that is has a binary operation, "implication", written $a\to b$ that as an operation $b\mapsto (a\to b)$ is [[right adjoint]] to the [[functor]] $-\circ a$. Here $M$ is considered as a [[thin category]] associated to the underlying [[poset]]. Additionally, for all $a\in M$, $a \leq a\circ a$, resulting in a [[monoidal category with diagonals]], where $\circ$ is the monoidal product.

The operation $\circ$ models [[intensional conjunction]], as $\to$ models implication, analogous to [[linear implication]] in [[linear logic]].

## Related concepts

* [[relevance monoidal category]]
* [[relevance logic]]
* [[Ackermann groupoid]] (where "groupoid" means [[magma]])
* [[Heyting algebra]]

## References

Church monoids were introduced in

* Robert K. Meyer, _Conservative extension in relevant implication_, Studia Logica volume 31 (1973) pp39–46, doi:[10.1007/BF02120525](https://doi.org/10.1007/BF02120525)

and named for [[Alonzo Church]].

[[!redirects Church monoids]]


Church numeral

# Church numerals

* toc
{: toc}

## Idea

The Church numerals are an encoding of the [[natural numbers]] into [[untyped lambda-calculus]].

## Definition

The $n$th Church numeral $\underline{n}$ is the operation of "iteration $n$ times", sending a function $f$ to its $n$th iterate.  Thus $\underline{0}(f)$ is the identity function, $\underline{1}(f) = f$, $\underline{2}(f) = f\circ f = \lambda x. f(f(x))$, and so on.

## Connection to realizability

The Church numeral $\underline{n}$ can be regarded as a [[realizer]] or "proof" that we can do induction up to $n$.  See [this discussion](https://nforum.ncatlab.org/discussion/8622/church-numerals-are-realizers/).

[[!redirects Church numerals]]


Church-Rosser theorem

#Contents#
* table of contents
{:toc}

## Idea

The Church–Rosser theorem states that the [[reduction rule]]s of the [[lambda calculus]] are [[confluent category|confluent]].

## References

* [[Masako Takahashi]], *Parallel Reductions in $\lambda$-Calculus*, Journal of Symbolic Computation **7** 2  (1989) 113-123 [[doi:10.1006/inco.1995.1057](https://doi.org/10.1006/inco.1995.1057)]

See also:

* Wikipedia, *[Church-Rosser theorem](https://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem)*


Church-Turing thesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Church-Turing thesis_ is a (mostly informal) statement about the nature of [[computability]]. It roughly asserts that there is, up to equivalence, only one single universal concept of computability.

Slightly more in detail, the (physical) _Church-Turing thesis_ says vaguely that what is [[computation|computable]] in the mathematical sense of [[computation]] is precisely what is "effectively" computable ([[computable physics|physically computable]]).

In interpreting this one has to be careful which concept of [[computation]] is used, there are two different main types:

[[!include computable mathematics -- table]]

Indeed, there are physical processes (described by the [[wave equation]]) which are not type-I computable ([Pour-El et al. 83](#PourEl83)), but they are type-II computable ([Weihrauch-Zhong 01](WeihrauchZhong01), [Weihrauch-Zhong 02](#WeihrauchZhong02), [Weihrauch-Zhong 06](#WeihrauchZhong06)). For more on this see at _[[computable physics]]_.

One then distinguishes:

* The **physical Church-Turing thesis**: What is [[computable physics|physically computable]] is also computable in the mathematical sense of [[computability]].

* The **strong physical Church-Turing thesis**: _Every [[real number]] found by [[experiment]] in the [[observable universe]] is a [[computable real number]]_.

  This strong version is often phrased as "the universe is a computer" or as "[digitial physics](https://en.wikipedia.org/wiki/Digital_physics)".

On the strong physical Church-Turing thesis see _[SEP -- Computation in Physical Systems -- Is every Physical system computational?](http://plato.stanford.edu/entries/computation-physicalsystems/#EvePhySysCom)_ for a general account and ([Waaldijk 03](#Waaldijk03)) for a more formal account emphasizing the role of [[intuitionistic mathematics|intuitionistic]]/[[constructive mathematics]].

More concretely, the kind of experiment proposed there to test whether non-computable sequences of events may appear in [[experiment]] has been claimed to have been carried out with [[quantum physics|quantum process]] in ([CDDS 10](#CDDS10)).

## Related concepts

* [[computability]]

* [[computable physics]]

## References

General reviews include

* Wikipedia, _[Church-Turing thesis](http://en.wikipedia.org/wiki/Church–Turing_thesis)_, _[Digital physics](https://en.wikipedia.org/wiki/Digital_physics)_

* Stanford Encyclopedia of Philosophy, _[The Church-Turing Thesis](http://plato.stanford.edu/entries/church-turing/)_, _[Computation in Physical Systems](http://plato.stanford.edu/entries/computation-physicalsystems/#EvePhySysCom)_

A discussion well-informed by theoretical [[computability]], [[realizability]] and [[intuitionistic mathematics|intuitionistic]](/[[constructive mathematics]] is in

* {#Waaldijk03} [[Frank Waaldijk]], section 7 of _On the foundations of constructive mathematics -- especially in relation to the theory of continuous functions_, 2003  ([pdf](http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf), [web discussion](http://fwaaldijk.wordpress.com/2014/01/14/experiment-to-disprove-strong-physical-church-turing-thesis/)).

Discussion of physical processes which are not type-I computable by [[partial recursive functions]] are found in

* {#PourEl83} Marian Boykan Pour-El, J. Ian Richards, _Computability and noncomputability in classical analysis, Trans. Amer. Math. Soc. 275 (1983) 539-560.

  Marian Pour-El, Ning Zhong, _The wave equation with computable initial data whose unique solution is nowhere computable_, Math. Logic Quart. 43 (1997) no. 4, 499-509.

Proof that these physics processes ([[wave equation]], [[Schrödinger equation]]) are however [[Type Two Theory of Effectivity|type-II computable]] is in

* {#WeihrauchZhong01} [[Klaus Weihrauch]],  Ning Zhong, _Is the linear Schrödinger Propagator Turing Computable?_, in Jens Blanck et al (eds.) _Computability and Complexity in Analysis: 4th International Workshop, CCA_, Springer 2001

* {#WeihrauchZhong02} [[Klaus Weihrauch]],  Ning Zhong, _Is wave propagation computable or can wave computers beat the Turing machine?_, Proc. of the London Math. Soc. (3) 85 (2002) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.5994))

* {#WeihrauchZhong06} [[Klaus Weihrauch]], _Computing Schrödinger propagators on Type-2 Turing machines_, Journal of Complexity
Volume 22, Issue 6, December 2006, Pages 918–935

Discussion of (non-)computability of quantum processes is in

* {#CDDS10} Cristian Calude, Michael Dinneen, Monica Dumitrescu, Karl Svozil, _Experimental Evidence of Quantum Randomness Incomputability_, Phys. Rev. A 82, 022102 (2010) ([arxiv:1004.1521](http://arxiv.org/abs/1004.1521), [web announcement](http://www.technologyreview.com/view/418445/first-evidence-that-quantum-processes-generate-truly-random-numbers/))

[[!redirects physical Church-Turing thesis]]
[[!redirects strong physical Church-Turing thesis]]


Chuu-Lian Terng

* [Wikipedia entry](https://en.wikipedia.org/wiki/Chuu-Lian_Terng)

* [InSpire page](https://inspirehep.net/authors/2364769)

## Selected writings

On [[natural bundles]]/[[natural operations]] in [[differential geometry]], via [[jet group]]-[[G-structure|structure]]:

* {#Terng78} [[Chuu-Lian Terng]], *Natural vector bundles and natural differential operators*, Amer. J. Math. **100** (1978) 775-828 [[doi:10.2307/2373910](https://doi.org/10.2307/2373910), [jstor:2373910](https://www.jstor.org/stable/2373910)]

category: people

[[!redirects Terng Chuu-Lian]]


Chu–Vandermonde identity

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

The Chu-Vandermonde identity is a basic identity in the [[combinatorics]] of [[binomial coefficients]]. It can be summarized as a [[polynomial]] identity

$$\sum_{j=0}^n \binom{x}{j} \binom{y}{n-j} = \binom{x+y}{n}$$

in the polynomial ring $\mathbb{C}[x, y]$, where we recall $\binom{x}{k} = \frac{x^{\underline{k}}}{k!} \coloneqq \frac{x (x-1) \ldots (x - k + 1)}{k!}$.

To prove it, it suffices to verify it for the values $(x, y) = (p, q)$ ranging over $\mathbb{N} \times \mathbb{N}$ (as $\mathbb{N}^2$ is [[Zariski topology|Zariski-dense]] in the [[affine space|affine plane]] $\mathbb{C}^2$). There one can resort to a structural proof or so-called "[[bijective proof]]", where we count the number of $n$-element [[subsets]] $N$ of a finite set $P + Q$, with ${|P|} = p$ and ${|Q|} = q$. Regarding $P$ and $Q$ as subsets of the [[coproduct]] $P + Q$, if the subset $N \cap P$ of $P$ has $j$ elements, then the subset $N \cap Q$ of $Q$ has $n-j$ elements. The subset $N$ determines and is uniquely determined by the pair $(N \cap P, N \cap Q)$. The number of such pairs is $\sum_{j=0}^n \binom{p}{j} \binom{q}{n-j}$, exactly as claimed.

(Structurally, we are relying on the fact that the category of [[finite sets]] is an [[extensive category]].)

There is more to be written on this in terms of [[species]], [[coalgebras]], etc.

[[!redirects Chu-Vandermonde formula]]

[[!redirects Chu-Vandermonde identity]]

[[!redirects Vandermonde identity]]
[[!redirects Vandermonde formula]]
[[!redirects Vandermonde convolution identity]]
[[!redirects Vandermonde convolution formula]]


Cihan Okay

* [personal page](https://www.cihanokay.com/)

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

## Selected writings

Discussion of [[quantum contextuality]] via [[singular cohomology]]:

* [[Cihan Okay]], [[Sam Roberts]], [[Stephen D. Bartlett]], [[Robert Raussendorf]], *Topological proofs of contextuality in quantum mechanics*, Quantum Information and Computation **17** (2017) 1135-1166 [[arXiv:1701.01888](https://arxiv.org/abs/1701.01888), [doi:10.26421/QIC17.13-14-5](https://doi.org/10.26421/QIC17.13-14-5)]

* {#OkayKharoofIpek22} [[Cihan Okay]], Aziz Kharoof, Selman Ipek, *Simplicial quantum contextuality*, Quantum **7** (2023) 1009 [[arXiv:2204.06648](https://arxiv.org/abs/2204.06648), [doi:10.22331/q-2023-05-22-1009](https://doi.org/10.22331/q-2023-05-22-1009)]

* Aziz Kharoof, [[Cihan Okay]], *Homotopical characterization of strongly contextual simplicial distributions on cone spaces* [[arXiv:2311.14111](https://arxiv.org/abs/2311.14111)]

* [[Cihan Okay]], *Simplicial Distributions and Contextuality*, [talk at](CQTS#OkayOct2023) [[CQTS]] (11 Oct 2023) [video:[YT](https://youtu.be/pgKYGUWl1kU)]

category: people


cinquefoil knot - SVG

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category: svg

[[!redirects cinquefoil - SVG]]


Ciprian Manolescu

* [personal page](https://web.stanford.edu/~cm5/)

## Selected writings

On [[triangulation theorems]]:

* {#Manolescu16} [[Ciprian Manolescu]], *$Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture*, J. Amer. Math. Soc. 29 (2016), 147-176  ([arXiv:1303.2354](https://arxiv.org/abs/1303.2354), [doi:10.1090/jams829]( https://doi.org/10.1090/jams829))

* [[Ciprian Manolescu]], *Triangulations of manifolds*, Notices of the International Congress of Chinese Mathematicians **2** 2 (2104) ([pdf](https://web.stanford.edu/~cm5/tm.pdf), [[Manolescu_TriangulationsOfManifolds.pdf:file]], [doi:10.4310/ICCM.2014.v2.n2.a2](https://dx.doi.org/10.4310/ICCM.2014.v2.n2.a2))

category: people


circle

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

# Contents
* table of contents
{: toc}

## Idea

The **circle** is a fantastic thing with lots and lots of [[stuff, structure, property|properties and extra structures]].  It is a:

* [[topological space]]
* [[smooth manifold]]
* [[Lie group]] (the [[circle group]])
* [[co-H-group]]

and it is one of the basic building blocks for lots of areas of mathematics, including:

* [[homotopy theory]]
* [[loop spaces]]
* [[knots]] and [[links]]

## Definition

We consider the circle first as a [[topological space]], then as the [[homotopy type]] represented by that space.

### As a topological space

+-- {: .num_defn #circle}
###### Definition
The two most common definitions of the circle as a [[topological space]] are:

1. It is the [[topological subspace]] of $\mathbb{C}$ consisting of those numbers of [[norm]]/[[absolute value]] $1$:

   $$
     U(1) \coloneqq \{z \in \mathbb{C} : {|z|} = 1\}
   $$

   (of course, $\mathbb{C}$ can be identified with $\mathbb{R}^2$ and an equivalent formulation of this definition given; also, to emphasise the reason for the notation $U(1)$, $\mathbb{C}$ can be replaced by $\mathbb{C}^* = GL_1(\mathbb{C})$)

2. It is the quotient of $\mathbb{R}$ by the integers:

   $$
   S^1 \coloneqq \mathbb{R}/\mathbb{Z}
   $$
=--

The standard equivalence of the two definitions is given by the map $\mathbb{R} \to \mathbb{C}$, $t \mapsto e^{2 \pi i t}$.

In constructive mathematics, the notions of real numbers and complex numbers split into multiple notions. In general, if the metric space of the real numbers is not [[sequentially Cauchy complete]], such as in the [[modulated Cantor real numbers]], then the two definitions cannot be proven to be equivalent, since the [[exponential function]] is not well defined on $\mathbb{C}$. Only if the real numbers are [[sequentially Cauchy complete]] are the two definitions the same.

### As a homotopy type

As the bare [[homotopy type]], the [[shape modality|shape]] $ʃ S^1 \simeq \mathbf{B}\mathbb{Z}$ of the [[cohesion|cohesive]] circle $S^1$ is equivalent to the [[homotopy pushout]]

$$
  ʃ S^1 \simeq * \coprod_{* \coprod * } *
  \,.
$$

In [[homotopy type theory]], this can be formalized as a [[higher inductive type]] generated by a point `base` and a path from `base` to itself; see [[circle type]] for more.

## Properties

The topological circle is a [[compact space|compact]], [[connected space|connected]] [[topological space]].  It is a $1$-[[dimension|dimensional]] [[smooth manifold]] (indeed, it is the only $1$-dimensional compact, connected smooth manifold).  It is **not** [[simply connected space|simply connected]].

The circle is a model for the [[classifying space]] for the [[abelian group]] $\mathbb{Z}$, the [[integer|integers]]. Equivalently, the circle is the [[Eilenberg-Mac Lane space]] $K({\mathbb{Z}},1)$. Explicitly, the first [[homotopy group]] $\pi_1(S^1)$ is the [[integer|integers]] $\mathbb{Z}$. But the higher [[homotopy groups]] $\pi_n(S^1) \simeq *$, $n \gt 1$ all vanish (and so is a [[homotopy type|homotopy 1-type]]). This can be deduced from the result that the loop space $\Omega S^1$ of the circle is the group ${\mathbb{Z}}$ of integers and that $S^1$ is path-connected. Proofs of this in [[homotopy type theory]] are in the [references](#references).

The result that $S^1\simeq K({\mathbb{Z}},1)$ holds in a general [[Grothendieck (∞,1)-topos]]. In fact, more generally, for $X$ a pointed object of a [[Grothendieck (∞,1)-topos]] ${\mathcal{H}}$, there is a natural equivalence between the [[suspension object]] $\Sigma X$ and the classifying space $ B{\mathbb{Z}}\wedge X$. In particular, when $X$ is specifically the 0-truncated two-point space $S^0$, we get $S^1\simeq K({\mathbb{Z}},1)$.

The [[product]] of the circle with itself is the ($2$)-[[torus]]

$$
  T = S^1 \times S^1
  \,.
$$

Generally, the $n$-torus $T^n$ is $(S^1)^n$.

## Related concepts

* [[unit circle]]

* [[circle action]]

* [[minimal simplicial circle]]

* [[circle type]]

* [[sphere]]

* [[pseudocircle]]

* [[conic section]]

  * [[ellipse]], [[parabola]], [[hyperbola]]

* [[circumference of a circle]]

* [[area of a circle]]

## References

A formalization of the [[shape modality|shape]] [[homotopy type]] $ʃ S^1 \simeq \mathbf{B}\mathbb{Z}$ of the circle as a [[higher inductive type]] in [[homotopy type theory]], along with a proof that $\Omega ʃS^1\simeq {\mathbb{Z}}$ (and hence $\pi_1(ʃS^1) \simeq \mathbb{Z}$):

* {#LicataShulman13} [[Dan Licata]], [[Mike Shulman]], *Calculating the Fundamental Group of the Circle in Homotopy Type Theory*, LICS '13: Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science June 2013 ([arXiv:1301.3443](http://arxiv.org/abs/1301.3443), [doi:10.1109/LICS.2013.28](https://doi.org/10.1109/LICS.2013.28))

  > blog announcement: [A formal proof that $\pi_1(S^1) = \mathbb{Z}$](https://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/)

* {#BezemBuchholtzGraysonShulman19} [[Marc Bezem]], [[Ulrik Buchholtz]], [[Daniel Grayson]], [[Michael Shulman]], _Construction of the Circle in UniMath_, Journal of Pure and Applied Algebra Volume 225, Issue 10, October 2021, 106687 ([arXiv:1910.01856](https://arxiv.org/abs/1910.01856))

The proof is formalized therein using the [[Agda]] [[proof assistant]].  See also

* The [[HoTT Book]], section 8.1

* The HoTT Coq library: [theories/Spaces/Circle.v](https://github.com/HoTT/Coq-HoTT/blob/V8.18/theories/Spaces/Circle.v), [theories/Homotopy/Pi1S1.v](https://github.com/HoTT/Coq-HoTT/blob/V8.18/theories/Homotopy/Pi1S1.v)

* The HoTT Agda library: [homotopy/LoopSpaceCircle.agda](https://github.com/HoTT/HoTT-Agda/blob/2.0/homotopy/LoopSpaceCircle.agda)

[[!redirects circle]]
[[!redirects circles]]
[[!redirects Circle]]
[[!redirects Circles]]


circle 2-bundle

A [[principal 2-bundle]] for structure [[2-group]] the [[circle 2-group]].

Also called a _circle-[[bundle gerbe]]_.

For more see at _[[circle n-bundle with connection]]_.

[[!redirects circle 2-bundles]]


circle action

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## Idea

An [[action]] by the [[circle group]].

## Related concepts

* [[free loop space]], [[Sullivan model of free loop spaces]]

* [[group action on n-spheres]]

## References

### Models in rational homotopy theory

Discussion of models in [[rational homotopy theory]] includes:

* [[Agusti Roig]], [[Martintxo Saralegi-Aranguren]], _Minimal Models for Non-Free Circle Actions_ ([arXiv:math/0004141](https://arxiv.org/abs/math/0004141))

* [[Martin Raussen]], _Circle Actions on Rational Homology Manifolds and deformations of rational homotopy theory_,  Transactions of the American Mathematical Society Vol. 347, No. 1 (Jan., 1995), pp. 137-153  ([jstor](http://www.jstor.org/stable/2154792))

See also

* G. Hector and [[Martintxo Saralegi-Aranguren]], _Intersection homology of $S^1$-actions_, Trans. Amer. Math. Soc. 338 (1993), 263-288  (doi:10.1090/S0002-9947-1993-1116314-7[pdf](http://www.ams.org/journals/tran/1993-338-01/S0002-9947-1993-1116314-7/S0002-9947-1993-1116314-7.pdf))

* G. Padilla, _Intersection Cohomology of $S^1$-Actions on Pseudomanifolds_, Indag. Math., New Ser. 15, No. 3, 383-412 (2004).

[[!redirects circle actions]]


circle bundle

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## Definition

A **circle bundle** is a [[principal bundle]] for the [[circle group]] $S^1$. Equivalently this is a $U(1)$-principal bundle, for the [[unitary group]] $U(1)$.

Under the canonical [[representation]] $\mathbf{B}U(1) \to Vect_{\mathbb{C}}$ the corresponding [[associated bundle]] is a complex [[line bundle]].

## Examples

* A basic example of a nontrivial [[circle group]]-[[principal bundle]] is the [[Hopf fibration]] $S^3 \to S^2$.

## Related concepts

* [[circle bundle with connection]], [[regular contact manifold]]

* [[prequantum circle bundle]]

* [[line bundle]]

* [[Seifert 3-manifold]]

[[!redirects circle bundles]]


circle constant

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## Idea ##

There are two circle constants used in [[geometry]], the more common circle constant $\pi$ representing the straight angle, and the less common circle constant $\tau = 2 \pi$ representing a full turn.

## History ##

Leonard Euler original definition of the circle constant $\pi$ was the constant valued as 6.283..., which is equivalent in today's notion as $\tau$. Euler's definition was often used in competition with the definition 3.1415.. until the late 1700s, when mathematicians collectively decided on standardizing $\pi$ as 3.1415...

However, in 2010, Michael Hartl proposed that the constant 6.283... which equals $2 \pi$ is much better for mathematics than $\pi$, as it would get rid of a significant number of factors of 2 from many equations, and is intuitively better for teaching and understanding subjects such as trigonometry and geometry, and proposed using the symbol $\tau$ for $2 \pi$. Since then, a few mathematicians such as Peter Harremoës have used $\tau$ instead of $\pi$ for research mathematics, and $\tau$ has been incorporated into a number of [[programming languages]] such as [[Python]] and [[Rust]].

## See also

* [[pi]]

* [[area of a circle]]

* [[circumference of a circle]]

## References

* Euler, Leonard (1746). Nova theoria lucis et colorum. Opuscula varii argumenti, p.169-244.

*  Segner, Johann Andreas von (1761). Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm (in Latin). Renger. p. 374. "Si autem π notet peripheriam circuli, cuius diameter eſt 2"

* Hartl, Michael (2010). [The Tau Manifesto](https://tauday.com/tau-manifesto)

* Tran, Andy (2017). Tau: The True Circle Constant. ([slides](https://www.maths.usyd.edu.au/u/sums/talks-2017/tau-true-circle.pdf))

* Harremoës, Peter (2017). "Bounds on tail probabilities for negative binomial distributions". Kybernetika. 52 (6): 943–966. arXiv:1601.05179. doi:10.14736/kyb-2016-6-0943. S2CID 119126029.


circle group

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## Definition

The **[[circle]] [[group]]** $\mathbb{T}$ is equivalently ([[isomorphism|isomorphically]])

* the [[quotient group]] $\mathbb{R}/\mathbb{Z}$ of the additive group of [[real numbers]] by the additive group of [[integers]], induced by the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{R}$;

* the [[unitary group]] $\mathrm{U}(1)$;

* the [[special orthogonal group]] $SO(2)$;

* the [[subgroup]] of the [[group of units]] $\mathbb{C}^\times$  of the [[field]] of [[complex numbers]] (its [[multiplicative group]]) given by those of any fixed positive modulus (standardly $1$).

## Properties

For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a [[short exact sequence]]

$$
  0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{T} \to 0
  \,,
$$

the "real [[exponential exact sequence]]".

(On the other hand, the last characterization is usually preferred when one wants to be concrete.)

A [[character]] of an [[abelian group]] $A$ is simply a [[homomorphism]] from $A$ to the circle group.

$U(1)$ is the compact [[real form]] of the [[multiplicative group]] $\mathbb{G}_m = \mathbb{C}^\times$ over the [[complex numbers]], see at _[form of an algebraic group -- Circle group and multiplicative group](form+of+an+algebraic+group#MultiplicativeAndCircleGroup)_.

## Related concepts

* A [[principal bundle]] with [[structure group]] the circle group is a [[circle bundle]]. The [[associated bundle]] under the standard [[linear representation]] of $U(1) \hookrightarrow \mathbb{C}$ is a [[complex line bundle]].

* [[cyclic group]], [[Q/Z]]

* [[circle 2-group]]

* [[circle]], [[unit circle]], [[circle type]], [[minimal simplicial circle]]

* [[torus]]

[[!include low dimensional rotation groups -- table]]

[[!redirects circle group]]
[[!redirects R/Z]]
[[!redirects U(1)]]
[[!redirects SO(2)]]

[[!redirects circle Lie group]]