C^∞-ring

> $\mathrm{C}^\infty$-rings should not be confused with [[C-infinity algebra|$\mathrm{C}_\infty$-algebras]].

***

+-- {: .rightHandSide}
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###Context###
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
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#Contents#
* tic
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## Idea

A **smooth algebra** or **$C^\infty$-ring** is an [[algebra]] $A$ over the reals $\mathbb{R}$ for which not only the product operation $\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ lifts to the algebra product $A \times A \to A$, but for which _every smooth map_ $f : \mathbb{R}^n \to \mathbb{R}^m$ ([[morphism]] in [[Diff]]) lifts to a map $A(f) : A^n \to A^m$ in a compatible way.

In short this means that $A$ is

* a [[product]]-preserving [[copresheaf]] on [[CartSp]];

* equivalently: an algebra for the [[Lawvere theory]] [[CartSp]];

The smoothness of such $C^\infty$-rings is witnessed by the fact that this Lawvere theory is even a [[Fermat theory]].

The [[opposite category]] of the [[category]] of $C^\infty$-rings is the category of [[smooth loci]]. This and its subcategories play a major role as [[site]]s for [[category of sheaves|categories of sheaves]] that serve as  [[Models for Smooth Infinitesimal Analysis|models]] for [[synthetic differential geometry]].

## Motivating example

For $X$ a [[smooth manifold]], the assignment

$$
  \mathbb{R}^n \mapsto C^\infty(X,\mathbb{R}^n)
   = Hom_{Diff}(X,\mathbb{R}^n)
$$

of the [[set]] of smooth $\mathbb{R}^n$-valued functions on $X$ is clearly covariant and hence yields a copresheaf on [[CartSp]] $\subset$ [[Diff]]: a [[functor]]

$$
  C^\infty(X,-) : CartSp \to Set
  \,.
$$

Since the [[hom-functor]] sends [[limit]]s to [[limit]]s in its second argument this is clearly [[product]] preserving.

$$
  C^\infty(X, \mathbb{R}^n \times \mathbb{R}^m)
  \simeq
  C^\infty(X,\mathbb{R}^n)
  \times
  C^\infty(X, \mathbb{R}^m)
$$

If as usual we write $C^\infty(X) := C^\infty(X,\mathbb{R})$ for the set of just $\mathbb{R}$-valued smooth functions, then the usual pointwise product of functions

$$
 \cdot :  C^\infty(X) \times C^\infty(X) \to C^\infty(X)
$$

can be regarded as the image of our copresheaf under the muliplication map $\mathbb{R} \times \mathbb{R} \stackrel{-\cdot -}{\to} \mathbb{R}$ on the algebra of real numbers:

$$
  \cdot
  :
  C^\infty(X)
  \times
  C^\infty(X)
  :=
  C^\infty(X,\mathbb{R})
  \times
  C^\infty(X,\mathbb{R})
  \simeq
  C^\infty(X,\mathbb{R}\times \mathbb{R})
  \stackrel{C^\infty(X,-\cdot-)}{\to}
  C^\infty(X,\mathbb{R})
  =:
  C^\infty(X)
  \,.
$$

## Definitions

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

[[CartSp]] is the full subcategory of [[Diff]]
on manifolds of the form $\mathbb{R}^n$.

=--

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

A $C^\infty$-algebra is a finite [[product]]-preserving [[copresheaf]] on [[CartSp]], i.e. a finite [[product]] preserving [[functor]]

$$
  A : CartSp \to Set
  \,.
$$

The category of such functors and [[natural transformation]]s between them we denote by $C^\infty Alg$.

=--

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

The standard name in the literature for generalized smooth algebras is
**$C^\infty$-rings**. Even though standard, this has the disadvantage that it collides badly with the use of $\infty$- for
[[higher category theory|higher categorical]] structures.

=--

### Tensor product

+-- {: .num_defn}
###### Definition (smooth tensor product)

The [[coproduct]] in $C^\infty Alg$
we call the **smooth tensor product**

$$
  \otimes_\infty : C^\infty Alg \times C^\infty Alg
   \to C^\infty Alg
  \,.
$$

More generally, for
$i : C \to A$ and $j : C \to B$ two morphisms in $C^\infty Alg$, we call the [[pushout]]

$$
  \array{
    C &\stackrel{i}{\to}& A
    \\
    \downarrow && \downarrow
    \\
    B &\stackrel{j}{\to}& A \otimes_C B
  }
$$

the **smooth tensor product over $C$** of $A$ and $B$.

=--

### Finitely generated $C^\infty$-rings

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

For $X$ a [[smooth manifold]], the smooth algebra
$C^\infty(X)$ is the functor

$$
  C^\infty(X) := Hom_{Diff}(X,-)
$$

=--

+-- {: .num_defn}
###### Definition
**(finitely generated and finitely presented $C^\infty$-rings)**

For $R$ a $C^\infty$-ring, and $I \in U(R)$ an ideal in the underlying ordinary ring, there is a canonical $C^\infty$-ring structure $R/I$ on the ordinary quotient ring $U(R)/I$.

A $C^\infty$-ring $R$ is called **finitely generated** if it is of the form $C^\infty(\mathbb{R}^n)/I$ for $n \in \mathbb{N}$ and $I $ an ideal in $U(C^\infty(\mathbb{R}^n))$.

It is **finitely presented** if also $I$ is finitely generated, as an ideal, $I = (i_1, \cdots, i_k)$ with $i_j \in U(R)$.

This is equivalent to $R$ being a [[pushout]] of the form

$$

  \array{
    C^\infty(\mathbb{R}^k) &\to& C^{\infty}(*) \simeq \mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    C^\infty(\mathbb{R}^n) &\to& R
  }
  \,.
$$

=--

+-- {: .num_defn}
###### Definition
**([[germ-determined ideal]])**

For $p \in \mathbb{R}^n$ let

$$
  \pi_p : C^\infty(\mathbb{R}^n) \to C^\infty_p(\mathbb{R}^n)
$$

be the natural projection onto the [smooth algebra of germs](#Germs) of functions at $p$.

A $C^\infty$-ring $C$ is called **fair** or **finitely generated and germ-determined** if it is finitely generated $C \simeq C^\infty(\mathbb{R}^n)/I$ and the ideal $I$ has the property that $f \in C^\infty(\mathbb{R}^n)$ is an element of $I$ if (and hence precisely if) for all $p \in \mathbb{R}^n$ the germ $\pi_p(f) \in C^\infty_p(\mathbb{R})^n$ is in the germ $\pi_p(I)$ of the ideal.

=--

### Internal $C^\infty$-rings

For any [[smooth topos]] $(\mathcal{T}, R)$, there is an [[internalization|internal]]
notion of generalized smooth algebra:

+-- {: .num_defn}
###### Definition (internal generalized smooth algebra)

For $(\mathcal{T}, R)$ a [[topos]] equipped with an [[internalization|internal]]
[[ring]] object $R$ (possibly but not necessarily a [[smooth topos]]), let
$CartSp(\mathcal{T},R)$ be the full [[subcategory]] of $\mathcal{T}$ on
objects of the form $R^n$ for $n \in \mathbb{N}$. Then a
**$(\mathcal{T},R)$-algebra** is a product-preserving functor
$A : CartSp(\mathcal{T}, R) \to Set$.
=--

All constructions on smooth algebras generalize to $(\mathcal{T},R)$-algebras.
In particular for $X \in \mathcal{T}$ any object we have the function
$(\mathcal{T},R)$-algebra

$$
  C(X) : R^n \mapsto \mathcal{T}(X,R^n)
  \,.
$$

The following remark asserts that when $\mathcal{T}$ is itself
a sufficiently nice [[category of sheaves]] on formal duals of $(Set,\mathbb{R})$-algebras,
then the internal notion of smooth function algebras on formal duals
of external smooth algebras reproduces these external smooth algebras.

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

Let $A$ be a finitely generated $C^\infty$-ring, $\ell A$ its incarnation as an object in
$\mathbb{L} = (C^\infty Ring^{fin})^{op}$ and $Y\ell A$ its incarnation in
$Sh(\mathbb{L}) \subset PSh(\mathbb{L})$, with $Y$ the [[Yoneda embedding]] and
using the assumption that the [[Grothendieck topology]] used to form $Sh(\mathbb{L})$
is [[subcanonical coverage|subcanonical]].

Also suppose that the line object $R$ is [[representable functor|represented]]
by $\ell C^\infty(\mathbb{R})$

Then we have for all
$A \in C^\infty Ring^{fin}$ that

$$
  C(Y\ell A) : R^n \mapsto A({*})^n
$$
=--

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

This is a straightforward manipulation:

$$
  \begin{aligned}
    Sh_{\mathbb{L}}(Y(\ell A), R^n)
    & =
    Sh_{\mathbb{L}}(Y(\ell A), Y(\ell C^\infty(\mathbb{R}^n)))
    \\
    & = PSh_{\mathbb{L}}(Y(\ell A), Y(\ell C^\infty(\mathbb{R}^n)))
    \\
    & \simeq \mathbb{L}(\ell A, \ell C^\infty(\mathbb{R}^n))
    \\
    & \simeq C^\infty Ring^{fin}(C^\infty(\mathbb{R}^n), A)
    \\
    & \simeq A({*})^n
  \end{aligned}
$$

Here

1. the first step expresses the nature of the line object in the models under consideration

1. the second step expresses that the embedding $Sh(\mathbb{L}) \to PSh(\mathbb{L})$
   is a [[full and faithful functor]]

1. the third step expresses that the [[Yoneda embedding]] is a [[full and faithful functor]]

1. the fourth step is the definition of $\mathbb{L}$ as the [[opposite category]] of
   $C^\infty Ring^{fin}$

1. the fifth step expresses that $C^\infty(R^n)$ is the free generalized smooth algebra
   on $n$ generators ([[Models for Smooth Infinitesimal Analysis|MSIA, chaper I, prop 1.1]])

=--

### Local $C^\infty$-algebra

The category [[CartSp]] carries a natural [[Grothendieck topology]].

A smooth algebra

$$
  A : CartSp \to Set
$$

is a [[locally algebra-ed topos|local algebra]] if $A$ sends [[covering]] families  to epimorphism families: for each covering $\{U_i \to U\}$ the morphism

$$
  \coprod_i A(U_i) \to A(U)
$$

is an [[epimorphism]].

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

Local smooth algebras are precisely the "local Archimedian" algebras (...).

=--

This is ([Bunge-Dubuc, prop. 2.1](#BungeDubuc)).

## Examples

### Functions on smooth manifolds

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

For $X$ a [[smooth manifold]], the smooth algebra
$C^\infty(X)$ is the functor

$$
  C^\infty(X) := Hom_{Diff}(X,-)
$$

=--

### Weil algebras: functions on small infinitesimal spaces

A _Weil algebra_ in this context is a finite-dimensional commutative $\mathbb{R}$-algebra $W$ with a maximal ideal $I$ such that $W/I \simeq \mathbb{R}$ and $I^n = 0$ for some $n \in \mathbb{N}$.

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

There is a unique $C^\infty$-ring structure on a Weil algebra $W$. It makes $W$ a finitely presented $C^\infty$-ring.

=--

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

The [[smooth loci]] corresponding to Weil algebras are [[infinitesimal space]]s. Weil algebras play a crucial role in the definition of [[smooth topos]]es.

=--

### Power series

...

### Functions on germs of manifolds {#Germs}

For $p \in \mathbb{R}^n$, the algebra of [[germ]]s of smooth $\mathbb{R}$-valued functions at $p$ carries an evident $C^\infty$-ring structure $C^\infty(\mathbb{R}^n)_p$.

With $I_p \subset C^\infty(\mathbb{R}^n)$ the ideal of functions that vanish on a neighbourhood of $p$ we have

$$
  C^\infty_p(\mathbb{R}^n) \simeq C^\infty(\mathbb{R}^n)/I_p
  \,,
$$

yielding a finitely generated but not (for $n \gt 0$) finitely presented $C^\infty$-ring.

## Properties

### Limits and colimits

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

All [[limit]]s and all [[directed colimit]]s in $C^\infty Ring$ are computed objectwise in $[CartSp,Set]$ as limits in [[Set]].

=--

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

As discussed at [[limits and colimits by example]], all [[limit]]s and [[colimit]]s in $[CartSp,Set]$ are computed objectwise, so the remaining question is if they preserve the property of functors $CartSp \to Set$ to preserved products. The claim follows from the observation that [[limit]]s and [[directed colimit]]s do commute with products.

=--

See also [[Models for Smooth Infinitesimal Analysis|MSIA, p. 22]].

### The underlying ordinary algebra {#UnderlyingAlgebra}

There is a [[stuff, structure, property|forgetful functor]]

$$
  U : C^\infty Alg \to Alg
$$

from generalized smooth algebras to ordinary algebras which is given by evaluation on $\mathbb{R}$

$$
  U : A \mapsto A(\mathbb{R})
$$

and equipping the set $A(\mathbb{R})$ with the algebra structure induced on it:

the product and sum on $A(\mathbb{R})$ is the image of the corresponding operations on the algebra $\mathbb{R}$

$$
  \cdot_A
  :
  A(\mathbb{R}) \times A(\mathbb{R})
  \stackrel{\simeq}{\to}
  A(\mathbb{R}\times \mathbb{R})
  \stackrel{A(\cdot)}{\to}
  A(\mathbb{R})
  \,.
$$

$$
  +_A :
  A(\mathbb{R}) \times A(\mathbb{R})
  \stackrel{\simeq}{\to}
  A(\mathbb{R} + \mathbb{R})
  \stackrel{A(\cdot)}{\to}
  A(\mathbb{R})
  \,.
$$

Moreover there is canonically a morphism of rings

$$
  \mathbb{R} \to A(\mathbb{R})
$$

given by

$$
  (* = \mathbb{R}^0 \stackrel{c}{\to} \mathbb{R})
  \mapsto
  (* = A(\mathbb{R}^0) \stackrel{A(c)}{\to} A(\mathbb{R}))
  \,.
$$

This makes $A(\mathbb{R})$ an $\mathbb{R}$-algebra.

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

The [[forgetful functor]] $U$ fits into an [[adjunction]]

$$
  (F \dashv U)
  :
  C^\infty Alg_{\mathbb{R}}
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  Alg_{\mathbb{R}}
  \,.
$$

=--

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

This statement may be understood as a special case of the following more general statement:

* If $S$, $T$ are [[finitary monad]]s and $f: S \to T$ is a monad morphism, then the relative forgetful functor
$$f^*: Alg_T \to Alg_S,$$
which [[pullback|pulls back]] a $T$-algebra $\xi: T X \to X$ to the $S$-algebra $\xi \circ f X: S X \to X$, admits a [[left adjoint]].

(In the case under discussion, $S$ is the free algebra monad on $Set$, $T$ is the free smooth algebra monad, and $f: S \to T$ is induced from the obvious inclusion $f(n): S(n) \to T(n)$ which interprets an $n$-ary algebra operation (in the theory $Th_S$) as a smooth operation in the theory $Th_T$. See [[finitary monad]] for discussion on the connection between finitary monads $T$ and Lawvere theories $Th_T$.)

The desired left adjoint $f_!$ takes an algebra $\theta: S X \to X$ to the [[reflexive coequalizer]] exhibited as a diagram

$$T S X \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} T X \stackrel{\pi}{\to} T \otimes_S X$$

in the category of $T$-algebras, where $\mu_T: T T \to T$ is the monad multiplication. The coequalizer is denoted $T \otimes_S X$ to emphasize the analogy with pushing forward $S$-modules $X$ along a ring homomorphism $f: S \to T$ to get $T$-modules; the proof below is an arrow-theoretic transcription of the usual proof of the adjunction between pushing forward and pulling back in the context of rings and modules.

A finitary monad $T$ [preserves reflexive coequalizers](http://ncatlab.org/nlab/show/reflexive+coequalizer#applications_4), so that there is a canonical isomorphism

$$T(T \otimes_S X) \cong (T T) \otimes_S X$$

It follows that $T$-algebra ($T$-module) maps $\bar{g}: T \otimes_S X \to Y$, i.e., maps that render commutative the diagram

$$\array{
T T \otimes_S X & \stackrel{T\bar{g}}{\to} & T Y \\
\mu \otimes_S X \downarrow & & \downarrow \xi \\
T \otimes_S X & \underset{\bar{g}}{\to} & Y
}$$

are in bijection with maps $g: T X \to Y$ that render commutative

$$\array{
T T X & \stackrel{T g}{\to} & T Y \\
\mu \downarrow & & \downarrow \xi \\
T X & \underset{g}{\to} & Y
}$$

(that is to say, $T$-algebra maps $g: T X \to Y$) which additionally coequalize the parallel pair in the diagram

$$T S X \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} T X \stackrel{g}{\to} Y$$

Since $f: S \to T$ is a monad morphism, we have commutativity of parallel squares in

$$\array{
T S X & \stackrel{\overset{(\mu_T \circ T f)X}{\to}}{\underset{T\theta}{\to}} & T X & \stackrel{g}{\to} & Y \\
f S X \uparrow & & \uparrow f X & & \\
S S X & \stackrel{\overset{\mu_S X}{\to}}{\underset{S \theta}{\to}} & S X
}$$

so that $g \circ f X$ coequalizes the bottom pair. However, because $g: T X \to Y$ is a $T$-algebra map, its pullback $g \circ f X: S X \to Y$ defines an $S$-algebra map $S X \to f^* Y$. This $S$-algebra map $g \circ f X$ factors through the coequalizer of the bottom pair of maps in $Alg_S$, i.e., factors uniquely through an $S$-algebra map $X \to f^*Y$. This establishes the adjunction $f_! \dashv f^*$.

=--

### Finitely presented $C^\infty$-rings

+-- {: .num_prop}
###### Proposition ([[Models for Smooth Infinitesimal Analysis|MSIA]], prop. 1.1)

$C^\infty(\mathbb{R}^n)$ is the free smooth algebra
on $n$ generators, in that for every $n \in \mathbb{N}$
and every smooth algebra $A$ there is an [[adjunction]]
isomorphism

$$
  Hom_{C^\infty Alg}(C^\infty(\mathbb{R}^n), A)
  \simeq
  Hom_{Alg}(\mathbb{R}[x_1,...,x_n], A(\mathbb{R}))
  \,.
$$

=--

* Every finitely presented $C^\infty$-ring is fair/germ determined.

We have a chain of inclusions

* finitely presented $C^\infty$-rings

* $\subset$ "good" $C^\infty$-rings

* $\subset$ fair $C^\infty$-rings

* $\subset$ finitely generated $C^\infty$-rings

### Points of smooth loci

An **$\mathbb{R}$-point** of a $C^\infty$-ring $C$ is a point $* \to \mathbb{L}(C)$ of the corresponding [[smooth locus]], i.e. a morphism $C \to \mathbb{R} \cong C^\infty(*)$.

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

Points of a $C^\infty$-ring are in bijection with points of the underlying $\mathbb{R}$-algebra $U(C)$, i.e. with ordinary $\mathbb{R}$-algebra morphisms $U(C) \to \mathbb{R}$.

=--

In particular every Weil algebra $W$ has a unique point $* \to \mathbb{L}(W)$: every Weil algebra is the algebra of functions on an _infinitesimal thickening_ of an ordinary point.

By the properties of $C^\infty(X)$ for $X$ a smooth manifold discussed below, the $\mathbb{R}$-points of $C^\infty(X)$ are precisely the ordinary points of the manifold $X$.

### Smooth function algebras on smooth manifolds
  {#SmoothFunctionAlgebrasOnSmoothManifolds}

+-- {: .num_prop}
###### Proposition ([[Models for Smooth Infinitesimal Analysis|MSIA]], prop. 2.5, 2.6 )

Let $f : X \to Z$ and $g : Y \to Z$ be  [[transversal maps]] of [[smooth manifold]]s. Then the functor $C^\infty(-)$ sends the [[pullback]]

$$
  \array{
    X \times_Z Y &\to& X
    \\
    \downarrow && \downarrow^f
    \\
    Y &\stackrel{g}{\to}& Z
  }
$$

to the [[pushout]]

$$
  \array{
    C^\infty(X) \otimes_{C^\infty(Z)} C^\infty(Y)
    =: & C^\infty(X \times_Z Y) &\leftarrow& C^\infty(X)
    \\
    & \uparrow && \uparrow^{f^*}
    \\
    & C^\infty(Y) &\stackrel{g^*}{\leftarrow}& C^\infty(Z)
  }
$$

=--

In particular this implies (for $Z = {*}$)that the the smooth tensor product of functions on $X$ and $Y$ is the algebra of functions on the [[product]] $X \times Y$:

$$
  C^\infty(X \times Y) \simeq C^\infty(X) \otimes_\infty C^\infty(Y)
  \,.
$$

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

The ordinary algebraic tensor product of $C^\infty(X)(\mathbb{R})$ and $C^\infty(Y)(\mathbb{R})$ regarded as ordinary algebras does _not_ in general satisfy this property. Rather one has an inclusion

$$
  C^\infty(X)(\mathbb{R}) \otimes C^\infty(Y)(\mathbb{R})
  \subset
  C^\infty(X \times Y)(\mathbb{R})
  \,.
$$

=--

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

In the context of [[geometric function theory]] the corresponding general statement (without the transversality condition) says that $C^\infty(X)$ is a "good" kind of function. The above equation is one sub-aspect of the one of the fundamental theorems of [[geometric infinity-function theory]].

=--

Turning this inclusion into an equivalence is usually called a _completion_ of the algebraic tensor product. Therefore we see:
+-- {: .standout}
The smooth tensor product is automatically the completed tensor product.
=--

In summary this yields the following characterization of smooth function algebras on manifolds.

+-- {: .num_theorem}
###### Theorem ([[Models for Smooth Infinitesimal Analysis|MSIA]], theorem 2.8)

The functor $C^\infty(-) = Hom_{Diff}(-,-) : Diff \to C^\infty Alg$

* is a [[full and faithful functor]]

* takes values in finitely presented smooth $C^\infty$-algebras

* sends transversal [[pullback]]s to [[coproduct]]s (and hence to the smooth tensor product).

=--

## Deformation theory of smooth algebras

> **under construction**

For $C$ any category whose objects we think of as "functions algebras on test spaces", such as $C = C^\infty Ring$, there is a general intrinsic notion of [[tangent complex]] and [[deformation theory]] of such objects.

As describe there, the key structure of interest from which all the other structure here is induced is the **tangent category**

$$
 T C \to C
 \,.
$$

This is $T C = Ab(Arr(C))$ the [[codomain fibration]] of $C$ "fiberwise stabilized", meaning that in each fiber one takes it to consist of $Ab(C/A)$, the abelian [[group object]]s in the [[overcategory]].

We now first recall what this means for ordinary rings and how it induces the ordinary notion of derivations and  modules for ordinary rings by setting $C = $ [[CRing]],  and then look at what it implies for $C^\infty$-rings by setting $C = C^\infty Rings$.

By an old argument by Quillen, for $C = $ [[CRing]] we have that $T C = Mod$ is the [[bifibration]] of [[module]]s over rings, there is a natural equivalence

$$
  Mod_A \stackrel{\simeq}{\to} Ab(C/A)
  \,.
$$

This is induced by the functor that sends
an $A$-module $N$ to the corresponding object in
the square-0-extension $R \oplus N \to R$. (See [[module]]).

From this structure alone a lot of further structure follows:

* a [[derivation]] $\delta A \to N$ is precisely a [[section]] of the corresponding morphism $A \oplus N \to A$ in $C/A$, in the category $C$ namely a ring homomorphism

  $$
    \array{
      A &&\stackrel{\delta}{\to}&& A \oplus N
      \\
      & {}_ {=}\searrow && \swarrow
      \\
      && A
    }
    \,.
  $$

* The forgetful functor $T Ring \simeq Mod \to Ring$ has a [[left adjoint]]

  $$
    \Omega_K^1 : Ring \to Mod
  $$

  that sends each ring to its module of [[Kähler differential]]s.

  The fact that it is left adjoint is the universal property
  of the Kähler differentials as the objects co-representing
  derivations

  $Hom_{Ab(Ring/R)}(\Omega_K^1(A),N) \simeq
  Hom_{Ring/A}(A,A \oplus N)$.

  So every derivation $\delta : A \to N$ uniquely corresponds to a module morphism $\Omega^1_K(A) \to N$, namely the one that sends
  $d a \mapsto \delta(a)$.

This abstract story remains precisely the same for $C^\infty$-rings (and in fact for everything else!) but what it means concretely changes.

The crucial observation is (as one can show) that an abelian group object in $C^\infty Ring/A$ is a square-0 extension $(A \oplus N)$ for $N$ an (ordinary) module of the _underlying_ $\mathbb{R}$-algebra $A$. This square-0-extension happens to be _uniquely_ equipped with the $C^\infty$-Ring-structure given by

$$
  (f \in C^\infty(\mathbb{R}^n) \to \mathbb{R})
  \mapsto
  \left(
     (a_1, n_1), \cdots, (a_n, n_n))
     \mapsto
      f(a_1, \cdots, a_n)
      +
      \sum_{i = 1^n} \frac{\partial f}{\partial x_i}(a_1, \cdots, a_n) n_i
  \right)
  \,.
$$

This uniquely induced smooth structure on objects in $Ab(C^\infty Ring/A)$ then in turn affects the nature of the notion of derivation and of Kähler differentials, as those are defined by general abstract reasoning from the former.

First of all it follows that a [[derivation]] -- by general abstract definition  a morphism of $C^\infty$-rings $Id \oplus \delta : A \to A \oplus N$ -- is a morphism that satisfies _for all_ $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ that

$$
  \delta : f(a_1, \cdots, a_n) \mapsto
  \sum_i \frac{\partial f}{\partial x_i}
  \delta a_i
  \,.
$$

For ordinary rings only the compatibility $\delta (a_1 \cdot a_2) = \delta (a_1) a_2 + a_1 \delta(a_2)$ with the single product operation is required. Here, however, compatibility with infinitely more operations $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ is demanded.

Accordingly, then, the Kähler differentials as defined with respect to such derivations are different from the purely ring-theoretic ones: they produce the _right_ notion of smooth 1-forms here, whereas the ring-theoretic one does not.

## Variants

The generalization of the notion of smooth algebra for  [[(∞,1)-category theory]] is

* [[smooth (∞,1)-algebra]].

The generalization to [[supergeometry]] is

* [[smooth superalgebra]].

## Related concepts

* [[germ-determined C-infinity ring]]

* [[Dubuc topos]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[derivations of smooth functions are vector fields]]

* [[EFC-algebra]]

## References {#References}

Textbook account

* [[Ieke Moerdijk]]. [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_ Springer (1991)

The concept of $C^\infty$-rings in particular and that of [[synthetic differential geometry]] in general was introduced in

* [[Bill Lawvere]], _Categorical dynamics_

  in [[Anders Kock]] (eds.) _Topos theoretic methods in geometry_, volume 30 of _Various Publ. Ser._, pages 1-28, Aarhus Univ. (1979)

but examples of the concept are older. A discussion from the point of view of [[functional analysis]] is in

* G. Kainz, A. Kriegl, [[Peter Michor]], _$C^\infty$-algebras from the functional analytic view point_ Journal of pure and applied algebra 46 (1987) ([pdf](http://www.mat.univie.ac.at/~michor/c-oo-alg.pdf))

A characterization of those $C^\infty$-rings that are algebras of smooth functions on some [[smooth manifold]] is given in

* [[Peter Michor]], [[Jiri Vanzura]], _Characterizing algebras of $C^\infty$-functions on manifolds_ ([pdf](http://www.emis.de/journals/CMUC/pdf/cmuc9603/michor.pdf))

Lawvere's ideas were later developed by [[Eduardo Dubuc]], [[Anders Kock]], [[Ieke Moerdijk]], [[Gonzalo Reyes]], and [[Gavin Wraith]].

Studies of the properties of $C^\infty$-rings include

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[RingsOfSmoothFunctionsI.pdf:file]]_ , Journal of Algebra 99 (1986)

The notion of the spectrum of a $C^\infty$-ring and that of $C^\infty$-[[schemes]] is discussed in

* [[Eduardo Dubuc]], _$C^\infty$-schemes_ Amer. J. Math. 103 (1981) ([pdf](http://www.math.ist.utl.pt/~jroquet/Dubuc.pdf) [JSTOR](http://www.jstor.org/stable/select/2374046)).

and more generally in

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _Rings of smooth functions and their localization II_ , in _Mathematical logic and theoretical computer science_ page 275 ([Google books](http://books.google.nl/books?id=w02XaOa_0HAC&pg=PA262&lpg=PA262&dq=smooth+zariski+topos&source=bl&ots=Hj0AN3Lz3y&sig=TqRwTXz_u5R30AN2udLNOMQ2Ly0&hl=nl&ei=mCWZS-eLJcaD-QalyZlS&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=smooth%20zariski%20topos&f=false))

* [[Marta Bunge]], [[Eduardo Dubuc]], _Archimedian local $C^\infty$-rings and models of synthetic differential geometry_ Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 3 (1986), p. 3-22 ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0)).
 {#BungeDubuc}

More recent developments along these lines are in

* [[Dominic Joyce]], _Algebraic geometry over $C^\infty$-rings_ ([arXiv:1001.0023](http://arxiv.org/abs/1001.0023))

The [[higher geometry]] generalization to a theory of [[derived smooth manifold]]s -- [[space]]s with [[structure sheaf]] taking values in [[simplicial C∞-ring]]s -- was initiated in

* [[David Spivak]], _Quasi-Smooth derived manifolds_
  ([pdf of original version](http://www.uoregon.edu/~dspivak/thesis2.pdf), [arXiv:0810.5174](http://arxiv.org/abs/0810.5174))

based on the general machinery of [[structured (∞,1)-topos]]es in

* [[Jacob Lurie]], _[[Structured Spaces]]_

where this is briefly mentioned in the very last paragraph.

See also the references at [[Fermat theory]], of which $C^\infty$-rings are a special case. And the references at [[smooth locus]], the formal dual of a $C^\infty$-ring.
And the references at [[super smooth topos]], which involves generalizations of $C^\infty$-rings to [[supergeometry]].

[[!redirects smooth algebra]]
[[!redirects smooth algebras]]
[[!redirects generalized smooth algebra]]
[[!redirects generalized smooth algebras]]
[[!redirects C-infinity ring]]
[[!redirects C-infinity-ring]]
[[!redirects C-infinity rings]]
[[!redirects C-infinity-rings]]
[[!redirects C-∞-ring]]
[[!redirects C-∞ ring]]
[[!redirects C-∞-rings]]
[[!redirects C-∞ rings]]
[[!redirects C^∞-ring]]
[[!redirects C^∞ ring]]
[[!redirects C^∞-rings]]
[[!redirects C^∞ rings]]


C_∞-algebra

[[!redirects C-infinity algebra]]

> $\mathrm{C}_\infty$-algebras should not be confused with [[C-infinity ring|$\mathrm{C}^\infty$-rings]].

***

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

#Contents#
* table of contents
{:toc}

## Idea

A _$C_\infty$-algebra_ is a [[commutative ring]] where the commutativity is relaxed up to [[coherence|coherent]] higher [[homotopy]].

## Related concepts

* [[A-∞ algebra]]

* [[E-∞ algebra]]

* [[L-∞ algebra]]

[[!redirects C-∞ algebra]]


Cabibbo anomaly

#Contents#
* table of contents
{:toc}

## Idea

(...)

apparent observed violation of [[unitary matrix|unitarity]] of the [[CKM matrix]]

(...)

## Statistical significance

[[statistical significance]]:

$ \gt 4\sigma$ ([Belfatto-Beradze-Berezhiani 19](#BelfattoBeradzeBerezhiani19))

## Related concepts

* [[flavor problem]]

* [[flavour anomaly]]

* [[(g-2) anomaly]]

* [[V_cb puzzle]]

## References

* {#BelfattoBeradzeBerezhiani19} Benedetta Belfatto, Revaz Beradze, Zurab Berezhiani, _The CKM unitarity problem: A trace of new physics at the TeV scale?_, Eur. Phys. J. C 80, 149 (2020)
([arXi:1906.02714](https://arxiv.org/abs/1906.02714), [arXiv:10.1140/epjc/s10052-020-7691-6](https://doi.org/10.1140/epjc/s10052-020-7691-6))

* Yuval Grossman, Emilie Passemar, Stefan Schacht, _On the Statistical Treatment of the Cabibbo Angle Anomaly_ ([arXiv:1911.07821](https://arxiv.org/abs/1911.07821))

* Matthew Kirk, _Cabibbo anomaly versus electroweak precision tests: An exploration of extensions of the Standard Model_, Phys. Rev. D 103, 035004 (2021) ([arXiv:2008.03261](https://arxiv.org/abs/2008.03261))

A [[leptoquark]] model meant to address all of the [[flavour anomalies]], the [[(g-2)-anomaly]] and the [[Cabibbo anomaly]] at once:

* {#MarzoccaTrifinopoulos21} [[David Marzocca]], Sokratis Trifinopoulos, *A Minimal Explanation of Flavour Anomalies: B-Meson Decays, Muon Magnetic Moment, and the Cabbibo Angle* ([arXiv:2104.05730](https://arxiv.org/abs/2104.05730))

See also:

* [[Andreas Crivellin]], *Explaining the Cabibbo Angle Anomaly* [[arXiv:2207.02507](https://arxiv.org/abs/2207.02507)]

[[!redirects Cabibbo angle anomaly]]

[[!redirects CKM unitarity problem]]


cactus language

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview).

[[!redirects Cactus Language]]


Cadabra

Cadabra is a mathematical computer software used for calculations with tensors, especially in high energy physics.

* [overview](http://www.swmath.org/software/97) at swMath.org

category: software


Cahier topos

see at _[[Cahiers topos]]_


Cahiers

__Cahiers de Topologie et Géométrie Différentielle Catégoriques__ (often referred simply as Cahiers) is one of the main journals primarily dedicated to category theory (see also [[TAC]]). It was created by the late [[Charles Ehresmann]] in 1958 and is presently edited by [[Andrée Charles Ehresmann]].

* the official site of the journal: [CahiersTGDC](http://cahierstgdc.com/)
* free issues 1966-2008 at [numdam](http://www.numdam.org/numdam-bin/feuilleter?j=CTGDC&sl=0)
* [indices of recent years](http://www.tac.mta.ca/tac/cahiers/index.html) at the _[[Theory and Applications of Categories]]_ website.

See also [[Cahiers topos]] which owes its name to this journal.

## Related entries

* [[list of journals publishing category theory]]

[[!redirects Cahiers de Topologie et Géométrie Différentielle Catégoriques]]

[[!redirects Cahiers de topologie et géométrie différentielle]]


Cahiers topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Cohesive toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Cahier topos_ is a [[cohesive topos]] that constitutes a [[Models for Smooth Infinitesimal Analysis|well-adapted model]] for [[synthetic differential geometry]] (a "[[smooth topos]]").

It is the [[sheaf topos]] on the [[site]] [[FormalCartSp]] of [[infinitesimal object|infinitesimally thickened]] [[Cartesian spaces]].

## Definition

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

Let [[FormalCartSp]] be the [[full subcategory]] of the category of [[smooth loci]] on those of the form

$$
  \mathbb{R}^n \times \ell W
  \,,
$$

consisting of a [[product]] of a [[Cartesian space]] with an [[infinitesimally thickened point]], i.e. a formal dual of a _Weil algebra_ .

Dually, the [[opposite category]] is the [[full subcategory]] $FormalCartSp^{op} \hookrightarrow SmoothAlg$ of [[smooth algebras]] on those of the form

$$
  C^\infty( \mathbb{R}^k \times \ell W)
  =
  C^\infty(\mathbb{R}^k) \otimes W
  \,.
$$

=--

This appears for instance in [Kock Reyes (1)](#KockReyes).

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

Define a structure of a [[site]] on [[FormalCartSp]] by declaring a [[covering]] family to be a family of the form

$$
  \{
    U_i \times \ell W \stackrel{p_i \times Id}{\to} U \times \ell W
  \}
$$

where $\{U_i \stackrel{p_i}{\to} U\}$ is an [[open cover]] of the [[Cartesian space]] $U$ by Cartesian spaces $U_i$.

=--

This appears as [Kock (5.1)](#Kock).

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

The **Cahiers topos** $\mathcal{CT}$ is the [[category of sheaves]] on this site:

$$
  \mathcal{CT} := Sh(FormalCartSp)
  \,.
$$

=--

This site of definition appears in [Kock, Reyes](#KockReyes). The original definition is due to [Dubuc 79](#Dubuc79)

## Properties

### Synthetic differential geometry

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

The Cahiers topos is a well-adapted model for [[synthetic differential geometry]].

=--

This is due to [Dubuc 79](#Dubuc79).

### Connectedness, locality and cohesion

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

The Cahiers topos is a [[cohesive topos]]. See [[synthetic differential infinity-groupoid]] for details.

=--

### Convenient vector spaces
 {#ConvenientVectorSpaces}

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

The [[category]] of [[convenient vector spaces]] with [[smooth functions]] between them embeds as a [[full subcategory]] into the Cahiers topos.

The embedding is given by sending a convenient vector space $V$ to the sheaf given by

$$
  V
  :
  \mathbb{R}^k \times \ell W
  \mapsto
  C^\infty(\mathbb{R}^k, V) \otimes W
  \,.
$$

=--

This result was announced in [Kock](#Kock). See the corrected proof in ([KockReyes](#KockReyes)).

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

Together with prop. \ref{ModelForSynthetic}
this means that the [[differential geometry]] on
convenient vector spaces may be treated synthetically
in the Cahiers topos.

=--

### Synthetic tangent bundles of smooth spaces

### Synthetic tangent spaces
 {#RelationToSyntheticTangentSpaces}

We discuss here induced [[synthetic tangent spaces]] of [[smooth spaces]] in the sense of [[diffeological spaces]] and more general [[sheaves]] on the site of [[smooth manifolds]] after their canonical embedding into the Cahiers topos.

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

Write $SmoothLoc$ for the [[category]] of [[smooth loci]]. Write

$$
  CartSp \hookrightarrow SmoothLoc
$$

for the [[full subcategory]] on the [[Cartesian spaces]] $\mathbb{R}^n$ ($n \in \mathbb{N}$). Write

$$
  InfThPoint \hookrightarrow SmoothLoc
$$

for the [[full subcategory]] on the [[infinitesimally thickened points]], and write

$$
  CartSp_{synthdiff} \hookrightarrow SmoothLoc
$$

for the full subcategory on those smooth loci which are the [[cartesian product]] of a [[Cartesian space]] $\mathbb{R}^n$ ($n \in \mathbb{N}$) and an [[infinitesimally thickened point]].

We regard [[CartSp]] as a [[site]] by equipping it with the [[good open cover]] [[coverage]]. We regard $InfThPoint$ as equipped with the trivial coverage and $CartSp_{synthdiff}$ as equipped with the induced product coverage.

The [[sheaf topos]] $Sh(CartSp)$ is that of _[[smooth spaces]]_. The sheaf topos $Sh(CartSp_{synthdiff})$ is the _Cahier topos_.

=--

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

We write

$$
  D \coloneqq \ell(\mathbb{R}[\epsilon]/(\epsilon^2))
  \in
  InfThPt \hookrightarrow CartSp_{synthdiff}
$$

for the [[infinitesimal space|infinitesimal interval]], the [[smooth locus]] dual to the [[smooth algebra]] "[[ring of dual numbers|of dual numbers]]".

=--

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

For $X \in Sh(CartSp_{synthdiff})$ any object in the Cahier topos, its **[[synthetic tangent bundle]]** in the sense of [[synthetic differential geometry]] is the [[internal hom]] space $X^D$, equipped with the projection map

$$
  X(\ast \to D) \colon X^D \to X
  \,.
$$

=--

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

The canonical inclusion functor $i \colon CartSp \to CartSp_{synthdiff}$ induces an [[adjoint pair]]

$$
  Sh(CartSp)
   \stackrel{\overset{i_!}{\to}}{\underset{i^\ast}{\leftarrow}}
  Sh(CartSp_{synthdiff})
$$

where $i^\ast$ is given by precomposing a [[presheaf]] on $CartSp_{synthdiff}$ with $i$. The [[left adjoint]] $i_!$ has the interpretation of the inclusion of [[smooth spaces]] as [[reduced objects]] in the Cahiers topos.

=--

This is discussed in more detail at _[[synthetic differential infinity-groupoid]]_.

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

For $X \in Sh(CartSp)$ a [[smooth space]], and for $\ell(W) \in InfThPoint$ an infinitesimally thickened point, the morphisms

$$
  \ell(W) \to i_! X
$$

in $Sh(CartSp_{synthdiff})$ are in natural [[bijection]] to [[equivalence classes]] of pairs of morphisms

$$
  \ell(W) \to \mathbb{R}^n \to X
$$

consisting of a morphism in $CartSp_{synth}$ on the left and a morphism in $Sh(CartSp)$ on the right (which live in different categories and hence are not composable, but usefully written in juxtaposition anyway). The [[equivalence relation]] relates two such pairs if there is a [[smooth function]] $\phi \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ such that in the diagram

$$
  \array{
    && \mathbb{R}^n
    \\
    & \nearrow & & \searrow
    \\
    \ell(W) && \downarrow^{\mathrlap{\phi}} && X
    \\
    & \searrow && \nearrow
    \\
    && \mathbb{R}^{n'}
  }
$$

the left triangle commutes in $CartSp_{synthdiff}$ and the right one in $Sh(CartSp)$.

=--

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

By general properties of [[left adjoints]] of functors of [[presheaves]], $i_! X$ is the [[left Kan extension]] of the presheaf $X$ along $i$. By the [[Yoneda lemma]] and the [[coend]] formula for these (as discussed there), we have that the set of maps $\ell(W) \to i_! X$ is naturally identified with

$$
  (i_! X)(\ell(W))
  =
  (Lan_i X)(\ell(W))
  =
  \int^{\mathbb{R}^n \in CartSp}
  Hom_{CartSp_{synthdiff}}(\ell(W), \mathbb{R}^n)
  \times
  X(\mathbb{R}^n)
  \,.
$$

Unwinding the definition of this [[coend]] as a [[coequalizer]] yields the above description of equivalence classes.

=--

## Variants and generalizations

* When restricting the [[site]] of infinitesimally thickened Cartesian spaces to that of plain [[Cartesian spaces]] one obtaines the topos discussed at _[[smooth space]]_. This is still a [[cohesive topos]], but no longer a model for [[synthetic differential geometry]].

* The [[(∞,1)-sheaf (∞,1)-topos]] over $CartSp_{th}$ is disucssed at [[synthetic differential ∞-groupoid]]. It contains that Cahiers topos as the sub-[[(n,1)-topos|(1,1)-topos]] of [[0-truncated]] objects.

* [[Dubuc topos]]

## References

The Cahiers topos was introduced in

* {#Dubuc79} [[Eduardo Dubuc]], *Sur les modèles de la géométrie différentielle synthétique*, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **20**  3 (1979) 231-279  [[numdam:CTGDC_1979__20_3_231_0](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)]

and got its name from this journal publication. The definition appears in theorem 4.10 there, which asserts that it is a well-adapted model for [[synthetic differential geometry]].

A review discussion is in of

* {#Kock} [[Anders Kock]], Section 5 of _Convenient vector spaces embed into the Cahiers topos_, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]], **27** 1 (1986) 3-17  [[numdam:CTGDC_1986__27_1_3_0](http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0)]

and with a corrected definition of the site of definition in

* {#KockReyes} [[Anders Kock]], [[Gonzalo Reyes]], _Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos_, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]], **28** 2 (1987)  99-110 [[numdam:CTGDC_1987__28_2_99_0](http://www.numdam.org/item?id=CTGDC_1987__28_2_99_0)]

where it is first observed (but apparently without proof) that [[ThCartSp]] is still a site of definition for the Cahiers topos (cf. [[formal smooth set]]).

With its original site of all [[formal smooth manifolds]], the Cahiers topos also appears briefly mentioned in:

*  {#Kock81} [[Anders Kock]], ex. 2 on p. 191 of: _Synthetic Differential Geometry_, Cambridge University Press (1981, 2006) [[pdf](http://home.imf.au.dk/kock/sdg99.pdf), [doi:10.1017/CBO9780511550812](https://doi.org/10.1017/CBO9780511550812)]

With an eye towards [[Frölicher space]]s the site is also considered in section 5 of

* Hirokazu Nishimura, _Beyond the Regnant Philosophy of Manifolds_ ([arXiv:0912.0827](http://arxiv.org/abs/0912.0827))

The [[(∞,1)-topos]] analog of the Cahiers topos ([[synthetic differential ∞-groupoids]]) is discussed in section 3.4 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_


Calabi-Yau algebra

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Calabi-Yau algebra_ is an algebraic incarnation of the notion of _[[Calabi-Yau manifold]]_  and [[higher algebra]]-version of the notion of [[Frobenius algebra]].

## Definition

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

For $A$ a [[dg-algebra]] and $N$ a [[dg-module|dg-bimodule]] over $A$, write

$$
  N^! := RHom_{A Bimod}(N, A \otimes A)
$$

for the dual $A$-[[bimodule]], where $RHom$ denotes the [[derived functor|right derived]] [[hom-functor]] with respect to the [[model structure on dg-modules]].

=--

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

A homologically smooth [[dg-algebra]] $A$ is a **Calabi-Yau algebra** of [[dimension]] $d$ if there is a [[quasi-isomorphism]] of $A$-[[bimodule]]s

$$
 f :  A \stackrel{\simeq}{\to} A^![d]
$$

such that

$$
  f \simeq f^![d]
  \,.
$$

=--

This is ([Ginzburg, def. 3.2.3](#Ginzburg)).

## Properties

### General

Let $X$ be a [[smooth scheme|smooth]] [[quasi-projective variety]]. Write $D^b(Coh X)$ for the [[derived category]] of bounded [[chain complex]]es of [[coherent sheaves]] over $X$.

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

An [[object]] $\mathcal{E} \in D^b(Coh X)$ is called a **tilting generator** if the [[Ext]]-functor satisfies

1. $Ext^i(\mathcal{E}, \mathcal{E}) = 0$ for all $i \gt 0$;

1. $Ext^\bullet(\mathcal{E},\mathcal{F}) = 0$ implies $\mathcal{F} = 0$;

1. the [[endomorphism]] algebra $End(\mathcal{E}) = Hom(\mathcal{E},\mathcal{E})$ has finite [[Hochschild cohomology|Hochschild]] [[dimension]].

=--

This appears as ([Ginzburg, def. 7.1.1](#Ginzburg)).

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

For $\mathcal{E}$ a tilting generator there is an [[equivalence of categories|equivalence]] of [[triangulated categories]]

$$
  D^b(Coh X) \stackrel{\simeq}{\to}  D^b(End(\mathcal{E})Mod)
$$

to the [[derived category]] of [[module]]s over $End(\mathcal{E})$.

=--

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

For $X$ smooth connected variety which is projective over an affine variety, let $\mathcal{E} in D^b(Coh X)$ be a tilting generator, def. \ref{TiltingGenerator}.

Then $End \mathcal{E}$ is a Calabi-Yau algebra of dimension $d$ precisely if $X$ is a [[Calabi-Yau manifold]] of dimension $d$.

=--

This appears as ([Ginzburg, prop. 3.3.1](#Ginzburg)).

#### $0$-Calabi-Yau algebras

A cochain dg-algebra over $k$ is $0$-Calabi-Yau iff it is Koszul and $Tor^0_A(k_A,{}_A k)$ is a symmetric coalgebra.  Proven in

*  J.-W. He, X.-F. Mao, _Connected cochain DG algebras of Calabi-Yau dimension 0_, Proc. Amer. Math. Soc. __145__ (2017) 937--953 [doi](https://doi.org/10.1090/proc/13081)

It follows that a Koszul dg-algebra is $0$-Calabi-Yau iff its Ext-algebra is symmetric [[Frobenius algebra|Frobenius]].

### Relation to 2d extended TQFT and the Cobordism hypothesis

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

Let $\mathbf{S}$ be a [[good symmetric monoidal (∞,1)-category|good]] [[symmetric monoidal (∞,1)-category]]. Write $Alg(\mathbf{S})$ for the [[symmetric monoidal (∞,n)-category|symmetric monoidal (∞,2)-category]] whose [[object]]s are [[algebra in an (∞,1)-category|algebra objects]] in $\mathbf{S}$ and whose [[morphisms]] are [[bimodule]] objects.

Then a Calabi-Yau object in $Alg(\mathbf{S})$ is an algebra object $A$ equipped with an $SO(2)$-equivariant morphism

$$
  tr \colon \int_{S^1} A \to 1
$$

from the [[Hochschild homology]] $\int_{S^1} A \simeq A \otimes_{A \otimes A} A$, satisfying the condition that the composite morphism

$$
  A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1
$$

exhibits $A$ as its own [[dual object]] $A^\vee$.

Such an algebra object is called a _[[Calabi-Yau algebra]] object_.

=--

This is ([Lurie 09, example 4.2.8](#Lurie09)).

### Classification of 2d TQFT

[[!include 2d TQFT -- table]]

## Related concepts

* [[Calabi-Yau object]]

  * **Calabi-Yau algebra**, [[Calabi-Yau manifold]]

  * [[Calabi-Yau category]]
* [[pre-Calabi-Yau algebra]]

## References

* {#Ginzburg} [[Victor Ginzburg]], _Calabi-Yau algebras_ ([arXiv:0612139](http://arxiv.org/abs/math/0612139))

* {#Lurie09} [[Jacob Lurie]], section 4.2 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

[[!redirects Calabi-Yau algebras]]

[[!redirects Calabi-Yau A-∞ algebra]]
[[!redirects Calabi-Yau A-∞ algebras]]


Calabi-Yau category

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _Calabi-Yau category_ is a [[horizontal categorification]] of that of [[Frobenius algebra]] -- a _Frobenius [[algebroid]]_. Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the [[derived category]] of [[coherent sheaves]] on a [[Calabi-Yau manifold]].

## Definition

### 1-categorical

A **Calabi-Yau category** is a [[Vect]]-[[enriched category]] $C$ equipped for each object $c \in C$ with a [[trace]]-like map

$$
  Tr_C : C(c,c) \to k
$$

to the ground field, such that for all objects $d \in C$ the induced pairing

$$
  \langle -,-\rangle_{c,d} :
  C(c,d) \otimes C(d,c) \to k
$$

given by

$$
  \langle f,g \rangle = Tr(g \circ f)
$$

is symmetric and non-degenerate.

[Question: this 1-categorical definition seems to allow for different
Frobenius structures on the endomorphism algebras of isomorphic objects.
Would it be better to define it as a dinatural transformation from
the Hom-functor to the constant functor with value the ground field $k$?]

A Calabi-Yau category with a single object is the same (or rather: is equivalently) the [[pointed object|pointed]] [[monoid]]  [[delooping]] of a [[Frobenius algebra]].

### $(\infty,1)$-categorical

A **Calabi-Yau $A_\infty$-category** of _dimension_ $d \in \mathbb{N}$ is an [[A-∞ category]] $C$ equipped with, for each pair $a,b$ of [[object]]s, a morphism of [[chain complex]]es

$$
  \langle -,-\rangle_{a,b}
  :
  C(a,b) \otimes C(b,a)
  \to
  k[d]
$$

such that

1. this is non-degenerate and is symmetric in that

   $$
    \langle - , - \rangle_{a,b} =  \langle - , - \rangle_{b,a} \circ \sigma_{a,b}
   $$

   for $\sigma_{a,b} : C(a,b)\otimes C(b,a) \to C(b,a) \otimes C(a,b)$
   the symmetry [[isomorphism]] of the [[symmetric monoidal category]]
   of [[chain complex]]es;

1. this is cyclically invariant in that for all elements $(\alpha_i)$
   is the respective hom-complexes we have

   $$
     \langle
       m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}),
       \alpha_{n-1}
     \rangle
     =
     (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|}
     \langle
       m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}),
       \alpha_0
     \rangle
     \,.
   $$

## Examples

### From Calabi-Yau varieties
 {#FromCYVarieties}

* Let $X$ be a smooth projective [[Calabi-Yau variety]] of dimension $d$.
  Write $D^b(X)$ for the bounded [[derived category]] of
  that of [[coherent sheaves]] on $X$.

  Then $D^b(X)$ is a CY $A_\infty$-category in a naive way:

  * the non-binary composition maps are all trivial;

  * the pairing is given by [[Serre duality]] (one needs also a choice of trivialization of the canonical bundle of $X$)

This is however not the morally correct CY $A_\infty$-structure associated with a Calabi-Yau. A correct choice is, for example, the [[Dolbeault cohomology|Dolbeault]] [[enhanced triangulated category|dg-enhancement]] of the derived category

([Costello 04, 7.2](#Costello04), [Costello 05, 2.2](#Costello05))

### From symplectic manifolds

The [[Fukaya category]] associated with a [[symplectic manifold]] $X$. But see [this MO discussion](http://mathoverflow.net/questions/13114/are-fukaya-categories-calabi-yau-categories) for more.

### From string topology

[[string topology]]: for $X$ a [[compact space|compact]] [[simply connected]] [[orientation|oriented]] [[manifold]], its cohomology $H^{\bullet}(X)$ is naturally a Calabi-Yau $A_\infty$-category with a single object. The $A_\infty$ structure comes from the [[homological perturbation lemma]]. One could also use the dg algebra of cochains $C^\bullet(X)$.

## Properties

### Classification of 2d TQFT

Calabi-Yau $A_\infty$-categories classify [[TCFT]]s. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the [[CFT]]s constructed from [[sigma-model]]s whose targets are [[Calabi-Yau spaces]].

[[!include 2d TQFT -- table]]

## Related concepts

* [[Calabi-Yau object]], [[Calabi-Yau algebra]], [[pre-Calabi-Yau algebra]]

## References

* {#Costello04} [[Kevin Costello]], _Topological conformal field theories and Calabi-Yau categories_ ([arXiv:math/0412149](https://arxiv.org/abs/math/0412149))

* {#Costello05} [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([arXiv:0509264](https://arxiv.org/abs/math/0509264))

* [[Maxim Kontsevich]], [[Yan Soibelman]]. _Notes on A-infinity algebras, A-infinity categories and non-commutative geometry_, [arXiv:math/0606241](https://arxiv.org/abs/math/0606241)

* Lee Cho, _Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras_ ([arXiv:1002.3653](http://arxiv.org/abs/1002.3653))

* {#Lurie09} [[Jacob Lurie]], section 4.2 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](https://arxiv.org/abs/0905.0465))

A relative version is defined for functors instead of categories,

* Christopher Brav , Tobias Dyckerhoff, _Relative Calabi–Yau structures_, Compositio Math. 155 (2019) 372--412 [doi](https://doi.org/10.1112/S0010437X1900702); _Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects_, Sel. Math. New Ser. 27, 63 (2021). [doi](https://doi.org/10.1007/s00029-021-00642-5)
* [[Bernhard Keller]], Yu Wang, _An introduction to relative Calabi-Yau structures_, [arXiv:2111.10771](https://arxiv.org/abs/2111.10771)

[[!redirects Calabi-Yau category]]
[[!redirects Calabi-Yau categories]]
[[!redirects Calabi–Yau category]]
[[!redirects Calabi–Yau categories]]
[[!redirects Calabi--Yau category]]
[[!redirects Calabi--Yau categories]]

[[!redirects CY category]]
[[!redirects CY categories]]
[[!redirects CY-category]]
[[!redirects CY-categories]]

[[!redirects Calabi-Yau A-infinity category]]
[[!redirects Calabi-Yau A-infinity categories]]
[[!redirects Calabi-Yau A-infinity-category]]
[[!redirects Calabi-Yau A-infinity-categories]]
[[!redirects Calabi-Yau A-∞ category]]
[[!redirects Calabi-Yau A-∞ categories]]
[[!redirects Calabi-Yau A-∞-category]]
[[!redirects Calabi-Yau A-∞-categories]]


Calabi-Yau cohomology

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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Calabi-Yau cohomology_ is [[complex oriented cohomology theory|complex oriented]] [[generalized cohomology theory]] whose associated [[formal group]] is an [[Artin-Mazur formal group]] $\Phi^n_X$ of a [[Calabi-Yau variety]] $X$ of [[dimension]] $n$. (Which means e.g. [[complex manifold|complex]] dimension if working over the [[complex numbers]]).

As special cases this includes:

* for $n = 0$: [[complex K-theory]];

* for $n = 1$: [[elliptic cohomology]];

* for $n = 2$: [[K3-cohomology]].

[[!include moduli of higher lines -- table]]

## Related concepts

* [[moduli space of Calabi-Yau spaces]]

## References

The general idea of [[Calabi-Yau cohomology]] apparently appears in

* {#Hopkins92} [[Michael Hopkins]], lectures at Midwest Topology Seminar, 1992

The suggestion that from the point of view of [[string theory]]/[[F-theory]] K3-cohomology, and more generally Calabi-Yau cohomology, is the required generalization of [[elliptic cohomology]] appears in

* {#Sati05} [[Hisham Sati]], p. 18 of _The Elliptic curves in gauge theory, string theory, and cohomology_, JHEP 0603 (2006) 096 ([arXiv:hep-th/0511087](http://arxiv.org/abs/hep-th/0511087))

See also:

* {#Nogami10} Jumpei Nogami, _On derived Calabi-Yau varieties_, University of Illinois at Chicago 2010 ([proquest](https://search.proquest.com/docview/814798658))

For more see the references at _[[K3-cohomology]]_.


Calabi-Yau manifolds in SU-bordism theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Since [[Calabi-Yau manifolds]] have, in particular [[special unitary group|SU]]-[[G-structure|structure]], they represent classes in [[SU-bordism theory]]. In fact, _all_ non-[[torsion element|torsion]] $SU$-bordism classes are represented by products of linear combinations of classes of  Calabi-Yau manifolds. Those in degree $4$ are spanned by the [[K3-surface]].

## Preliminaries

We write $\Omega^{SU}_\bullet$ for the [[SU-bordism ring]].

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

The [[kernel]] of the [[forgetful functor|forgetful]] morphism

$$
  \Omega^{SU}_\bullet
  \longrightarrow
  \Omega^{\mathrm{U}}_\bullet
$$

from the [[SU-bordism ring]] to the [[complex bordism ring]], is pure [[torsion subgroup|torsion]].

=--

([CLP 19, Thm. 5.8a](#CLP19))

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

The [[torsion subgroup]] of the [[SU-bordism ring]] is concentrated in degrees $8k+1$ and $8k+2$, for $k \in \mathbb{N}$.

=--

([CLP 19, Thm. 5.11a](#CLP19))

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

Every [[torsion element]] in the [[SU-bordism ring]] $\Omega^{SU}_\bullet$ has [[order of an element|order]] 2.

=--

([CLP 19, Thm. 5.8b](#CLP19))

+-- {: .num_prop #SUBordismRingAwayFromTwo}
###### Proposition
**([[SU-bordism ring]] [[localization of a ring|away from 2]] is [[polynomial algebra]])**

The [[SU-bordism ring]] with [[localization of a ring|2 inverted]] is the [[polynomial algebra]] over $\mathbb{Z}\big[\tfrac{1}{2}\big]$ on one generator in every even degree $\geq 4$:

$$
  \Omega^{SU}\big[\tfrac{1}{2}\big]
  \;\simeq\;
  \mathbb{Z}
  \big[
    \tfrac{1}{2}
  \big]
  \big[
    \{
      y_{2i+4}
    \}_{i \in \mathbb{N}}
  \big]
  \,.
$$

=--

(due to [Novikov 62](MSU#Novikov62), review in [LLP 17, Thm. 1.2](#LLP17))

## Statement

+-- {: .num_prop #K3SurfaceSpansSUBordismRingInDegree4}
###### Proposition
**([[K3-surface]] [[linear span|spans]] [[SU-bordism ring]] in degree 4)**

The degree-4 generator $y_4 \in \Omega^{SU}_4$ in the [[SU-bordism ring]] (Prop. \ref{SUBordismRingAwayFromTwo}) is represented by minus the class of any (non-[[torus]]) [[K3-surface]]:

$$
  \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle
  \,.
$$

=--

([LLP 17, Lemma 1.5, Example 3.1](#LLP17), [CLP 19, Theorem 13.5a](#CLP19))

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] generate the [[SU-bordism ring]] [[localization of a ring|away from 2]])**

The [[SU-bordism ring]] [[localization of a ring|away from 2]] is multiplicatively generated by [[Calabi-Yau manifolds]].

=--

([LLP 17, Theorem 2.4](#LLP17))

In particular:

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] in complex dim $\leq 4$ generating the [[SU-bordism ring]] in $deg \leq 8$ [[localization of a ring|away from 2]])**

There are [[Calabi-Yau manifolds]] of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. \ref{SUBordismRingAwayFromTwo}.

=--

([CLP 19, Theorem 13.5b,c](#CLP19))

## References

* {#LLP17} [[Ivan Limonchenko]], [[Zhi Lu]], [[Taras Panov]], _Calabi-Yau hypersurfaces and SU-bordism_, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 ([arXiv:1712.07350](https://arxiv.org/abs/1712.07350))

Survey:

* {#CLP19} Georgy Chernykh, [[Ivan Limonchenko]], [[Taras Panov]], _$SU$-bordism: structure results and geometric representatives_, Russian Math. Surveys 74 (2019), no. 3, 461-524 ([arXiv:1903.07178](https://arxiv.org/abs/1903.07178))

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


Calabi-Yau object

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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

See at _[cobordism hypothesis -- For non-compact cobordisms](cobordism+hypothesis#ForNoncompactCobordisms)_.

## Definition

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

For $C$ a [[symmetric monoidal (infinity,n)-category|symmetric monoidal (infinity,2)-category]], a **Calabi-Yau object** in $C$ is

* a [[dualizable object]] $X$;

* a morphism $\eta\colon\operatorname{dim}(X) \equiv ev_X \circ coev_X \longrightarrow Id_x$ in $\Omega_x C$ which is [[equivariance|equivariant]] with respect to [the canonical](cobordism%20hypothesis#TheCanonicalOnAction) [[∞-action]] of the [[circle group]] $SO(2)$ on $dim(X)$ and which is the [[unit of an adjunction|counit]] for an [[adjunction]] between the [[evaluation map]] $ev_X$ and [[coevaluation map]] $coev_X$.

=--

This is ([Lurie 09, def. 4.2.6](#Lurie09)).

## Examples

### Calabi-Yau algebras
 {#CYAlgebra}

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

Let $\mathbf{S}$ be a [[good monoidal (∞,1)-category|good]] [[symmetric monoidal (∞,1)-category]]. Write $Alg(\mathbf{S})$ for the [[symmetric monoidal (∞,n)-category|symmetric monoidal (∞,2)-category]] whose [[object]]s are [[algebra in an (∞,1)-category|algebra objects]] in $\mathbf{S}$ and whose [[morphisms]] are [[bimodule]] objects.

Then a Calabi-Yau object in $Alg(\mathbf{S})$ is an algebra object $A$ equipped with an $SO(2)$-equivariant morphism

$$
  tr \colon \int_{S^1} A \to 1
$$

from the [[Hochschild homology]] $\int_{S^1} A \simeq A \otimes_{A \otimes A} A$, satisfying the condition that the composite morphism

$$
  A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1
$$

exhibits $A$ as its own [[dual object]] $A^\vee$.

Such an algebra object is called a _[[Calabi-Yau algebra]] object_.

=--

This is ([Lurie 09, example 4.2.8](#Lurie09)).

## Properties

### Relation to extended 2d TQFT (TCFT) and the Cobordism hypothesis

A version of the [[cobordism hypothesis]] says that symmetric monoidal $(\infty,2)$-functors

$$
  Z : Bord_2^{nc} \to \mathcal{C}
$$

out of a version of the [[(infinity,n)-category of cobordisms|(infinity,2)-category of cobordisms]] where all 2-cobordisms have at least one outgoing (ingoing) boundary component, are equivalently given by their value on the point, which is a Calabi-Yau object in $\mathcal{C}$.

This is ([Lurie 09, theorem 4.2.11](#Lurie09)).

Here the trace condition translates to the [[cobordism]] which is the "disappearance of a circle".

$$
  \array{
     && \longleftarrow
    \\
    & \swarrow && \nwarrow
    \\
    & \searrow && \nearrow
    \\
    && \longrightarrow
  }
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
   \ast
$$

Its would-be [[adjoint]], the "appearance of a circle" is not included in $Bord_2^{nc}$.

This is closely related to the description of [[2d TQFT]] as [[TCFT]]s ([Lurie 09, theorem 4.2.13](#Lurie09)).

[[!include 2d TQFT -- table]]

## Related concepts

* [[Calabi-Yau category]]

* [[3d Calabi-Yau object]]

## References

* {#Lurie09} [[Jacob Lurie]], section 4.2 of _[[On the Classification of Topological Field Theories]]_ ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465))

[[!redirects Calabi-Yau objects]]


Calabi-Yau variety

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An $n$-dimensional _Calabi-Yau variety_ is an $n$-dimensional [[Kähler manifold]] with ([[holomorphic line bundle|holomorphically]], rather than just topologically) trivial [[canonical bundle]]. This is equivalent to saying that it is real [[Riemannian manifold]] of even [[dimension]] $2 N$ which has [[special holonomy]] in the [[subgroup]] $SU(N)\subset O(2 N, \mathbb{R})$.

For [[compact topological space|compact]] [[Kähler manifolds]], [[Yau's theorem]] (also known as the  [[Calabi conjecture]]) states any of the above conditions implies the vanishing of the first [[Chern class]].

+--{.query}
Is it also true for non-compact?
=--

Note that $c_1(X) = 0$ implies in general that the canonical bundle is *topologically trivial*. But if $X$ is a simply connected compact Kähler manifold, $c_1(X) = 0$ implies further that the canonical bundle is *holomorphically trivial*.

+--{.query}
The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?
=--

## Definition

A Calabi-Yau variety can be described algebraically as a [[smooth scheme|smooth]] [[proper scheme|proper]] [[variety]] $X$ of [[dimension]] $n$ over a [[field]] $k$ (not necessarily [[algebraically closed field|algebraically closed]] and not necessarily of [[characteristic]] $0$) in which $\omega_X=\wedge^n\Omega^1\simeq \mathcal{O}_X$ and also $H^j(X, \mathcal{O}_X)=0$ for all $1\leq j \leq n-1$.

If the base field is $\mathbb{C}$, then one can form the analyticification of $X$ and obtain a [[compact space|compact]] [[manifold]] that satisfies the first given definition.

Beware that there are slightly different (and inequivalent) definitions in use. Notably in some contexts only the trivialization of the [[canonical bundle]] is required, but not the vanishing of the $H^{0 \lt \bullet \lt n}(X,\mathcal{O}_X)$. To be explicit on this one sometimes speaks for emphasis of "strict" CY varieties when including this condition.

## Examples

* in [[dimension]] 1: an [[elliptic curve]] is a pointed Calabi-Yau $1$-fold.

* In [[dimension]] 2: [[K3 surface]].

## Properties

### In terms of $G$-structure

Calabi-Yau structure is equivalently [[integrability of G-structure|integrable]] [[G-structure]] for $G = $ [[SU(n)]].

Details are in [Prins 16, Prop. 1.3.2](#Prins16).
See also [Vezzoni 06, p. 24](#Vezzoni06).

### In SU-bordism theory
 {#InSUBordismTheory}

We discuss the classes of Calabi-Yau manifolds seen in [[SU-bordism theory]]. For more see _[[Calabi-Yau manifolds in SU-bordism theory]]_.

+-- {: .num_prop #K3SurfaceSpansSUBordismRingInDegree4}
###### Proposition
**([[K3-surface]] spans [[SU-bordism ring]] in degree 4)**

The degree-4 generator $y_4 \in \Omega^{SU}_4$ in the [[SU-bordism ring]] (Prop. \ref{SUBordismRingAwayFromTwo}) is represented by minus the class of any (non-[[torus]]) [[K3-surface]]:

$$
  \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle
  \,.
$$

=--

([LLP 17, Lemma 1.5, Example 3.1](#LLP17), [CLP 19, Theorem 13.5a](#CLP19))

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] generate the [[SU-bordism ring]] [[localization of a ring|away from 2]])**

The [[SU-bordism ring]] [[localization of a ring|away from 2]] is multiplicatively generated by [[Calabi-Yau manifolds]].

=--

([LLP 17, Theorem 2.4](#LLP17))

+-- {: .num_prop }
###### Proposition
**([[Calabi-Yau manifolds]] in complex dim $\leq 4$ generate the [[SU-bordism ring]] in $deg \leq 8$ [[localization of a ring|away from 2]])**

There are [[Calabi-Yau manifolds]] of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. \ref{SUBordismRingAwayFromTwo}.

=--

([CLP 19, Theorem 13.5b,c](#CLP19))

### Artin-Mazur formal group

Over an [[algebraically closed field]] of [[positive number|positive]] [[characteristic]] an $n$-dimensional Calabi-Yau variety $X$ has an [[Artin-Mazur formal group]] $\Phi^n_X$ which gives the [[deformation theory]] of the trivial [[line n-bundle]] over $X$.

See also ([Geer-Katsura 03](#GeerKatsura03)).

### As supersymmetric compactification spaces in string theory

* [[supersymmetry and Calabi-Yau manifolds]]

* [[heterotic string theory on CY3-manifolds]]

## Related concepts

[[!include special holonomy table]]

* [[Calabi-Yau object]]

  * [[Calabi-Yau algebra]], **Calabi-Yau manifold**, [[generalized Calabi-Yau manifold]]

* [[Calabi-Yau cohomology]]

* [[moduli space of Calabi-Yau spaces]]

* [[SU-cobordism theory]]

## References

The original articles:

* [[Shing-Tung Yau]], *On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation*, Communications on Pure and Applied Mathematics,
**31** 3 (1978) 339-411 [[doi:10.1002/cpa.3160310304](https://doi.org/10.1002/cpa.3160310304)]

Surveys and review:

* [[Shing-Tung Yau]], _[Calabi-Yau manifold](http://www.scholarpedia.org/article/Calabi-Yau_manifold)_, Scholarpedia **4** 86524 (2009) [[doi](https://doi.org/10.4249/scholarpedia.6524)]

Motivated from the relation between [[supersymmetry and Calabi-Yau manifolds]]:

* [[Tristan Hübsch]], _Calabi-Yau Manifolds -- A Bestiary for Physicists_, World Scientific 1992 ([doi:10.1142/1410](https://doi.org/10.1142/1410))

In terms of [[G-structure]]:

* {#Vezzoni06} Luigi Vezzoni, _The geometry of some special $SU(n)$-structure_, 2006 ([pdf](https://core.ac.uk/download/pdf/14699671.pdf), [[VezzoniSUnStructure.pdf:file]])

* {#Prins16} [[Daniël Prins]], Section 1.3 of: _On flux vacua, $SU(n)$-structures and generalised complex geometry_, Université Claude Bernard -- Lyon I, 2015.  ([arXiv:1602.05415](https://arxiv.org/abs/1602.05415), [tel:01280717](https://tel.archives-ouvertes.fr/tel-01280717))

Discussion of the case of [[positive number|positive]] [[characteristic]] includes

* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_ ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))

The following page collects information on Calabi-Yau manifolds with an eye to application in [[string theory]] (e.g. [[supersymmetry and Calabi-Yau manifolds]]):

* [[Sheldon Katz]], [[Rolf Schimmrigk]], Andreas Wißkirchen, _[CALABI-YAU](http://www.th.physik.uni-bonn.de/th/Supplements/cy.html)_

Discussion of the relation between the various shades of definitions includes

* MathOverflow: _[Calabi-Yau manifolds](http://mathoverflow.net/q/42707/381)_

Mathematical review of the relation to  [[quiver representations]] and [[mirror symmetry]] includes

* Yang-Hui He, _Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants_ ([arXiv:1611.09398](https://arxiv.org/abs/1611.09398))

Discussion of CYs in [[positive characteristic]] includes

* [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, I_ ([arXiv:hep-th/0012233](http://arxiv.org/abs/hep-th/0012233))

* [[Philip Candelas]], [[Xenia de la Ossa]], Fernando Rodriguez-Villegas, _Calabi-Yau Manifolds Over Finite Fields, II_ ([arXiv:hep-th/0402133](http://arxiv.org/abs/hep-th/0402133))

Discussion of [[Calabi-Yau orbifolds]]:

* [[Dominic Joyce]], _On the topology of desingularizations of Calabi-Yau orbifolds_ ([arXiv:math/9806146](https://arxiv.org/abs/math/9806146), [spire:485280](https://inspirehep.net/literature/485280))

* [[Dominic Joyce]], _Deforming Calabi-Yau orbifolds_, Asian Journal of Mathematics 3.4 (1999): 853-868 ([doi:10.4310/AJM.1999.v3.n4.a7](https://dx.doi.org/10.4310/AJM.1999.v3.n4.a7) [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0004/AJM-1999-0003-0004-a007.pdf))

* [[Dominic Joyce]], Section 6.5.1 of: _Compact Manifolds with Special Holonomy_, Oxford Mathematical Monographs, Oxford University Press (2000) ([ISBN:9780198506010](https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=de&lang=en&))

* Wei-Dong Ruan, Yuguang Zhang, _Convergence of Calabi-Yau manifolds_, Advances in Mathematics Volume 228, Issue 3, 20 October 2011, Pages 1543-1589 ([arXiv:0905.3424](https://arxiv.org/abs/0905.3424), [doi:10.1016/j.aim.2011.06.023](https://doi.org/10.1016/j.aim.2011.06.023))

* Ronan J. Conlon, Anda Degeratu, Frédéric Rochon, _Quasi-asymptotically conical Calabi-Yau manifolds_. Geom. Topol. 23 (2019) 29-100 ([arXiv:1611.04410](https://arxiv.org/abs/1611.04410))

and in view of [[mirror symmetry]]:

* Shi-Shyr Roan, _The mirror of Calabi-Yau orbifold_,  International Journal of Mathematics Vol. 02, No. 04, pp. 439-455 (1991)  ([doi:10.1142/S0129167X91000259](https://doi.org/10.1142/S0129167X91000259))

* Alan Stapledon, _New mirror pairs of Calabi-Yau orbifolds_,  Adv. Math. 230 (2012), no. 4-6, 1557-1596 ([arXiv:1011.5006](https://arxiv.org/abs/1011.5006))

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

* {#LLP17} [[Ivan Limonchenko]], [[Zhi Lu]], [[Taras Panov]], _Calabi-Yau hypersurfaces and SU-bordism_, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 ([arXiv:1712.07350](https://arxiv.org/abs/1712.07350))

* {#CLP19} Georgy Chernykh, [[Ivan Limonchenko]], [[Taras Panov]], _$SU$-bordism: structure results and geometric representatives_, Russian Math. Surveys 74 (2019), no. 3, 461-524 ([arXiv:1903.07178](https://arxiv.org/abs/1903.07178))

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

[[!redirects CY]]
[[!redirects CYs]]

[[!redirects Calabi-Yau]]
[[!redirects Calabi–Yau]]
[[!redirects Calabi--Yau]]

[[!redirects Calabi-Yau structure]]
[[!redirects Calabi-Yau structures]]

[[!redirects Calabi-Yau space]]
[[!redirects Calabi-Yau spaces]]
[[!redirects Calabi–Yau space]]
[[!redirects Calabi–Yau spaces]]
[[!redirects Calabi--Yau space]]
[[!redirects Calabi--Yau spaces]]

[[!redirects Calabi-Yau manifold]]
[[!redirects Calabi-Yau manifolds]]
[[!redirects Calabi–Yau manifold]]
[[!redirects Calabi–Yau manifolds]]
[[!redirects Calabi--Yau manifold]]
[[!redirects Calabi--Yau manifolds]]

[[!redirects Calabi-Yau orbifold]]
[[!redirects Calabi-Yau orbifolds]]
[[!redirects Calabi–Yau orbifold]]
[[!redirects Calabi–Yau orbifolds]]
[[!redirects Calabi--Yau orbifold]]
[[!redirects Calabi--Yau orbifolds]]

[[!redirects Calabi-Yau variety]]
[[!redirects Calabi-Yau varieties]]
[[!redirects Calabi–Yau variety]]
[[!redirects Calabi–Yau varieties]]
[[!redirects Calabi--Yau variety]]
[[!redirects Calabi--Yau varieties]]


calculus

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#### Analysis
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#Contents#
* table of contents
{:toc}

## Calculus

In various areas of [[mathematics]] there are some standard objects and operations on them and often reasonings and rules of those are expressed in symbolic (typically) sequences which are usually viewed as acts of calculations. Calculus (Latin: 'pebble', 'stone', as for example a bead on an abacus) would be then a set of rules for those calculations. There are for example [[propositional calculus]], [[predicate logic|predicate calculus]], [[sequent calculus]], [[deduction calculus]] as forms of [[logic]], [[relational calculus]] at the interface of logic and [[set theory]], the [[lambda calculus]] in [[type theory]], the probabilistic calculus, the [[matrix calculus]], [[Schubert calculus]], famously [[differential calculus]], and various variants in [[analysis]] like [[tensor calculus]], [[functional calculus]], [[variational calculus]], [[umbral calculus]] etc.. A version of differential calculus in [[homotopy theory]] is [[Goodwillie calculus]]. Then there is [[calculus of fractions]] in [[localization]] theory.

In many, perhaps most Anglophone university curricula, the term "calculus" is a standard abbreviation for the standard basic course in [[differential calculus]] and [[integral calculus]]. A less ambiguous and once common term for such a course is _infinitesimal calculus_[^1] (even where actual infinitesimals were never actually introduced); also still in frequent use these days is simply "the calculus". From the point of view of research mathematics this is the (usually nonrigorous) introduction into the subject properly called mathematical analysis or simply _[[analysis]]_. Thus, while "calculus" has a standard meaning for Anglophone undergraduate students, "calculus" in the context of research mathematics is considered a rather ambiguous and overloaded term, although "the calculus" without further qualifiers may still used by mathematicians today to refer to the differential and integral calculus.

[^1]: Even if actual infinitesimals are not discussed in classical approaches to the differential calculus, they are nevertheless there *implicitly*, inasmuch as a derivative in any incarnation is a rate of infinitesimal change. From this point of view, the term 'infinitesimal calculus' or 'infinitesimal analysis' is an eminently reasonable description of differential/integral calculus, that accurately situates it within the vast body of general mathematical [[analysis]] which deals with limit processes more generally.

## Formal calculus

The problem with the ambiguity of the term _calculus_ extends somewhat to "formal calculus". The name is sometimes used for symbolic logical systems where the deducibility is given by syntactic rules, hence "formal".

More often the term "formal" pertains to various calculi which are akin to infinitesimal calculus, but without the analytic tools like limiting procedure, but rather applying formal rules of differentiation, integration and so on. Many
examples from abstract derivations and differential calculi on noncommutative associative algebras, regular differential operators on algebraic [[scheme]]s to Goodwillie calculus of functors belong to this wide group. [[Joyal]]'s [[species]] are a particular formal tool which can (among other things) relating calculi with power series and applying those to [[combinatorics]]. Particularly often one says formal calculus for formal rules of differentiation and integration with formal power series, formal Laurent series and generalizations. Thus formal calculi define the infinitesimal calculus and its generalizations and analogues without usage of proper mathematical analysis. This, of course, can be done for quite a different (sometimes smaller, sometimes larger) class of formal objects replacing functions which the operations are applied upon.

In order to mark the subject and ideas, the $n$Lab tag [formal calculus](/nlab/list/formal+calculus) will be used for the analogues of infinitesimal calculus defined in areas or in a manner not using mathematical analysis. The tag will not be used for entries on various "calculi" which are not analogues of infinitesimal calculus.

## References

* [[Michael Spivak]], _[[Calculus on Manifolds]]_ (1971)

Discussion of the history of differential calculus with emphasis on its roots all the way back in [[Zeno's paradoxes of motion]] is in

* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949

Closely related to the notion of "a calculus" is the notion of "an algebra", for example we have "[[relational calculus]]" and "the algebra of relations", or we have [[partial combinatory algebra|combinatory algebra]] and "the calculus of combinators". Thus "calculus" and "algebra" in these contexts are often used interchangeably, both referring to modes of reckoning by symbolic manipulations in a formal system. For some discussion of possible distinctions between the two terms, see this MathOverflow discussion:

* Steffen Jensen (https://mathoverflow.net/users/8797/steffen-jensen), Difference between a 'calculus' and an 'algebra', URL (version: 2011-12-08): [https://mathoverflow.net/q/36758](https://mathoverflow.net/q/36758).

category: disambiguation, analysis, formal calculus

[[!redirects calculi]]
[[!redirects formal calculus]]
[[!redirects formal calculi]]


calculus of constructions

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### Context
#### Type theory
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#### Foundations
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[[!include foundations - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The **Calculus of Constructions** (CoC) is a [[type theory]] [[formal system]] for [[constructive mathematics|constructive]] [[proof]] in [[natural deduction]] style.
This calculus goes back to [[Thierry Coquand]] and [[Gérard Huet]]

When supplemented by [[inductive types]], it become the __Calculus of Inductive Constructions__ (CIC).  Sometimes [[coinductive types]] are included as well and one speaks of the __Calculus of (co)inductive Constructions__. This is what the [[Coq]] software implements.

More in detail, the _Calculus of (co)Inductive Constructions_ is

1. a [[pure type system]], hence

   1. a system of [[natural deduction]] with [[dependent types]];

   1. with the natural-deduction rule for [[dependent product types]] specified;

1. with a rule for how to introduce new natural-deduction rules for arbitrary ([[coinductive type|co]])[[inductive types]].

1. and with [[universes]]:

   a cumulative hierarchy of [[predicative mathematics|predicative]] [[types of types]]

   and an impredicative [[type of propositions]].

All of the other standard [[type formation|type formations]] are subsumed by the existence of arbitrary inductive types, notably the _[[empty type]]_, _[[dependent sum types]]_ and _[[identity types]]_ are special inductive types. Specifying these hence makes the calculus of constructions be an [[intensional type theory|intensional]] [[dependent type theory]].

## Related concepts

* [[initiality conjecture]]

* [[Coq]]

* [[computational type theory]]

## References

### General

Original articles:

*  [[Thierry Coquand]], [[Gérard Huet]], _The Calculus of Constructions_, INRIA Rapport **530** (1986) [[inria:00076024](http://hal.inria.fr/inria-00076024/en/), [pdf](http://hal.inria.fr/docs/00/07/60/24/PDF/RR-0530.pdf)]

* [[Christine Paulin-Mohring]], *Inductive definitions in the system Coq -- Rules and Properties*, in: *Typed Lambda Calculi and Applications* TLCA 1993, Lecture Notes in Computer Science **664** Springer (1993) [[doi:10.1007/BFb0037116](https://doi.org/10.1007/BFb0037116)]

  > (adding [[inductive types]])

Review and surveys:

* [[Zhaohui Luo]], *Computation and Reasoning -- A Type Theory for Computer Science*, Clarendon Press (1994) $[$[ISBN:9780198538356](https://global.oup.com/academic/product/computation-and-reasoning-9780198538356), [pdf](https://www.researchgate.net/profile/Zhaohui-Luo-4/publication/243770570_Computation_and_Reasoning_A_Type_Theory_for_Computer_Science/links/54d0dc6b0cf29ca81103f26a/Computation-and-Reasoning-A-Type-Theory-for-Computer-Science.pdf)$]$

* M. Bunder, [[Jonathan Seldin]], *Variants of the basic calculus of constructions*, Journal of Applied Logic
**2** 2 (2004) 191-217 [[doi:10.1016/j.jal.2004.02.004](https://doi.org/10.1016/j.jal.2004.02.004)]

* [[Christine Paulin-Mohring]], *Introduction to the Calculus of Inductive Constructions*, contribution to: *Vienna Summer of Logic* (2014) [[hal:01094195](https://hal.inria.fr/hal-01094195), [pdf](https://hal.inria.fr/hal-01094195/document), [pdf slides](https://easychair.org/smart-program/VSL2014/APPA-invited-slides-6.pdf)]

* Frade, _Calculus of inductive constructions_ ([pdf](http://www3.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf))

* Wikipedia, _[Calculus of constructions](http://en.wikipedia.org/wiki/Calculus_of_constructions)_

A [[categorical semantics]] for CoC is discussed in

* [[Martin Hyland]], [[Andy Pitts]], _The Theory of Constructions: Categorical Semantics and Topos-theoretic Models_ , Cont. Math. **92** (1989) pp.137-199. ([draft](https://www.dpmms.cam.ac.uk/~martin/Research/Pub81-90/hp89.pdf))

For specifics of the implementation in [[Coq]] see

*  [[Coq]] manual, _[Calculus of Inductive Constructions](http://coq.inria.fr/doc/Reference-Manual006.html)_

[[!include history of inductive types -- references]]

[[!redirects calculus of inductive constructions]]
[[!redirects calculus of coinductive constructions]]
[[!redirects calculus of (co)inductive constructions]]


calculus of fractions

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#### Homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea ##

A class $W$ of [[weak equivalence]]s in a [[category]] $C$ is said to admit a **calculus of fractions** if it satisfies some axioms ensuring that its [[localization]] $C[W^{-1}]$ can be constructed in a particularly simple way using 'one-step generalized morphisms.'  These axioms are a categorical analogue of the notion of a [[multiplicative system]] at which one can localize a ring.

Since composition in a category is generally non-commutative, we distinguish 'left' and 'right' calculi of fractions, just as for localization of non-commutative rings.  In either case $C[W^{-1}]$ is referred to as a [[category of fractions]], since its morphisms are two-step zigzags (either $\overset{f}{\to} \overset{w}{\leftarrow}$ or $ \overset{w}{\leftarrow} \overset{f}{\to}$, depending on the handedness of the calculus) in which $w\in W$, which we can think of  as 'fractions' $w^{-1} f$ or $f w^{-1}$.  One sometimes also says that $(C,W)$ 'admits a category of fractions.'

## Definition ##

A pair $(C,W)$ of a [[category]] $C$ and a class of [[morphism]]s $W$ is said to admit a **calculus of right fractions** if the following properties hold.

* $W$ is a [[wide subcategory]] of $C$ (that is, $W$ contains all [[identity morphisms]] and is closed under [[composition]]).

* (right Ore condition) Given a [[morphism]] $v \colon x\to z$ in $W$ and any morphism $f \colon y\to z$, there is a morphism $v' \colon w \to y$ in $W$ and a morphism $f' \colon w \to x$ in $C$ such that the following [[commuting diagram|diagram commutes]]:

$$
\begin{matrix}
  w & \overset{f'}{\longrightarrow} & x
  \\
  \mathllap{{}^{v'}}
  \Big\downarrow
  &&
  \Big\downarrow\mathrlap{{}^{v}}
  \\
  y &\underset{f}{\longrightarrow} & z
\end{matrix}
$$

* (right cancellability) Given a morphism $v \colon y\to z$ in $W$ and a [[pair]] of [[parallel morphisms]] $f,g \colon  x\to y$ such that $v\circ f = v \circ g$, there is a morphism $v' \colon w\to x$ in $W$ such that $f\circ v' = g \circ v'$:
$$
  w \xrightarrow{v'} x
    \underoverset{g}{f}{\rightrightarrows} y
  \xrightarrow{v} z\,.
$$

One may also say that $W$ is a **right Ore system** in $C$ (although this is potentially confusing since the Ore condition is only part of the definition), or that $(C,W)$ **admits a category of right fractions**.  If $(C^{op}, W^{op})$ admits a calculus of right fractions, we say that $(C,W)$ admits a **calculus of left fractions**.  Unfortunately there is no uniformity regarding the choice of 'left' versus 'right;' some authors use 'left' where we use 'right' and vice versa.

### Additional conditions

It is common to assume additional closure conditions on $W$ which make no difference to the localization.  For example, one often assumes that $W$ contains all isomorphisms in $C$.  One can also assume the [[2-out-of-3 property]] (so that $(C,W)$ is a [[category with weak equivalences]]) or the stronger [[2-out-of-6 property]] (so that $(C,W)$ is a [[homotopical category]]).  Note that the 2-out-of-3 property includes closure under composition, and the 2-out-of-6 property together with containment of all identities implies containment of all isomorphisms.

In the presence of either sort of calculus of fractions, the 2-out-of-6 property is equivalent to _saturation_ of $W$, i.e. that any morphism in $C$ which becomes an isomorphism in $C[W^{-1}]$ is already in $W$.  Therefore, in this case we may equivalently call $(C,W)$ _saturated_.  See [[2-out-of-6 property]] for a proof, taken from 7.1.20 of [[Categories and Sheaves]] (where a pair $(C,W)$ admitting a calculus of left fractions is called a _right multiplicative system_).

## Construction of the localization ##

Suppose that $(C,W)$ admits a calculus of right fractions.  Then the [[localization]] of $C$ at $W$ can be realized by taking the same objects as in $C$ and the [[hom-set]] $C[W^{-1}](a,b)$ to be the set of [[equivalence classes]] of [[span|spans]] whose left leg is in $W$, under the equivalence relation where $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ is equivalent to $a\stackrel{w}\leftarrow a''\stackrel{g}\rightarrow b$ iff there exists an object $\bar{a}$
and morphisms $s:\bar{a}\to a'$, $t:\bar{a}\to a''$ such that $f\circ s = g\circ t$, $v\circ s = w\circ t$, and $v\circ s = w\circ t$ is in $W$.  We denote the equivalence class of $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ by $f\circ v^{-1}$.

These equivalence classes compose as follows: take a representative $a\stackrel{v}\leftarrow a'\stackrel{f}\rightarrow b$ and a representative $b\stackrel{u}\leftarrow b'\stackrel{h}\rightarrow c$; then by the Ore condition there exist morphisms $z:d\to a'$ and $k:d\to b'$, where $z\in W$, such that $f\circ z= u\circ k$.  The composition $(h\circ u^{-1})\circ (f\circ v^{-1})$ is the equivalence class of the span $a\stackrel{v\circ z}\leftarrow d\stackrel{h\circ   k}\rightarrow c$.  One proves that this definition does not depend on the choice of representatives, and that it is associative with units $1_a\circ 1^{-1}_a$.  Obviously, the localization functor $C\to C[W^{-1}]$ sends a morphism $p : a\to b$ to $p\circ 1^{-1}_a$.

If instead $(C,W)$ admits a calculus of left fractions, the hom-sets of $C[W^{-1}]$ are equivalence classes of [[cospan]]s (spans in opposite category).  In fact, we can realize $C[W^{-1}]$ as $(C^{\mathrm{op}}[(W^{\mathrm{op}})^{-1}])^{\mathrm{op}}$.  Note that _two_ dualizations are involved, in order to get the cospans to be pointing in the correct direction.

An equivalent way to say this is that if $W$ admits a calculus of right fractions, then the hom-sets in $C[W^{-1}]$ are obtained as the colimit over maps in $C$ out of a $W$-replacement of the source object:
$$
  Hom_{C[W^{-1}]}(X,Y) = \underset{X' \stackrel{p \in W}{\to}X}{colim} Hom_C(X',Y).
$$
Dually, if $W$ admits a calculus of left fractions,  we instead take the colimit over maps into a $W$-replacement under the target object:
$$
  Hom_{C[W^{-1}]}(X,Y) = \underset{Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X,Y').
$$
If $W$ admits both a left and a right calculus of fractions, then both prescriptions coincide and are equivalent to taking a colimit over replacements of both objects:
$$
  \begin{aligned}
    Hom_{C[W^{-1}]}(X,Y) &=
    \underset{X' \stackrel{p \in W}{\to}X}{colim} Hom_C(X',Y)
    \\
      &= \underset{Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X,Y')
    \\
      &=
      \underset{X' \stackrel{p \in W}{\to}X,
        Y \stackrel{i \in W}{\to} Y'}{colim} Hom_C(X',Y')
  \end{aligned}
  \,.
$$

## Properties of the Localization ##

One important consequence of this construction is that when $W$ admits a calculus of right fractions, the localization functor $Q:C\to C[W^{-1}]$ is left [[exact functor|exact]], and therefore preserves all finite [[limit]]s existing in $C$.  Dually, if $W$ admits a calculus of left fractions, then $Q$ is right exact and preserves finite colimits.

Another important fact is that if

* $W$ admits a calculus of right fractions,
* $C$ admits all small [[filtered colimits]], and
* for all $X \in C$ the category $W/X$, whose objects are morphisms $X'\to X$ in $W$ and whose morphisms are commutative triangles, is [[cofinally small category|cofinally small]],

then $C[W^{-1}]$ is [[locally small category|locally small]].  In this case a left [[derived functor]] of any functor $F:C\to A$, where $A$ admits small filtered colimits, can be constructed as the colimit
$$
  R_W F(X)
  =
  \underset{X \stackrel{s \in W}{\to} X'}{\colim} F(X').
$$

## Examples ##

* If $C$ is a [[category of fibrant objects]] and $\pi C$ its
category of morphisms modulo [[homotopy]], the collection of
weak equivalences in $\pi C$ admits a calculus of right fractions.  The corresponding localization is the homotopy category $\Ho(C)$ of $C$.  Note that this example does not satisfy the 2-out-of-3 property.

* Given a [[null system]] $N$ in a [[triangulated category]] $C$,
the collection of morphisms $f : X \to Y$ in $C$ such that
there is a distinguished triangle $X \to Y \to Z$ where $Z \in N$
admits calculi of both left and right fractions.

* For $S$ a [[site]], the collection of [[local epimorphism]]s
in $C = [S^{op},Set]$ with respect to the given [[Grothendieck topology]] on $S$ admits a calculus of right fractions.  In this case the localization is the [[category of sheaves]] on $S$.

## Related concepts

* [[localization]]

* [[colocalization]]

## References ##

> See also references at *[[category of fractions]]*.

The above definition is due to

* [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_, _Ergebnisse der Mathematik und ihrer Grenzgebiete_, Band 35. Springer, New York (1967).

See also

* [[Frank Adams]], part III, section 14 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Borceux94} [[Francis Borceux]], vol 1, chapter 5 of: _[[Handbook of Categorical Algebra]], Cambridge University Press (1994)

* {#Fritz11} [[Tobias Fritz]], *Categories of Fractions Revisited*, Morfismos **15**  2 (2011) 19-38 [[arXiv:0803.2587](https://arxiv.org/abs/0803.2587), [morfismos:vol15-n2-3](https://morfismos.math.cinvestav.mx/sites/default/files/Upload/vol15-n2-3.pdf)]

Generalization from categories to [[quasi-categories]] ("$(\infty,1)$-calculus of fractions", cf. at *[[localization of an (infinity,1)-category|localization of an $(\infty,1)$-category]]*):

* [[Daniel Carranza]], [[Chris Kapulkin]], [[Zachery Lindsey]], *Calculus of Fractions for Quasicategories* [[arXiv:2306.02218](https://arxiv.org/abs/2306.02218)]

Exposition:

* [[Chris Kapulkin]], *Calculus of Fractions for Quasicategories (Part I)*, [talk at](CQTS#KapulkinOct2023) [[CQTS]] (18 Oct 2023) [video:[YT](https://youtu.be/96ViSKAuApc)]

* [[Daniel Carranza]], *Calculus of Fractions for Quasicategories (Part II)*, [talk at](CQTS#CarranzaOct2023) [[CQTS]] (25 Oct 2023) [video:[YT](https://youtu.be/Z41YDb99cZk)]

[[!redirects left calculus of fractions]]
[[!redirects right calculus of fractions]]
[[!redirects calculus of left fractions]]
[[!redirects calculus of right fractions]]


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=--
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This entry is about the book

* [[Pierre Gabriel]], [[Michel Zisman]]:

  **Calculus of fractions and homotopy theory**

  Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35,

  Springer (1967)

  [doi:10.1007/978-3-642-85844-4](https://link.springer.com/book/10.1007/978-3-642-85844-4)

  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf)

on [[homotopy theory]], notably [[simplicial homotopy theory]], and specifically the [[calculus of fractions]] for constructing [[homotopy categories]].

This book first introduced a general [[localization]] and [[reflective localization]] of a [[category]] at  an arbitrary [[class]] of [[morphisms]] (nowadays sometimes called the _Gabriel-Zisman localization_ (see also _[[Gabriel localization]]_)); however the [[size issues]] are not discussed and the formalism of [[Grothendieck universe|universes]] is understood as an excuse. The special case of [[calculus of fractions|categories of fractions]] is treated in detail.

The book has large historical importance for a clean and innovative formalism treating the interaction of [[category theory]] (including [[adjoint functors]], [[Kan extensions]], [[strict 2-category|strict 2-categories]]), [[simplicial set|simplicial]] methods and [[homotopy theory]]. An important version of a definition of a [[homotopy category]] by the abstract categorical localization by the class of [[weak equivalences]] is introduced.

It has been proved that the [[homotopy categories]] of [[CW complexes]] and of [[simplicial sets]] are equivalent. The notion of a _fundamental category_ of a simplicial set (now sometimes also called the [[homotopy category]]), refining the notion of [[fundamental groupoid]], is defined using [[adjoint functor|adjointness]].

An important method for studying [[cofibrations]] of [[simplicial sets]], namely the notion of an [[anodyne extension]], is introduced for the first time and effectively used.

The book is written in a recognizable abstract, clean and precise language with economic, rather short and formal, formulations.

category: reference

[[!redirects Gabriel-Zisman]]
[[!redirects Gabriel–Zisman]]
[[!redirects Gabriel--Zisman]]
[[!redirects Gabriel and Zisman]]
[[!redirects Calculus of fractions and homotopy theory]]

[[!redirects Calculus of Fractions and Homotopy Theory]]


Calculus on Manifolds

This page collects material related to the textbook

* [[Michael Spivak]],

  _Calculus on Manifolds_ -- A Modern Approach To Classical Theorems Of Advanced Calculus

  (1971)

  [pdf](http://www.strangebeautiful.com/other-texts/spivak-calc-manifolds.pdf), [pdf](https://www.math.ncku.edu.tw/~rchen/2015%20Teaching/Calculus%20on%20manifolds%20-%20Spivak,%20M..pdf)

on [[calculus]] and [[differential geometry]] on [[differentiable manifold|differentiable]]/[[smooth manifolds]].

category: reference


Calder Daenzer

* [webpage](http://www.math.psu.edu/daenzer/)

## Selected writings

On [[T-duality]] and [[Langlands dual groups]]:

* {#DaenzerErp} [[Calder Daenzer]], [[Erik Van Erp]], *T-Duality for Langlands Dual Groups*, Advances in Theoretical and Mathematical Physics **18** 6 (2014) 1267–1285 [[arXiv:1211.0763](http://arxiv.org/abs/1211.0763), [doi:10.4310/ATMP.2014.v18.n6.a2](https://dx.doi.org/10.4310/ATMP.2014.v18.n6.a2)]

category: people