cartesian monoidal category

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## Definition

A __cartesian monoidal category__ (usually just called a **[[cartesian category]]**), is a [[monoidal category]] whose monoidal structure is given by the category-theoretic [[product]] (and so whose unit is a [[terminal object]]).

A cartesian monoidal category which is also [[closed monoidal category|closed]] is called a [[cartesian closed category]].

A [[strong monoidal functor]] between cartesian categories is called a [[cartesian functor]].

Any category with finite products can be considered as a cartesian monoidal category (as long as we have either (1) a specified product for each pair of objects, (2) a global [[axiom of choice]], or (3) we allow the monoidal product to be an [[anafunctor]]).

The term __cartesian category__ usually means a category with finite products but can also mean a [[finitely complete category]], so we avoid that term.

## Properties

Among general [[monoidal categories]], the cartesian monoidal categories have a number of special and important properties, such as the existence of [[diagonal maps]] $\Delta_x : x \to x\otimes x$ and augmentations $e_x: x \to I$ for any object $x$, cf. the *[[structural rules]]* in the corresponding [[internal type theory]]: In applications to [[computer science]] we can think of $\Delta$ as 'duplicating data' and $e$ as 'deleting' data.  These maps make any object into a [[comonoid]]: (see also at *[comonoid -- In a cartesian monoidal category](comonoid#InACartesianMonoidalCategory)*):

\begin{example}\label{ComonoidObjects}
**(comonoid objects)**
\linebreak
In fact, any object in a cartesian monoidal category becomes a comonoid in a *unique* way, and this is automatically a [[cocommutative comonoid]] [[structure]].  Furthermore, this comonoid structure is 'natural', meaning that the all morphisms between objects preserve their comonoid structure.  In other words, any morphism $f: x \to y$ is a comonoid homomorphism:
\[       (f \otimes f) \circ \Delta_x = \Delta_y \circ f, \quad i_x = i_y \circ f .\]

Moreover, one can show (e.g. [Fox 1976](#Fox76) or [Heunen-Vicary 2012, p. 79 (p. 85 of the pdf)](#HeunenVicary12)) that any [[symmetric monoidal category]] equipped with suitably well-behaved diagonal and augmentation maps _must_ in fact be cartesian monoidal.   More precisely: suppose $C$ is a symmetric monoidal category equipped with monoidal natural transformations
$$           \Delta_x : x \to x \otimes x $$
and
$$             e_x : x \to I   $$
such that the following composites are identity morphisms:
$$       x \stackrel{\Delta_x}{\longrightarrow} x \otimes x
\stackrel{e_x \otimes 1}{\longrightarrow} I \otimes x
\stackrel{\ell_x}{\longrightarrow} x $$
$$       x \stackrel{\Delta_x}{\longrightarrow} x \otimes x
\stackrel{1 \otimes e_x}{\longrightarrow} x \otimes I
\stackrel{r_x}{\longrightarrow} x $$
where $r$, $\ell$ are the right and left unitors.   Then $C$ is cartesian.

Heuristically: a symmetric monoidal category is cartesian if we can duplicate and delete data, and 'duplicating a piece of data and then deleting one copy is the same as not doing anything'.
\end{example}

A related theorem describes cartesian monoidal categories as monoidal categories satisfying two properties involving the [[unit object]].  First, we say a monoidal category $C$ is [[semicartesian monoidal category|semicartesian]] if the unit for the tensor product is [[terminal object|terminal]].   If this is true, any tensor product of objects $x \otimes y$ comes equipped with morphisms
$$ p_x : x \otimes y  \to x $$
$$ p_y : x \otimes y \to y$$
given by
$$ x \otimes y \stackrel{1 \otimes e_y}{\longrightarrow} x \otimes I \stackrel{r_x}{\longrightarrow} x $$
and
$$ x \otimes y \stackrel{e_x \otimes 1}{\longrightarrow} I \otimes y \stackrel{\ell_y}{\longrightarrow} y $$
respectively, where $e$ stands for the unique morphism to the terminal object and $r$, $\ell$ are the right and left unitors.  We can thus ask whether $p_x$ and $p_y$ make $x \otimes y$ into the [[product]] of $x$ and $y$.  If so, it is a theorem that $C$ is a cartesian monoidal category.  (This theorem is probably in Eilenberg and Kelly's paper on closed categories, but they may not have been the first to note it.)

Cartesian categories may be [[freely generated]] from sets, categories, signatures, etc., as explained at [[free cartesian category]].

## Free and cofree cartesian monoidal categories

As outlined above, cartesianity is an algebraic structure on top of a monoidal structure. This means that the [[2-category]] of cartesian monoidal categories $\mathbf{CartMonCat}$ is [[monadic]] over $\mathbf{MonCat}$, i.e. there is a [[free functor|free]]-[[forgetful functor|forgetful]] adjunction
\[
U \dashv F : \mathbf{CartMonCat} \leftrightarrows \mathbf{MonCat}
\]
Thomas Fox showed in ([Fox 1976](#Fox76)) that $\mathbf{Cart}$ is also comonadic, that is, that $U$ admits a right adjoint $C \dashv U$, given by constructing the category of comonoids in a given monoidal category $\mathbf{M}$.

## Related concepts

* [[monoidal category]]

* [[symmetric monoidal category]]

* [[semicartesian monoidal category]]

* [[cocartesian monoidal category]]

* [[monoidal category with diagonals]]

* [[relevance monoidal category]]

* [[cartesian monoidal dagger category]]

* [[cartesian monoidal (∞,1)-category]]

* [[cocartesian monoidal preordered object]]

## References

The characterization of cartesian monoidal categories as [[symmetric monoidal categories]] with [[extra structure]]:

* {#Fox76} [[Thomas Fox]], *Coalgebras and cartesian categories*, Communications in Algebra **4** 7 (1976) 665-667 ([doi:10.1080/00927877608822127](https://doi.org/10.1080/00927877608822127), [pdf](https://www.tandfonline.com/doi/pdf/10.1080/00927877608822127))

Discussion with an eye towards [[finite quantum mechanics in terms of dagger-compact categories]] is in

* {#HeunenVicary12} [[Chris Heunen]], [[Jamie Vicary]], _Lectures on categorical quantum mechanics_, 2012 ([pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf))

* Kosta Došen and Zoran Petrić. _The maximality of cartesian categories_. Mathematical Logic Quarterly: Mathematical Logic Quarterly 47.1 (2001): 137-144.

[[!redirects cartesian monoidal category]]
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[[!redirects Cartesian monoidal category]]
[[!redirects Cartesian monoidal categories]]

[[!redirects cartesian monoidal structure]]
[[!redirects cartesian monoidal structures]]

[[!redirects Cartesian monoidal structure]]
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cartesian monoidal dagger category

[[!redirects cartesian monoidal dagger categories]]

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## Definition ##

A **cartesian monoidal dagger category** is [[category]] which is both a [[dagger category]] and a [[cartesian monoidal category]] in a compatible way, namely a [[monoidal dagger category]] $(C, \times, 1)$ with

* a morphism $p_A \in Hom_C(A \times B,A)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
* a morphism $p_B \in Hom_C(A \times B,B)$ for $A \in Ob(C)$ and $B \in Ob(C)$.
* a morphism $p_{A \times B} \in Hom_C(D,A \times B)$ for an object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,A)$ and $d_B \in Hom_C(D,B)$
* a morphism $!_A \in Hom_C(A,1)$ for every object $A \in C$

such that

* for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,B)$ and $d_B \in Hom_C(D,B)$, $p_A \circ d_{A \times B} = d_A$
* for every object $D \in Ob(C)$ and morphisms $d_A \in Hom_C(D,B)$ and $d_B \in Hom_C(D,B)$, $p_B \circ d_{A \otimes B} = d_B$
* for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $f \in Hom_C(A,B)$, $!_B \circ f = !_A$.

In a cartesian monoidal dagger category, the tensor product is called a **cartesian product** and the tensor unit is called a **terminal object**.

## Examples ##

* [[semiadditive dagger category]]

## See also ##

* [[dagger category]]

* [[monoidal dagger category]]

* [[cartesian monoidal category]]


cartesian monoidal preordered object

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## Definition

In a [[finitely complete category]] $C$, a **cartesian monoidal preordered object** $X$ is a [[preordered object]] with internal preorder $R\stackrel{(s,t)}\hookrightarrow X \times X$ and a [[monoid object]] in $C$ with multiplication $(-)\wedge(-):X \times X \to X$ and a [[global element|global]] unit $\top:* \to X$, where $*$ is the [[terminal object]] in $C$, with

* a function $\tau:(* \to X) \to (* \to R)$ such that for all [[global elements]] $a:* \to X$, $s \circ \tau(a) = a$ and $t \circ \tau(a) = \top$.

* functions
$$\lambda_l:((* \to X) \times (* \to X)) \to (* \to R)$$
$$\lambda_r:((* \to X) \times (* \to X)) \to (* \to R)$$
such that for all [[global elements]] $a:* \to X$ and $b:* \to X$, $s \circ \lambda_l(a,b) = a \wedge b$, $t \circ \lambda_l(a,b) = a$, $s \circ \lambda_r(a,b) = a \wedge b$, and $t \circ \lambda_r(a,b) = b$.

## See also

* [[preordered object]]

* [[monoid object]]

* [[cocartesian monoidal preordered object]]

* [[bicartesian preordered object]]

* [[meet-semilattice object]]

* [[cartesian monoidal category]]

[[!redirects cartesian monoidal preordered objects]]


Cartesian morphism

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## Idea

Given a [[functor]] $p\colon  X \to Y$ between categories one may ask for each [[morphism]] $f\colon  y_1 \to y_2$ that, if given a lift of its target
$$
  \array{
    X &&& && \hat y_2
    \\
    \downarrow^p
    \\
    Y &&& y_1 &\stackrel{f}{\to}& y_2
  }
$$

there be a _universal_ lift of $f$

$$
  \array{
    X &&& \hat y_1 &\stackrel{\hat f}{\to}& \hat y_2
    \\
    \downarrow^p
    \\
    Y &&& y_1 &\stackrel{f}{\to}& y_2
  }
  \,.
$$

There may also be other lifts of $f$, but the universal one is essentially unique, as usual for anything having a [[universal property]].  Specifically, $\hat f$ in $X$ is essentially uniquely determined by its target $\hat y_2$ and its image $f = p(\hat f)$ in $Y$, and is called a __cartesian morphism__.  A morphism which is cartesian relative to $p^{op}\colon X^{op}\to Y^{op}$ is called __opcartesian__ or __cocartesian__.

If there are enough cartesian morphisms in $Y$, they may be used to define [[functor]]s

$$
  f^* : X_{y_2} \to X_{y_1}
$$

between the [[fiber]]s of $p$ over $y_1$ and $y_2$.

This way a functor $p : X \to Y$ with enough Cartesian morphisms -- called a [[Cartesian fibration]] or [[Grothendieck fibration]] -- determines and is determined by a fiber-assigning functor $Y \to Cat^{op}$.

This has its analog in [[higher category theory|higher categories]].

## Definition

### In categories

#### Traditional definition

+-- {: .num_defn}
###### Definition
**(cartesian morphism)**

Let $p : X \to Y$ be a [[functor]]. A [[morphism]] $f : x_1 \to x_2$ in the [[category]] $X$ is **strongly cartesian** with respect to $p$ (nowadays often just *cartesian*), or (strongly) **$p$-cartesian** if for every $x'\in X$,
for every $h:x'\to x_2$ and every
$u:p(x')\to p(x_1)$ such that $p(h) = p(f) u$, there exists a unique $v:x'\to x_1$ such that $h = f v$ and $u = p(v)$:

$$
  \array{
    \forall x'
    \\
    \downarrow^{\mathrlap{\exists! v}} & \searrow^{\mathrlap{\forall h}}
    \\
    x_1 &\stackrel{f}{\to}& x_2
  }
  \;\;\;
  \;\;\;
  \stackrel{p}{\mapsto}
  \;\;\;
  \;\;\;
  \array{
    p(x')
    \\
    \downarrow^{\mathrlap{\forall u}} & \searrow^{\mathrlap{p(h)}}
    \\
    p(x_1) &\stackrel{p(f)}{\to}& p(x_2)
  }
$$

In imprecise words: for all commuting triangles in $Y$ (involving $p(f)$ as above) and all lifts through $p$ of its 2-[[horn]] to $X$ (involving $f$ as above), there is a unique refinement to a lift of the entire commuting triangle.

We can make this definition slightly more explicit by working with the fibres of $p$: let $X_x$ denote $p^{-1}(y)$, the set of objects living over $y$; and, for $f : y \to y'$, $x : X_y$ and $x' : X_{y'}$, let $X_f(x,x')$ denote $p^{-1}(f) \cap X(x,x')$, the set of morphisms from $x$ to $x'$ that $p$ maps to $f$.

Then for $x : X_y$, $x' : X_{y'}$ and $f : y' \to y$, a morphism $\bar f : X_f(x',x)$ is *cartesian* iff for any $u : y'' \to y'$, $x'' : X_{y''}$ and $h : X_{f \circ u}(x'', x)$, there is a unique $\bar u : X_u(x'',x')$ with $h = \bar f \circ \bar u$. This can be expressed in the following diagram, where the upper objects and morphisms live in $X$ over their corresponding data in $Y$:

\begin{tikzcd}
x'' \arrow[d, dotted, maps to] \arrow[rd, "\exists ! \bar u", dashed] \arrow[rrd, "h", bend left] &                                                   &                              \\
y'' \arrow[rd, "u"]                                                                               & x' \arrow[d, dotted, maps to] \arrow[r, "\bar f"] & x \arrow[d, dotted, maps to] \\
                                                                                & y' \arrow[r, "f"]                                 & y
\end{tikzcd}

If we pass to the [[nerve]] $N(X)$ and $N(Y)$ of the categories, then in terms of diagrams in [[sSet]] this means that the morphism $f : x \to y$ is $p$-cartesian precisely if for all [[horn]] inclusions

$$
  \array{
    \Delta^{\{1,2\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda_2[2] &\to& N(X)
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[2] &\to& N(Y)
  }
$$

such that the last edge of the 2-[[horn]] is the given edge $f$, a unique lift $\sigma$

$$
  \array{
    \Delta^{\{1,2\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda_2[2] &\to& N(X)
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}}
    \\
    \Delta[2] &\to& N(Y)
  }
$$

exists.

=--

\begin{remark}\label{WeakCartesianMorphisms}
**(weak/local Cartesian morphisms)**
\linebreak
There is a weaker universal property, originally devised by Grothendieck and Gabriel, where one requires the above lifting property only for $u = id_{p(x_1)}$.  Morphisms satisfying this universal property have in recent years been called *[[locally Cartesian morphisms]]*, although historically they have been called simply *Cartesian*, or sometimes *weak Cartesian*. For the case of Grothendieck [[fibered categories]] the notion of weak Cartesian morphisms already coincides with that of actual Cartesian morphisms.

Equivalently, a morphism $f:x\to x^\prime$ is called *locally $p$-Cartesian* (relative to a functor $X\to Y$) if it is Cartesian with respect to the projection functor $p_{pf}:X_{pf}\to \mathbf{2}$ where $X_{pf}$ is the (homotopy) pullback of $X$ along the functor $c_{pf}:\mathbf{2}\to Y$ classifying the arrow $pf:px\to px^\prime$ in $Y$.
\end{remark}

+-- {: .num_defn}
###### Definition
**(Grothendieck fibration)**

If for every morphism in $Y$ and every lift of its target there is at least one lift which has as its target the chosen one and is a $p$-cartesian morphism in the strong sense, one says that $p$ is a __fibered category__ (also called [[Grothendieck fibration]]). Equivalently,

* ($p$ is [[prefibered category]]) for every morphism in $Y$ and every lift of its target there is at least one lift through $p$ which has as its target the chosen one and is a $p$-cartesian morphism in the weak sense

*and*

* the composition of every two composable $p$-cartesian morphisms (in the weak sense) is a $p$-cartesian morphism (in the weak sense).
=--

#### Reformulations {#CartInOrdCatReformulation}

We discuss equivalent reformulations of the above definition of Cartesian
morphism that lend themselves better to generalization to
[[higher category theory]].

For the following, we need this notation: let

* $X/x_2$ by the [[overcategory]] of $X$ over the object $x_2$;

* $Y/p(x_2)$ the corresponding [[overcategory]] of $Y$ over $p(x_2)$;

* $X/f$ the category whose objects

  $$
    Obj(X/f) =
    \left\{

    \array{
       && a
       \\
       &\swarrow && \searrow
       \\
       x_1 &&\stackrel{f}{\to}&& x_2
    }
    \right\}
  $$

  are objects $a$ of $X$ equipped with morphisms to $x_1$ and $x_2$ such that the obvious triangle commutes, and whose morphisms are morphisms between these tip objects such that all diagrams in sight commute.

* similarly for $Y/p(f)$.

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

The condition that $f \in Mor X$ is a Cartesian morphism with respect to $p : X \to Y $  is equivalent to the condition that the functor

$$
  \phi : X/f \to X/{x_2} \times_{Y/{p(x_2)}} Y/p(f)
$$

into the (strict) [[pullback]] of the obvious projection $X/{x_2} \to Y/p(x_2)$ along the projection $Y/p(f) \to Y/p(x_2)$ induced by the commutativity of

$$
  \array{
    X/f &\stackrel{\phi_2}{\to}& Y/p(f)
    \\
    {}^{\mathllap{\phi_1}}\downarrow && \downarrow
    \\
    X/{x_2} &\to& Y/{p(x_2)}
  }
$$

is a [[k-surjective functor|surjective equivalence]], and this in turn is equivalent to it being an [[isomorphism]] of categories.

=--

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

It is immediate to see that $\phi$ being an isomorphism of categories is equivalent to the condition that $f$ is a Cartesian morphism. We discuss that just the condition that $\phi$ is a surjective equivalence already implies that it is an isomorphism of categories.

So assume now that $\phi$ is a surjective equivalence.

Notice that objects in the pullback category are compatible pairs

$$
  \left(
  \left(
  \array{
     && a
     \\
     &&& \searrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
  \right) \in X/_{x_2}
  \;\;\;\;\;\;\,,\;\;\;\;\;\;
  \left(
  \array{
     && b
     \\
     & \swarrow && \searrow
     \\
     p(x_1) &&\stackrel{p(f)}{\to}&& p(x_2)
  }
  \right)
   \in Y/p(f)
  \right)
  \,.
$$

We have that $\phi$ being _surjective_ on object means that every such pair is in the image of some object

$$
  \left(
  \array{
     && a
     \\
     &{}^{\mathllap{g}}\swarrow&& \searrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
  \right)
  \in X_{f}
  \,,
$$

and hence that every filler _exists_ . Assume two such fillers $g$ and $g'$. Then by the fact that an [[equivalence of categories]] is a surjection (even an isomorphism) on corresponding hom-sets, it follows that there exists (even uniquely) a morphism in $X/f$ connecting them

$$
  \array{
     && a
     \\
     &{}^{\mathllap{g}}\swarrow & \downarrow^{\mathrlap{h}} & \searrow
     \\
     && a
    \\
     \downarrow & {}^{\mathllap{g'}}\swarrow && \searrow & \downarrow
     \\
     x_1 &&\stackrel{f}{\to}&& x_2
  }
$$

such that this maps under $\phi$ to the identity morphism in the pullback category. But in particular this maps to the morphism

$$
  \array{
     && a
     \\
     && \downarrow^{\mathrlap{h}} & \searrow
     \\
     && a
    \\
     & && \searrow & \downarrow
     \\
      &&&& x_2
  }
$$

in $X/{x_2}$ and evidently is the identity there if and only if $h$ is the identity. Hence this maps also to the identity in the pullback category if and only if $h$ is the identity. So $h$ must be the identity. So if two lifts of an object through the surjective equivalence $\phi$ exist, they must already be equal. Hence the surjective equivalence $\phi$ is even an isomorphism on objects and hence an isomorphism of categories.

=--

### In $(\infty,1)$-categories

The notion of cartesian morphism generalizes from [[category theory]] to [[(∞,1)-category theory]]. The definition given above can be rephrased as a pullback relation between homsets, which we can take as an abstract definition

+-- {: .num_defn}
###### Definition
**(cartesian edge in an $(\infty,1)$-category)**

Let $p : X \to Y$ be a functor of [[(∞,1)-categories]]. Then, a [[morphism]] $f : x \to x'$ in $X$ is **$p$-cartesian** if and only if the commutative square induced by the distributivity of $p$ over composition

  \begin{centre}
    \begin{tikzcd}
      X(a,x) \arrow[r, "{X(a,f)}"] \arrow[d, "p"]
      & X(a,x') \arrow[d, "p"]
      \\ Y(p(a),p(x)) \arrow[r, "{Y(p(a),p(f))}"]
      & Y(p(a),p(x'))
    \end{tikzcd}
  \end{centre}

is a pullback square for every $a \in X$.

=--

Similarly, the reformulation in terms of slice/cone categories generalizes directly, and is indeed equivalent to the definition given above:

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

Let $p : X \to Y$ be a functor of [[(∞,1)-categories]]. Then, a [[morphism]] $f : x \to x'$ in $X$ is $p$-cartesian iff the commutative square

  \begin{centre}
    \begin{tikzcd}
      X_{/x} \arrow[r, "{f_!}"] \arrow[d, "p"]
      & X_{/x'} \arrow[d, "p"]
      \\ Y_{/p(x)} \arrow[r, "{p(f)_!}"]
      & Y_{/p(x')}
    \end{tikzcd}
  \end{centre}

is a pullback square.

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

The definition of being $p$-cartesian can be described as a pullback square of hom-functors. By the contravariant [[(∞,1)-Grothendieck construction]], $X(-, x)$ classifies the [[right fibration]], $X_{/x} \to X$, and $Y(p-, p(x))$ classifies the pullback of $Y_{/p(x)} \to Y$ along $p$; that is by the [[comma category]] $(p \downarrow p(x)) \to Y$. There is a commutative diagram

  \begin{centre}
    \begin{tikzcd}
      X_{/x} \arrow[r, "{f_!}"] \arrow[d]
      & X_{/x'} \arrow[d]
      \\ (p \downarrow p(x)) \arrow[r] \arrow[d]
      & (p \downarrow p(x')) \arrow[r] \arrow[d]
      & X \arrow[d, "p"]
      \\ Y_{/p(x)} \arrow[r, "{p(f)_!}"]
      & Y_{/p(x')} \arrow[r] & Y
    \end{tikzcd}
  \end{centre}

The bottom-right square and wide rectangle are pullbacks by construction. By the [[pasting law]], the bottom-left square is a pullback, and thus the top-left square is a pullback if and only if the tall rectangle is.

By the [[(∞,1)-Grothendieck construction]], the top-left square is a pullback iff $f$ is $p$-cartesian, so the proposition follows.

=--

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

If $f : x \to y$, then $X_{/f} \to X_{/x}$ is an [[equivalence of (∞,1)-categories]], and composing its inverse with $X_{/f} \to X_{/x}$ is the [[dependent sum]] $f_! : X_{/x} \to X_{/y}$.

=--

To make this concrete, we can also discuss adaptations of the abstract idea to two different models of [[(∞,1)-category]] theory:  [[quasi-categories]] and [[simplicially enriched categories|sSet categories]].

#### In quasi-categories

We formulate a notion **cartesian edge** or _cartesian morphism_ in a [[simplicial set]] $X$ relative to a morphism $p : X \to Y$ of simplicial sets. In the case that these simplicial sets are [[quasi-category|quasi-categories]] -- i.e. simplicial set incarnations of [[(∞,1)-category|(∞,1)-categories]] -- this yields a notion of cartesian morphisms in $(\infty,1)$-categories.

Let $p : X \to Y$ be a morphism of [[simplicial set]]s.
Let $f : x_1 \to x_2$ be an edge in $X$, i.e. a morphism $f : \Delta^1 \to X$.

Recall the notion of [[over quasi-category]] obtained from the notion of [[join of quasi-categories]]. Using this we obtain [[simplicial set]]s $X/f$, $X/{x_2}$, $S/p(f)$ and $S/p(x_2)$ in generalization of the categories considered in the above definition of cartesian morphisms in categories.

+-- {: .num_defn}
###### Definition
**(cartesian edge in a simplicial set)**

Let $p : X \to Y$ be an [[inner Kan fibration]] of simplicial sets.

Then a [[morphism]] $f : x \to y$ in $X$ is **$p$-cartesian** if the induced morphism

$$
  X_{/f} \to X_{/y} \times_{Y_{/p(y)}} Y_{/p(f)}
$$

into the [[pullback]] in [[sSet]]
is an acyclic [[Kan fibration]].

=--

This is [[Higher Topos Theory|HTT, def 2.4.1.1]].

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

The morphism $f : x \to y$ as above, for $p : X \to Y$ an [[inner fibration]], is $p$-cartesian precisely if for all $n \geq 2$ and all right outer [[horn]] inclusions

$$
  \array{
    \Delta^{\{n-1,n\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda[n]_n &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\to& Y
  }
$$

(with $\Lambda[n]_n$ the $n$th [[horn]] of the $n$-[[simplex]]) such that the last edge of the horn is the given edge $f$, a lift $\sigma$

$$
  \array{
    \Delta^{\{n-1,n\}}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda[n]_n &\to& X
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}}
    \\
    \Delta[n] &\to& Y
  }
$$

exists.

=--

This is [[Higher Topos Theory|HTT remark 2.4.1.4]].

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

This means that an [[inner fibration]] $p : X \to Y$ with a collection of $p$-cartesian morphisms in $X$ specified satisfies the same kind of condition as a _[[right fibration]]_ , the only difference being that not _all_ right outer horns inclusion are required to have lifts, but only those where the last edge of the horn maps to a cartesian morphism.

In this sense a [[Cartesian fibration]] is a generalization of a [[right fibration]].

=--

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

If $p : X \to Y$ is an [[inner fibration]] of [[quasi-categories]] then a morphism $f : x \to y$ in $X$ is $p$-Cartesian precisely if
for all objects $a$ in $X$ the diagram

  $$
    \array{
      Hom_X(a,x) &\stackrel{Hom_X(a,f)}{\to}& Hom_X(a,y)
      \\
      \downarrow && \downarrow
      \\
      Hom_Y(p(a), p(x)) &\stackrel{Hom_Y(p(a), p(f))}{\to}&
      Hom_Y(p(a), p(y))
    }
  $$

  of [[hom-object in a quasi-category|hom-objects in a quasi-category]] is
  a [[homotopy pullback]] square (in [[sSet]] equipped with its [[model structure on simplicial sets|standard model structure]]).

=--

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

This is [[Higher Topos Theory|HTT, prop. 2.4.4.3]].

=--

#### In $sSet$-categories

Let $C$ and $D$ be [[simplicially enriched categories]] and $F : C \to D$ a [[sSet]]-[[enriched functor]].

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

A morphism $f : x \to y \in C$ is **$F$-cartesian** if it is so under the [[homotopy coherent nerve]] $N : sSet Cat \to sSet$ in the sense of quasi-categories above, i.e. if

$$
  N(C)_{/f} \to N(C)_{/y} \times_{N(D)_{/F(y)}} N(D)_{/F(f)}
$$

is an acyclic [[Kan fibration]].

=--

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

If $C$ and $D$ are enriched in [[Kan complex]]es and if $F$ is hom-wise a [[Kan fibration]], then

* $N(F) : N(C) \to N(D)$ is an [[inner fibration]];

* a morphism $f :x \to y$ in $N(C)$ is an $N(F)$-cartesian morphism precisely if for all objects $a$ in $C$ the diagram

  $$
    \array{
      C(a,x) &\stackrel{C(a,f)}{\to}& C(a,y)
      \\
      \downarrow && \downarrow
      \\
      D(F(a), F(x)) &\stackrel{C(F(a), F(f))}{\to}&
      D(F(a), F(y))
    }
  $$

  is a [[homotopy pullback]] square in [[sSet]] equipped with its [[model structure on simplicial sets|standard model structure]].

=--

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

This is [[Higher Topos Theory|HTT, prop. 2.4.1.10]].

=--

## Properties

### Pullback along Cartesian morphisms

+-- {: .num_lemma}
###### Lemma

For $p : \mathcal{E} \to \mathcal{C}$ a functor, if in a diagram

$$
  \array{
    A &\stackrel{f}{\to}& B
    \\
    {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{h}}
    \\
    C &\stackrel{k}{\to}& D
  }
$$

in $\mathcal{E}$ the two vertical morphisms are vertical with respect to $p$ (meaning that $p(g) = Id_p(A)$ and $p(h) = Id(B)$) and if the two horizontal morphisms are $p$-Cartesian morphisms, then this square is a [[pullback]] square.

=--

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

If

$$
  \array{
    Q &\stackrel{}{\to}& B
    \\
    \downarrow && \downarrow^{\mathrlap{h}}
    \\
    C &\stackrel{k}{\to}& D
  }
$$

is another cone over $C \to D \leftarrow B$, then its image under $p$
is

$$
  \array{
    p(Q)
    \\
    \downarrow & \searrow
    \\
    p(C) &\stackrel{p(k)}{\to}& D
  }
  \,.
$$

Since $p(f) = p(k)$, another lift of the right horn of this is given by

$$
  \array{
    Q
    \\
    & \searrow
    \\
    A &\stackrel{f}{\to}& B
  }
$$

which gives a unique filler $Q \to A$ by the fact that $f$ is Cartesian.

But this produces now two fillers -- namely the original $Q \to C$ and the $Q \to A \to C$ just obtained -- of the horn

$$
  \array{
    Q &\to& B
    \\
    && \downarrow
    \\
    C &\stackrel{k}{\to}& D
  }
$$

over

$$
  \array{
    p(Q)
    \\
    \downarrow & \searrow
    \\
    p(C) &\stackrel{p(k)}{\to}& D
  }
  \,.
$$

Since $k$ is Cartesian, these two fillers must be equal. This means that the morphism $Q \to A$ is a cone morphism and unique as such. Hence the original square is a pullback.

=--

This appears as [[Elephant|Elephant, lemma 1.3.3]].

### Cartesian morphisms and equivalences

+-- {: .num_lemma}
###### Observation

For $C$ a [[category]], a [[morphism]] in $C$ is cartesian with respect to the [[terminal object|terminal]] [[functor]] $C \to *$ precisely if it is an [[isomorphism]].

In particular all [[identity]] morphisms are cartesian.

=--

This is trivial to see. The analog statement holds also for [[quasi-categories]], where it is rather more nontrivial and quite useful:

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

For $C$ a [[quasi-category]], a morphism in $C$ is cartesian with respect to the [[terminal object|terminal]] morphism $C \to *$ precisely if it is an [[equivalence in a quasi-category|equivalence]].

More generally, for $p : X \to Y$ an [[inner fibration]], a morphism $f$ in $X$ is an [[equivalence in a quasi-category|equivalence]] precisely if it is $p$-cartesian and $p(f)$ is an equivalence in $Y$.

=--

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

The first statement is a proposition of [[Andre Joyal]], slightly reformulated in the language of cartesian morphisms. It appears as [[Higher Topos Theory|HTT, prop 1.2.4.3]]. A proof appears below [[Higher Topos Theory|HTT, corollary 2.1.2.2]].

The second statement is [[Higher Topos Theory|HTT, prop. 2.4.1.5]].

=--

## Related pages

* A Cartesian morphism is the special case of a [[strictly final lift]] of a structured [[sink]] when the sink is a singleton.

## References

Original reference:

* [[Alexander Grothendieck]], §VI.5 of: _Revêtements Étales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61_ ([[SGA 1]]) , LNM **224** Springer (1971) [updated version with comments by M. Raynaud: [arxiv.0206203](http://arxiv.org/abs/math/0206203)]

Review:

* [[Angelo Vistoli]], Def. 3.1 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

For the 1-categorical case see for instance section B1.3 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

The $(\infty,1)$-categorical version is in section 2.4 of

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

See also the references at [[Grothendieck fibration]].

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cartesian multicategory

# Cartesian multicategories
* table of contents
{: toc}

## Idea

A **cartesian multicategory** is a [[multicategory]]-like structure which is related to categories with finite [[products]] in the same way that ordinary multicategories are related to [[monoidal categories]].

## Definition

### Explicit

A **cartesian multicategory** is a [[symmetric multicategory]] equipped with:

* *duplication* or *diagonal* or *contraction* operations
  $$ hom(c_0,\dots,c_{k-1},c_k,c_k,c_{k+1},\dots,c_{n-1} \;;\; d) \to hom(c_0,\dots,c_{k-1},c_k,c_{k+1},\dots,c_{n-1} \;;\; d)$$

* *deletion* or *projection* or *weakening* operations
  $$ hom(c_0,\dots,c_{k-1},c_{k+1},\dots,c_{n-1} \;;\; d) \to hom(c_0,\dots,c_{k-1},c_k,c_{k+1},\dots,c_{n-1} \;;\; d) $$

which satisfy certain evident axioms.

### As a generalized multicategory

A cartesian multicategory can equivalently be defined as a [[generalized multicategory]] relative to the monad on [[Cat]] (or more precisely [[Prof]]) whose algebras are categories with (strict) [[finite products]].

### As a free category with finite products

A cartesian multicategory can also be defined as a category with specified [[finite products]] whose set of objects under the "product" operation is a free monoid on specified generators.

When there is exactly one such generator, this recovers the definition of a [[Lawvere theory]]; thus a cartesian multicategory may be considered a "colored" or "many-object" Lawvere theory.  Note, though, that the morphisms of cartesian multicategories are more restrictive than morphisms of finite-product categories; they are required to take generators to generators.

### Alternative presentation

The composition structure of a cartesian multicategory can also be equivalently presented in the style usually used for abstract clones. The symmetry, contraction and weakening are derivable rather than required.
In this style, a  cartesian multicategory is given by a collection of objects, and for every sequence $c_1\dots c_{n+1}$ of objects, a collection of multimorphisms $hom(c_1\dots c_{n}\;;\;c_{n+1})$, and

* for every sequence $c_{1}\dots c_{n}$ and $i\leq n$, a projection map
  \[ \pi_{i} \in hom(c_{1} \dots c_{n} \;;\; c_{i}) \]

* for sequences $c_1\dots c_m$, $d_1\dots d_{n}$ and object $e$, a composition map
  \[ \circ : hom(d_1\dots d_{n} \;;\; e)\times \prod_{i=1}^{n} hom(c_1\dots c_m \;;\; d_i)\to hom(c_1\dots c_m,e) \]

all such that

* identity laws: $f\circ (\pi_i)_i = f$ and $\pi_i \circ (f_0\dots f_{n-1})=f_i$;

* associativity law: $f\circ (g_1\circ (h_1\dots h_{n})\dots g_{m}\circ (h_1\dots h_{n})) = f \circ (g_1\dots g_{m}) \circ (h_1\dots h_{n})$.

## Representability

A cartesian multicategory, like an ordinary multicategory, is **[[representable multicategory|representable]]** if for any finite list $(c_1,\dots,c_n)$ of objects there exists an object "$c_1 \times \cdots \times c_n$" and a morphism $c_1,\dots,c_n \to c_1 \times \cdots \times c_n$ which is universal, in that the following induced functions are all bijections:

$$ hom(d_1,\dots,d_k, c_1 \times \cdots \times c_n , e_1,\dots,e_m \;;\; f) \to
hom(d_1,\dots,d_k, c_1,\dots,c_n, e_1,\dots,e_m \;;\; f)
$$

Just as representable (symmetric) multicategories are equivalent to (symmetric) monoidal categories, representable cartesian multicategories are equivalent to [[cartesian monoidal categories]].

More abstractly, representable cartesian multicategories are the algebras for a [[colax-idempotent 2-monad]] on the 2-category of cartesian multicategories, whose algebras are categories with finite products.

## Syntax, semantics, and type theory

Small cartesian multicategories may be considered presentations of "many-sorted finite-product theories".  On the spectrum between syntax and semantics, they sit in between a fully syntactic presentation (such as a [[type theory]]) and a fully incarnated semantic one (the [[walking structure|free category]] with finite products generated by a model of some theory).

Cartesian multicategories are also a natural place to talk about the semantics of [[type theory]].  Every type theory (with [[contraction rule|contraction]] and [[weakening rule|weakening]], and without [[dependent types]]) gives rise to a cartesian multicategory whose objects are its types and whose multimorphisms are its terms.  This is in contrast to the usual construction of an ordinary category from a type theory, in which we have to take the objects to be [[contexts]] in order to recapture all the terms.  This usual [[category of contexts]] can be recaptured as the free category-with-finite-products generated by this cartesian multicategory of types.  Similarly, we can talk about models of such a type theory in any cartesian multicategory.

## Related pages

* [[multicategory]]
* [[symmetric multicategory]]
* [[generalized multicategory]]
* [[Lawvere theory]]
* A cartesian multicategory with one object is called a [[clone]].
* [[sequential multicategory]]

## References

* [[Claudio Pisani]], _Sequential multicategories_, ([arXiv:1402.0253](https://arxiv.org/abs/1402.0253))
* Nathanael Arkor, and Dylan McDermott. "Abstract Clones for Abstract Syntax." 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2021. ([arXiv](https://arxiv.org/abs/2105.00969))

[[!redirects cartesian multicategory]]
[[!redirects cartesian multicategories]]
[[!redirects finite-product multicategory]]
[[!redirects finite-product multicategories]]


cartesian natural transformation

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

# Cartesian natural transformation

* table of contents
{: toc}

## Definition

Let $F,G:C\to D$ be [[functors]].  A [[natural transformation]] $\alpha:F\to G$ is **cartesian** if for any [[morphism]] $f:x\to y$ in $C$, the naturality square

$$\array{ F x & \overset{F f}{\to} & F y\\
  ^{\alpha_x}\downarrow & & \downarrow^{\alpha_y} \\
  G x & \underset{G f}{\to} & G y}
$$
is a [[pullback]].

There is an evident generalization to natural transformations between [[higher category theory|higher categories]].

## Properties

* Given a functor $G:C\to D$, if $C$ has a [[terminal object]] $1$, then to give a functor $F$ and an cartesian natural transformation $F\to G$ is equivalent to giving a single object $F1$ and a morphism $F1 \to G1$.  The rest of $F$ can then be constructed uniquely by taking pullbacks.  This construction is important in the theory of [[clubs]].

[[!include colimits of equifibered transformations -- section]]

## Related pages

* [[cartesian monad]]
* [[club]]
* [[polynomial functor]]
* [[van Kampen colimit]]

## References

In the context of [[category theory]] the concept is discussed in

* [[Aurelio Carboni]] and [[Peter Johnstone]], *Connected limits, familial representability and Artin glueing*, Mathematical Structures in Computer Science, Vol. 5 Iss. 4, Cambridge U. Press (December 1995), 441-459.
([doi:10.1017/S0960129500001183](https://doi.org/10.1017/S0960129500001183) [web](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/div-classtitleconnected-limits-familial-representability-and-artin-glueingdiv/54E93903F7D7321B98B64AE7CB3E7AE0))

* [[Tom Leinster]], *Higher Operads, Higher Categories*, Cambridge University Press 2003. ([arXiv:math/0305049](https://arxiv.org/abs/math/0305049))

* [[Mark Weber]]. _Generic morphisms, parametric representations and weakly Cartesian monads_. Theory Appl. Categ 13.14 (2004): 191-234.

* {#Koslowski} Juergen Koslowski, *A monadic approach to polycategories*, [TAC](http://www.tac.mta.ca/tac/volumes/14/7/14-07abs.html) Vol. 14, 2005, No. 7, pp 125-156.

* [[Nicola Gambino]] and [[Joachim Kock]]. _Polynomial functors and polynomial monads_. Mathematical proceedings of the cambridge philosophical society. Vol. 154. No. 1. Cambridge University Press, 2013.

In the context of [[(infinity,1)-categories]] (with an eye towards [[(infinity,1)-toposes]]) the concept is called **equifibered natural transformations** and is considered in

* {#Rezk10} [[Charles Rezk]], p. 9 of _Toposes and homotopy toposes_ (2010) ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

* {#Rezk14} [[Charles Rezk]], p. 2 of _When are homotopy colimits compatible with homotopy base change?_ (2014) ([pdf](http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf))

The name "equifibered" comes from the fact that since $\alpha_x$ is a pullback of $\alpha_y$, they must have [[isomorphism|isomorphic]] [[fibers]].  (Of course, if $C$ is not [[connected category|connected]], then being equifibered does not imply that *all* components of $\alpha$ have isomorphic fibers.)

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cartesian object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

## Definition

A **cartesian object** in a [[2-category]] with finite [[2-limit|2-products]] is an object $X$ such that the [[diagonal morphism]] $X \to X \times X$ and the unique map $X \to 1$ have [[right adjoints]].

For example, a cartesian object in [[Cat]] is precisely a category with finite [[products]].

## Properties

Any category with finite products becomes a [[symmetric monoidal category]] by making a choice of products. The symmetric monoidal structure is canonical in the sense that for any other choice of products, there is a canonical isomorphism between the two symmetric monoidal categories. Specifically, there is an invertible strong symmetric monoidal functor whose underlying functor is the identity and whose structure maps are given by the universal property of products.

This fact generalizes to cartesian objects:

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

A cartesian object in a 2-category $C$ with finite 2-products is a symmetric [[pseudomonoid]] in $C$ in a canonical way.

=--

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

The idea of the proof is to reduce to the case of Cat using the [[Yoneda lemma for bicategories|2-categorical Yoneda embedding]].

Let $X$ be a cartesian object in $C$, and make a choice of  right adjoints to the diagonal and terminal maps. Because 2-functors preserve adjunctions and the 2-Yoneda embedding $C \to [C^{op}, Cat]$, $A \mapsto C(-,A)$ preserves 2-products, $C(-,X)$ is a cartesian object with a choice of right adjoints in the 2-category of 2-presheaves on $C$. In particular, since 2-products in this 2-category are computed pointwise, $C(Y,X)$ is a cartesian object with a choice of right adjoints in Cat, and hence has the structure of a symmetric pseudomonoid in Cat, for each object $Y$ of $C$. Then, by similar reasoning, $C(-,X)$ is a symmetric pseudomonoid in the 2-presheaf 2-category. Finally, because the 2-Yoneda embedding is locally fully faithful and in particular locally [[conservative functor|reflects isomorphisms]], $X$ is a symmetric pseudomonoid in $C$.

For any other choice of rights adjoints for the cartesian object $X$, there are unique invertible 2-cells between them that commute with the units and counits of the adjunctions. Threading these through the 2-Yoneda embedding as before, the 2-cells define the comparison iso-cells of a strong morphism of symmetric pseudomonoids whose underlying morphism between the carrier objects is the identity.

=--

## References

* Carboni, Kelly, Wood, _A 2-categorical approach to change of base and geometric morphisms I_ (1991) ([Numdam](http://www.numdam.org/item/?id=CTGDC_1991__32_1_47_0)), Section 5: Cartesian objects in $F$ and various morphisms between them

[[!redirects cartesian objects]]


cartesian power

Given a [[set]] $S$ and a [[cardinal number]] $\kappa$, the
$\kappa$th __cartesian power__ $S^\kappa$ of $S$ is the $\kappa$-fold [[cartesian product]] of $S$ with itself.

In particular, the $S^2$ is the [[cartesian square]] of $S$, the set of [[ordered pairs]] of elements of $S$; and $S^{\aleph_0}$ is the set of [[infinite sequences]] of elements of $S$.

The concept generalises from [[Set]] to any [[category]] $C$ with all [[products]]; $S$ becomes an [[object]] of $C$, but $\kappa$ remains a cardinal number (still essentially an object of $Set$).

If we think of $\kappa$ as a full-fledged [[set]] in its own right (rather than just its cardinal number), then we are talking about a _[[function set]]_, and the generalisation is to [[cartesian closed categories]].

## See also

* [[n-ary operation]]
* [[multicategory]]
* [[multimorphism]]

[[!redirects cartesian power]]
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[[!redirects Cartesian power]]
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cartesian product

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In the strict sense of the word, a _cartesian product_ is a [[product]] in [[Set]], the [[category]] of [[sets]]. Hence for $S_1$ and $S_2$ two [[sets]], their cartesian product is the set denoted $S_1\times S_2$ whose [[elements]] are [[ordered pairs]] of elements in $S_1$ and $S_2$, respectively.

The abstract concept of such [[products]] generalizes from [[Set]] to any other [[category]] (as a special case of a category-theoretic _[[limit]]_), only that in general products of any given [[objects]] may or may not actually exist in that category.

(If they all exist, then one speaks of a [[cartesian monoidal category]]. Especially if the category is also a [[monoidal category]] with respect to some other [[tensor product]], then one says "Cartesian product" to distinguish the two. For instance one speaks of the cartesian product on [[Cat]] and on [[2Cat]] in contrast to the [[Gray tensor product]].)

A product [[created limit|created]] by the [[forgetful functor]] of a [[concrete category]], especially in [[algebra]], is often called a [[direct product]] (for instance a [[direct product of groups]]).

## Definition
 {#Definition}

### In Sets

Given any family $(A_i)_{i:I}$ of sets, the __cartesian product__ $\prod_i A_i$ of the family is the set of all [[functions]] $f$ from the index set $I$ with $f_j$ in $A_j$ for each $j$ in $I$.

As stated, the [[target]] of such a function depends on the argument, which is natural in dependent [[type theory]]; but if you don't like this, then define $\prod_i A_i$ to be the set of those functions $f$ from $I$ to the [[disjoint union]] $\biguplus_i A_i$ such that $f_j \in A_j$ (treating $A_j$ as a [[subset]] of $\biguplus_i A_i$ as usual) for each $j$ in $I$.

In traditional forms of [[set theory]], one can also take the target of $f$ to be the [[union]] $\bigcup_i A_i$ or even the class of all objects (equivalently, leave it unspecified).

### In a general category
 {#InAGeneraCategory}

Given any [[category]] $\mathcal{C}$, and any set $\{X_i\}_{i \in I}$ of its [[objects]], then the product of all these objects is, if it exists, an object denoted

$$
  \underset{i \in I}{\prod} X_i
  \in
  \mathcal{C}
$$

and equipped with [[morphisms]]

$$
  p_i \;\colon\; \left(\underset{i \in I}{\prod} X_i\right)
   \longrightarrow
  X_i
$$

(the _[[projections]]_), for each $i \in I$, such that it is [[universal property|universal with this property]], i.e. such that given any other object $Q \in \mathcal{C}$ with morphisms

$$
  Q \overset{f_i}{\longrightarrow}  X_i
$$

for $i \in I$, then there is a _unique_ morphism

$$
  (f_i)_{i \in I}
  \;\colon\;
  Q \longrightarrow \underset{i \in I}{\prod} X_i
$$

which factors the $f_i$ through the $p_i$, i.e. such that all these [[commuting diagram|diagrams commute]]:

$$
  \array{
    Q
    \\
    {}^{\mathllap{(f_i)_{i \in I}}}\downarrow & \searrow^{\mathrlap{f_i}}
    \\
    \underset{i \in I}{\prod} X_i &\underset{p_i}{\longrightarrow}& X_i
  }
$$

for all $i \in I$.

In the case of a binary product it is also denoted by "$(-)\times(-)$":

$$
  \array{
    && Q
    \\
    & \swarrow &\downarrow^{\mathrlap{\exists !}}& \searrow
    \\
    X_1 &\overset{p_1}{\longleftarrow}& X_1 \times X_2 &\overset{p_2}{\longrightarrow}& X_2
  }
  \,.
$$

An important special case is a [[power]] of an object $X$, where (referring to notation above) we take all $X_i$ to be $X$ and form

$$X^I = \prod_{i \in I} X$$

Referring to notation above, if we take $Q = X$ and all $f_i: Q \to X_i$ to be the [[identity morphism]] $1_X: X \to X$, then the map

$$(f_i)_{i \in I} = (1_X)_{i \in I}: X \to X^I$$

is called the *[[diagonal morphism]]*, typically denoted by $\delta_X$ or $\Delta_X$. For example, we have a diagonal map

$$\delta_X: X \to X \times X$$

for binary products.

### As a functor

If a [[category]] $C$ admits $I$-fold products (products over an indexing set $I$), then a product [[functor]]

$$\prod_I = \prod_{i \in I}: C^I \to C$$

can be defined, taking a collection of [[objects]] $(c_i)_{i \in I}$ to their product, and taking a collection of [[morphisms]] $(f_i: c_i \to d_i)_{i \in I}$ to a morphism

$$\prod_{i \in I} f_i: \prod_{i \in I} c_i \to \prod_{i \in I} d_i$$

defined to be $(f_i \circ p_i: \prod_{i \in I} c_i \to d_i)_{i \in I}$, using the notation above. (N.B.: the [[Cat|category of (small) categories]] has itself small products, and the notation $C^I$ makes reference to this fact. With some care, we can remove the restriction to [[small categories]].)

As a [[functor]], $\prod_I: C^I \to C$ is [[right adjoint]] to the diagonal functor $\Delta: C \to C^I$. The [[adjunction unit|unit of the adjunction]] $1_C \to \prod \Delta$, at the component $c \in Ob(C)$, is the [[diagonal morphism]]

$$\delta_c: c \to c^I$$

and the [[counit]] $\Delta \prod \to 1_{C^I}$ is the $I$-tuple of projection maps:

$$p_i: \prod_{i \in I} c_i \to c_i$$

Following the yoga of [[adjunctions]], we can write the "tupling" $(f_i)_{i \in I}: Q \to \prod_{i \in X_i}$ in terms of the product functor and the unit/diagonal map:

$$(f_i)_{i \in I} = \left(Q \stackrel{\delta_Q}{\to} \prod_{i \in I} Q \stackrel{\prod_{i \in I} f_i}{\to} \prod_{i \in I} X_i\right)$$

while in the other direction, as stated in the [[universal property]] above, we can retrieve the components $f_i: Q \to X_i$ of a general map $f: Q \to \prod_{i \in I} X_i$ in terms of the diagonal functor and the counit/projection maps. It boils down to saying

$$f_i = \left(Q \stackrel{f}{\to} \prod_{i \in I} X_i \stackrel{p_i}{\to} X_i\right)$$

## Examples

### Products of sets

Given [[sets]] $A$ and $B$, the cartesian product of the binary family $(A,B)$ is written $A \times B$; its elements $(a,b)$ are called __[[ordered pairs]]__.  (In [[set theory]], one often makes a special definition for this case, defining
$$ (a,b) = \{\{a\},\{a,b\}\} $$
rather than as a function so that ordered pairs can then be used in the definition of function.  From a structural perspective, however, this is unnecessary.)

Given sets $A_1$ through $A_n$, the cartesian product of the $n$-ary family $(A_1,\ldots,A_n)$ is written $\prod_{i=1}^n A_i$; its elements $(a_1,\ldots,a_n)$ are called __ordered $n$-[[tuples]]__.

Given sets $A_1$, $A_2$, etc, the cartesian product of the countably infinitary family $(A_1,A_2,\ldots)$ is written $\prod_{i=1}^\infty A_i$; its elements $(a_1,a_2,\ldots,)$ are called __infinite [[sequences]]__.

Given a set $A$, the cartesian product of the unary family $(A)$ may be identified with $A$ itself; that is, we identify the __ordered [[singleton]]__ $(a)$ with $a$.

The cartesian product of the empty family $()$ is the [[point]], a set whose only element is the __[[empty list]]__ $()$; we often call this set $1$ (or $\pt$, when we\'re Urs) and write its element as $*$.

### Further examples

* In categories of [[algebras]], products are constructed in the "obvious" manner; see for example [[direct product group]].

For [[algebraic categories]] like [[Grp]], where the objects are sets equipped with (globally defined) specified operations that satisfy specified universally quantified identities, products are always constructed in the way you'd expect, viz. by taking products at the level of the underlying sets and endowing them with operations defined in the evident pointwise fashion, just like the way it works in $Grp$. More commentary on this in more general contexts will be given below.

* For a non-example: The category [[Field]] of [[fields]] does not admit [[products]], even though some might consider that an algebraic example (in some loose or informal sense).

* In categories of topological type, products are again constructed in a common sense manner; see for instance [[product topological space]] for a prototype.

For [[topological categories]], which include examples like [[Top]] and [[Preord]], products (e.g. [[product topological spaces]]) are always constructed in the way one expects, by taking products at the level of the underlying sets and endowing them with an initial lift structure; e.g., in the case of $Top$, the smallest or initial topology for which the projection maps are continuous.

This principle continues to hold even for *infinitary* products, where the correct structure might not be obvious without guidance from categorical considerations. For example, for topological spaces, for an infinite product $\prod_{i \in I} X_i$, we use the [[product topology]], and not say the [[box topology]] which might be the first thing one would try.

* In any [[presheaf category]], i.e., a category of type $Set^C$ where $C$ is a [[small category]], products are calculated in obvious pointwise fashion, where $(F \times G)(c) = F(c) \times G(c)$ with the pointwise product projection maps. This even works if $C$ is [[large category|large]] (making $Set^C$ 'very large').

* If $D$ has products and $C \hookrightarrow D$ is a [[reflective subcategory]], products in $C$ exist and are calculated as they are in $D$. For example, this applies to any [[Grothendieck topos]], as a category of [[sheaves]] on a [[site]] forming a reflective subcategory of a category of presheaves (on the underlying category of the site).

  * For instance, the category [[Pos]] of [[posets]] is a reflective subcategory of [[Preord]].
  * Any [[locally presentable category]] can be exhibited as a reflective subcategory of a presheaf topos; as such we know how to calculate products there.

* Generalizing the last example still further, if $U: A \to X$ is a [[monadic functor]], then products and more generally [[limits]] in $A$ are created (or [[reflected limit|reflected]]) from products/limits in $X$. In other words, products of algebras over a [[monad]] $T: X \to X$ are given by products in $X$, equipped with the evident "pointwise" $T$-algebra structure.

* If $D$ has products and $i: C \hookrightarrow D$ is a *[[coreflective subcategory|coreflective]]* subcategory, where $i$ has a right adjoint (co-reflector) $r$, then products in $C$ can be formed by taking products in $D$ and then applying $r$.

  * For example, the category $CG$ of [[compactly generated spaces]] is coreflective in [[Top]]. The product in $CG$ does not coincide in general with the topological product: one may have to add to the product topology still more sets, just enough to ensure that the topology is compactly generated.

* Products in $C^{op}$ are given by [[coproducts]] in $C$.

  * In some cases, this formal tautology gives the only sensible way to construct products. For example, to form products $X \times Y$ in the category [[Loc]] of [[locales]], one must take the formal dual of the coproduct (denoted $\mathcal{O}(X) \otimes \mathcal{O}(Y)$) in the category of [[frames]].

  * Similarly, the product of two [[affine schemes]] is the formal dual of the coproduct of their corresponding coordinate rings, again given by a tensor product.

* Relative to a [[symmetric monoidal category]] $C$ with monoidal product $\otimes$, products in the category of [[cocommutative comonoids]] are given by the tensor product $X \otimes Y$ in the underlying smc category.

  * For instance, this is how products of [[cocommutative coalgebras]] are formed. This applies for instance to cocommutative [[Hopf algebras]].
  * In a [[cartesian bicategory]] such as $Rel$, the accompanying category of objects and *maps* (= left adjoint 1-cells) coincides with the category of cocommutative comonoids and comonoid maps between them (which is $Set$ in the case of $Rel$). Thus, at the level of objects, the usual tensor product in $Rel$ coincides with the cartesian product in $Set$.

* For categories $C$ enriched in the category of [[commutative monoids]], [[finite products]] are [[biproducts]] and hence coincide with coproducts in a controlled fashion (via [[matrices]]). However, this is usually not the case for infinite products. For such examples, finite products are [[absolute limits]].

* In the category [[SupLat]] of [[sup-lattices]], arbitrary products coincide with coproducts (and in fact the functor $X \mapsto X^{op}$ that takes a sup-lattice to its posetal opposite is part of a perfect duality $SupLat^{op} \to SupLat$, taking a sup-lattice map $f: X \to Y$ to $g^{op}: Y^{op} \to X^{op}$ where $f \dashv g$). This example descends to $Rel$ itself, so we conclude that products in $Rel$ are actually given by coproducts in $Rel$ which in turn are given by coproducts in $Set$.

* [[product of simplices]]

* In the [[Cat]] the cartesian product is the [[product of categories]].

* If either [[projection]] out of a Cartesian product is regarded as a ([[fiber bundle|fiber]]) [[bundle]] map, it is called a _[[trivial fiber bundle]]_.

## Properties

### Relation to type theory

Under the [[relation between category theory and type theory]] products in a category correspond to [[product types]] in [[type theory]].

[[!include product natural deduction - table]]

### Foundational status

In [[material set theory]], the existence of binary cartesian products follows from the [[axiom of pairing]] and the axiom of [[weak replacement]] (which is very weak).  In [[structural set theory]], their existence generally must be stated as an axiom: the __axiom of products__.  See [[ordered pair]] for more details.

## Related entries

* [[cartesian monoidal category]]

* [[finite product]]

* [[tensor product]]

* [[sum]]

* [[external product]]

* [[restricted product]]

* [[product type]]

* [[base change]]

  * [[dependent sum]], [[dependent product]]

  * [[dependent sum type]], [[dependent product type]]

## References

Textbook account:

* [[Francis Borceux]], Section 2.1 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

category: foundational axiom

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cartesian reflector

The term *cartesian reflector* (A4.3.1) is used in the [[Elephant]] to say *idempotent  cartesian monad*. This is because an idempotent monad is called a *reflector* (A1.1.1), there.


cartesian site

> This entry is about a general class of [[sites]]. For the specific site *[[CartSp]]* of [[cartesian spaces]], see there.

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## Defintion

A _cartesian site_ is a [[site]] whose underlying [[category]] is [[finitely complete category|finitely complete]] (which the [[Elephant]] calls a [[cartesian category]]).

Cartesian sites play a central role in the construction of [[classifying topos]].

## Examples

The [[syntactic site]] of any [[theory]] is a cartesian site.

## References

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_


cartesian space

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#### Differential geometry
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# Contents
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## Idea
  {#Idea}

A __Cartesian space__ is a [[finite product|finite]] [[Cartesian product]] of the [[real line]] $\mathbb{R}$ with itself. Hence, a Cartesian space has the form $\mathbb{R}^n$ where $n$ is some [[natural number]] (possibly [[zero]]).

What exactly this means depends on which [[category]] the [[real line]] $\mathbb{R}$ is being considered as an [[object]] of.

For instance, if $\mathbb{R}$ is regarded as a [[topological space]] (hence an object in the category [[Top]]), then $\mathbb{R}^n$ is a [[Euclidean space]]. Another possibility is to regard $\mathbb{R}$ as a [[topological manifold]], [[differentiable manifold]] or [[smooth manifold]] (hence an object in the categories [[TopMfd]] or [[Diff]]), in which case $\mathbb{R}^n$ is the archetypical such [[manifold]] of [[dimension of a manifold|dimension]] $n$ (the basic [[coordinate chart]]).

The Cartesian space $\mathbb{R}^n$ with its standard [[topological space|topology]]/[[smooth structure]] is also called *real $n$-dimensional space*.

One could also regard $\mathbb{R}$ as the [[real vector space]] of [[dimension of a vector space|dimension]] 1, in which case $\mathbb{R}^n$ would be the real vector space of dimension $n$ (maybe understood as equipped with the [[linear basis]] canonically induced by this presentation), hence [[generalized the|the]] "real $n$-dimensional vector space". However, a vector space would typically not be referred to as a "Cartesian space".

But in between the last two examples, regarding $\mathbb{R}^n$ as an *[[affine space]]* makes it the basis of "[analytic geometry](https://en.m.wikipedia.org/wiki/Analytic_geometry)" in the sense originally due to [[René Descartes]] ([[Cartesian geometry]]), and this is where the term "Cartesian space" originates from.

One might also speak of the [[complexified]] cartesian space $\mathbb{C}^n$ as a [[complex manifold]], or indeed of the cartesian space $K^n$ for any [[field]] $K$, maybe regarded as an [[analytic space]], though this is not common terminology.

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

In particular:

*  $\mathbb{R}^0$ is the [[point]],
*  $\mathbb{R}^1$ is the [[real line]],
*  $\mathbb{R}^2$ is the real [[plane]], which may be identified (in two canonical ways) with the [[complex number|complex plane]] $\mathbb{C}$.

=--

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

Cartesian spaces carry plenty of further canonical structure:

* It is canonically a [[metric space]] and the [[Euclidean topology]] is the corresponding [[metric space|metric topology]].

* There is a canonical [[smooth structure]] on $\mathbb{R}^n$ that makes it a [[smooth manifold]].

* A Cartesian space is canonically a[[vector space]] over the [[field]] of [[real number]]s.

Sometimes one is interested in allowing $n$ to take other values, in which case one wants a [[product]] in some category that might not be the Cartesian product on underlying sets.

For example, if one is studying Cartesian spaces as [[inner product space]]s, then one might want an $\aleph_0$-dimensional Cartesian space to be the $\aleph_0$-dimensional [[Hilbert space]] $l^2$, which is a proper subset of the cartesian product $\mathbb{R}^{\aleph_0}$.

=--

## Properties

### Topological structure

The [[open n-ball]] is [[homeomorphic]] [[Cartesian space]] $\mathbb{R}^n$

$$
  \mathbb{B}^n \simeq \mathbb{R}^n
  \,.
$$

* [[topological invariance of dimension]]

### Smooth structures

For all $n \in \mathbb{N}$, the [[open n-ball]] with its standard [[smooth structure]] is [[diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^n$ with its standard smooth structure

$$
  \mathbb{B}^n \simeq \mathbb{R}^n
  \,.
$$

In fact, in $d \neq 4$ there is no choice:

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

For $n \in \mathbb{N}$ a [[natural number]] with $n \neq 4$, there is a unique (up to [[isomorphism]]) smooth structure on the [[Cartesian space]] $\mathbb{R}^n$.

=--

This was shown in ([Stallings](#Stallings)).

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###### Theorem

In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist [[exotic smooth structure]]s.

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

In [[dimension]] $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is [[homeomorphic]] to $\mathbb{B}^d$ is also [[diffeomorphic]] to it.

=--

See the first page of ([Ozols](#Ozols)) for a list of references.

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

In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$.

=--

## The category of Cartesian spaces

See [[CartSp]].

## Related concepts

* [[super Cartesian space]]

* [[polydisk]]

* [[affine space]]

In [[complex geometry]] for purposes of [[Cech cohomology]] the role of Cartesian spaces is played by [[Stein manifolds]].

* [[topological invariance of dimension]]

## References

Named after [[René Descartes]].

* {#Stallings} [[John Stallings]], _The piecewise linear structure of Euclidean space_ , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. ([pdf](http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=2118140))

* {#Ozols} V. Ozols, _Largest normal neighbourhoods_ ,
Proceedings of the American Mathematical Society
Vol. 61, No. 1 (Nov., 1976), pp. 99-101  ([jstor](http://www.jstor.org/stable/2041672))

There are various slight variations of the category $CartSp$ that one can consider without changing its basic properties as a category of test spaces for [[generalized smooth space]]s. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see [[diffeological space]].

The [[site]] $ThCartSp$ of infinitesimally thickened Cartesian spaces is known as the site for the [[Cahiers topos]]. It is considered

in detal in section 5 of

* [[Anders Kock]], _Convenient vector spaces embed into the Cahiers topos_ ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0))

and briefly mentioned in example 2) on p. 191 of

* [[Anders Kock]], _Synthetic differential geometry_ ([pdf](http://home.imf.au.dk/kock/sdg99.pdf))

following the original article

* [[Eduardo Dubuc]], _Sur les modeles de la geometrie differentielle synthetique_ ([numdam](http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0)).

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))

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cartesian square

 _Cartesian square_ is another term for [[pullback]] [[diagram]].

Similarly,  *homotopy cartesian square* is another term for *[[homotopy pullback]]* diagram.

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cartesian theory > history

< [[cartesian theory]]


Cartier divisor

For the moment see at [[divisor (algebraic geometry)]]

## References

* [[The Stacks Project]], _[53.2 Effective Cartier Divisors](http://stacks.math.columbia.edu/tag/083A)_

[[!redirects Cartier divisors]]


Cartier duality

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## Idea

_Cartier [[duality]]_ is a refinement of [[Pontryagin duality]] from [[topological groups]] to [[group schemes]].

## Definition

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

Let $G$ be a [[finite group scheme]] over $k$, regarded as a
[[sheaf]] of groups $G \in Sh(Ring^{op}_k)$. Write $\mathbb{G}_m$ for the [[multiplicative group]], similarly regarded.

Then the _Cartier dual_ $\widehat G$ is the [[internal hom]]

$$
  \widehat G \coloneqq [G,\mathbb{G}_{m}]
$$

of group [[homomorphisms]], hence the [[sheaf]] which to $R \in Ring_k^{op}$ assigns the [[set]]

$$
  \widehat G \;\colon\; R \mapsto Hom_{Grp/Spec R}(G \times Spec R, \mathbb{G}_m \times Spec R)
$$

of group homomorphisms over $Spec(R)$

=--

This appears for instance as ([Polishchuk, (10.1.11)](#Polishchuk)).

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

Cartier duality is indeed a duality in that for any finite commutative group scheme $G$ there is an [[isomorphism]]

$$
  \widehat{\widehat{G}} \simeq G
$$

of the double Cartier dual with the original group scheme.

=--

(e.g. [Polishchuk, right above (10.1.11)](#Polishchuk), [Hida 00, theorem 1.7.1](#Hida00))

## Related concepts

* [[dual abelian group scheme]]

* [[Picard scheme]]

* [[Poincaré line bundle]]

## References

The classical textbook account is in chapter 1 of

* [[Jean Dieudonné]], _Introduction to the theory of formal groups_, Marcel Dekker, New York 1973.

and a more recent textbook account is in section 10.1 of

* [[Alexander Polishchuk]], _Abelian Varieties, Theta Functions and the Fourier Transform_
 {#Polishchuk}

or section 1.7 of

*  Haruzo Hida, _Geometric Modular Forms and Elliptic Curves_, 2000, World scientific
 {#Hida00}

lecture notes include

* [pdf](http://www.math.ethz.ch/~pink/ftp/FGS/Lecture11.pdf)

Generalization beyond finite group schemes is  discussed in

* Amelia Álvarez Sánchez, Carlos Sancho de Salas, Pedro Sancho de Salas, _Functorial Cartier duality ([arXiv:0709.3735](http://arxiv.org/abs/0709.3735))

and in

* [[Dima Arinkin]], appendix in [[Ron Donagi]], [[Tony Pantev]], _Torus fibrations, gerbes, and duality_, ([arXiv:math/0306213](http://arxiv.org/abs/math/0306213))

Discussion in the context of [[higher algebra]] ([[brave new algebra]]) is in

* [[Jacob Lurie]], section 5.3 of _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_
* [[Jacob Lurie]], section 3 of _[[Elliptic Cohomology I]]_

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Cartier module

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## Definition

Let $k$ be a [[perfect field]] of [[characteristic]] $p\neq 0$. Let $W$ be the [[ring]] of [[Witt vectors]] over $k$. A **Cartier module** is a pair $(M, f)$ where $M$ is a [[free construction|free]] $W$-[[module]] of finite [[rank]] and $f:M\to M$ is a semi-[[linear map|linear]] [[endomorphism]] in the following sense: $f(a\cdot m)=\phi(a)f(a)$ where $\phi$ is the [[Frobenius map]].

Cartier modules form a [[category]] by taking [[morphisms]] to be in the [[Mod|category of W-modules]] that also respect the extra $f$ data.

### Examples

* $(W, \phi)$ is a Cartier module

* If $G$ is a [[p-divisible group]] of height $h$, then the [[Dieudonne module]] $D(G)$ is a free $W$-module of rank $h$. The natural action of Frobenius turns $D(G)$ into a Cartier module.

* If $X$ [[proper map|proper]], [[smooth scheme|smooth]] [[scheme]] over $k$ of dimension $n$, then all $H^m_{crys}(X/W)/torsion $ with the action of pullback by Frobenius $F^*$ is a Cartier module when $m$<$n$.

## Slope Decomposition

Consider the Cartier module $(M, f)$. Let $K$ be the [[fraction field]] of $W$. Define the finite dimensional [[vector space]] $V=M\otimes_W K$. Extend $f$ linearly to $V$. Note that $f$ preserves the $W$-lattice $M$ inside $V$ by construction.

Define $A=K[T]$ to be the noncommutative [[polynomial]] ring with commutative relation $Ta=\phi(a)T$. This allows us to define a left $A$-[[action]] on $V$ by $T\cdot v=f(v)$.

Define $U_{r,s}$ to be the left $A$-module $A/A\cdot (T^s-p^r)$. This is the canonical $A$-module of pure [[slope]] $r/s$ and multiplicity $s$. It is a $K$-vector space of dimension $s$.

When $r\geq 0$ $T$ preserves the $W$ lattice $W[t]/W[t]\cdot(T^s-p^r)\subset U_{r,s}$. We have that $U_{r,s}$ is simple if and only if $(r,s)=1$. It is a theorem of Dieudonne and Manin that when $k$ is algebraically closed there is a unique choice of integers $r_i, s_i$ with $s_i\geq 1$ such that $r_1/s_1$ < $r_2/s_2$ < $\cdots$ < $r_i/s_i$ where $V$ decomposes as a direct sum $\bigoplus_{i=1}^t V_{r_i/s_i}$ where $V_{r_i/s_i}$ is noncanonically isomorphic as an $A$-module to $U_{r_i, s_i}$. This is called the *slope decomposition* of $V$.

The $r_i/s_i$ are called the [[slope]]s of $V$ with multiplicity $s_i$. Up to noncanonical isomorphism $V$ is completely determined by knowledge of the slopes and multiplicities.

### Examples

* Let $k=\mathbb{F}_q$ with $q=p^a$. Given a Cartier module $(M, F)$, the slopes of $(M\otimes_{W(k)}W(\overline{k}), F)$ are the $p$-adic valuations (chosen so $\nu(q)=1$) of the eigenvalues of the linear endomorphism $F^a$ of $M$, and the multiplicity is the (algebraic) multiplicity of this eigenvalue.

* In the second example above, if $G$ a [[p-divisible group]], then $(D(G), F)$ has all slopes in $[0,1)$.

* In the third example above if $X$ is projective, then since $F_*\circ F^* = p^n$, all the slopes of $H^m_{crys}(X/W)/torsion$ lie in $[0,n]$.

## References

* [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)

* Pierre Berthelot, _Slopes of Frobenius in Crystalline Cohomology_, Proceedings of Symposia in Pure Mathematics Vol 29, 1975.

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Cartier operator

#Contents#
* table of contents
{:toc}

## Notation

For the following discussion we suppose $\pi: X\to S$ is a [[smooth scheme|smooth]] map of [[scheme]]s. Let $d:\mathcal{O}_X\to \Omega^1_{X/S}$ be the standard differential. It is an integrable [[flat connection|connection]] on $\mathcal{O}_X$. We'll define the (relative) de Rham cohomology to be the [[hyper-derived functor]] pushforward applied to the de Rham complex $\mathbf{R}^q\pi_*(\Omega_{X/S}^\bullet)$.

We will denote the relative Frobenius map $F: X\to X^{(p)}$ where $X^{(p)}$ is the pullback of the structure map and the absolute Frobenius $F_{ab}: S\to S$, i.e. $X^{(p)}=X\otimes_{\pi^{-1}(\mathcal{O}_S)} \pi^{-1}(\mathcal{O}_S)$. If $\mathcal{A}^\bullet$ is a complex of sheaves, then we denote by $\mathcal{H}^q(\mathcal{A}^\bullet)$ the sheaf that is obtained by taking cohomology with respect to the maps of the complex.

## The Cartier Isomorphism

Suppose that $S$ is an $\mathbb{F}_p$-scheme and $X/S$ is smooth, then there is a unique map of graded $\mathcal{O}_{X^{(p)}}$-algebras $C^{-1}: \Omega^i_{X^{(p)}/S} \stackrel{\sim}{\to} \mathcal{H}^i(F_*\Omega^\bullet_{X/S})$ that satisfies:

$C^{-1}(1)=1$,

$C^{-1}(\omega\wedge \tau)=C^{-1}(\omega)\wedge C^{-1}(\tau)$

and $C^{-1}(d F_{ab}^{-1}(f))=[f^{p-1}d f]$ in $\mathcal{H}^1(F_*\Omega^\bullet_{X/S})$.

The inverse of this isomorphism is the traditional Cartier isomorphism.

The construction is quite simple. First note that we can immediately reduce to constructing $C^{-1}$ for $i=1$. This is because if $C^{-1}(1)=1$ and $C^{-1}$ is $\mathcal{O}_{X^{(p)}}$-linear it is determined for $i=0$. Likewise, if $C^{-1}$ is determined for $i=1$, then the case $i\geq 1$ is determined from the second property.

Now to construct for $i=1$ we just note that such a map is equivalent to a $(\pi^{(p)})^{-1}$-linear derivation $\mathcal{O}_{X^{(p)}}\to \mathcal{H}^1(F_*\Omega_{X/S}^\bullet)$. This is equivalent to defining a map on local sections $\delta: \mathcal{O}_X\times \pi^{-1}(\mathcal{O}_S)\to \mathcal{H}^1(\Omega_{X/S}^\bullet)$ that is biadditive and satisfies the extra three properties

$\delta(f s, s')=\delta(f, s^p s')$,

$\delta(g f, s)=g^p\delta(f,s)+f^p\delta(g,s)$ and

$\delta(f,1)=[f^{p-1}d f]$.

Now define the map explicitly by $\delta(f,s)=[s f^{p-1}d f]$. It can be checked that this map satisfies all the properties listed and is indeed an isomorphism. This is $C^{-1}$, the inverse of the Cartier isomorphism.

## Relation to the Hodge-de Rham Spectral Sequence

For this discussion let's assume that $X/k$ is proper and smooth. Deligne and Illusie had an insight that the degeneration of the Hodge-de Rham spectral sequence (HdR SS) is closely related to the Cartier isomorphism. Recall that the HdR SS is formed by taking the spectral sequence associated to hypercohomology $E_1^{p,q}=H^q(X, \Omega^p_{X/k})\Rightarrow H^{p+q}_{dR}(X/k)$.

Now notice that if we form the complex $\bigoplus_{i}\Omega^i_{X^{(p)}}[-i]$ which is $\Omega^i$ in degree $i$ and $d=0$ everywhere, then the left side of the inverse Cartier isomorphism is exactly $\mathcal{H}^i(\bigoplus_{i}\Omega^i_{X^{(p)}}[-i])$. Likewise, the right side is $\mathcal{H}^i$ of the complex $F_*\Omega_{X/k}^\bullet$. We can think of both of these complexes as living in $\mathbf{D}(X^{(p)}):=\mathbf{D}^b_{qCoh}(X^{(p)})$.

We can ask whether or not there is some map in the derived category $\phi: \bigoplus_{i}\Omega^i_{X^{(p)}}[-i] \to F_*\Omega_{X/k}^\bullet$ with the property that $\mathcal{H}^i(\phi)=C^{-1}$ for all $i$. It turns out this is a sufficient condition for convergence of the HdR SS. This is just because we get a string of isomorphisms

$\mathbf{H}^n(X, \Omega_X^\bullet)=\mathbf{H}^n(X^{(p)}, F_*\Omega_X^\bullet)\simeq \bigoplus_{i} H^{n-i}(X^{(p)}, \Omega_X^i)$. Thus the dimensions of the $k$-vector spaces at the $E_1$ term match the dimensions at the $E_\infty$ term. Since everything is a $k$-vector space this is all that is needed for degeneration (there can be no non-trivial quotients without dimension decreasing).

## The Generalization to Non-commutative algebra

See Kaledin Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie on the arxiv

## References

Nilpotent Connections and the Monodromy Theorem by Nicholas M. Katz

Relevements modulo $p^2$ et decomposition du complexe de de Rham by Deligne and Illusie

=--


cartographic group

The [[group]]

$$\mathcal{C}_2 = \langle\sigma_0,\sigma_1,\sigma_2 \mid \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = (\sigma_0\sigma_2)^2 = 1\rangle$$

is called the **cartographic group** (of dimension 2), while its [[index of a subgroup|index]] 2 [[subgroup]]

$$\mathcal{C}_2^+ = \langle\rho_0,\rho_1,\rho_2 \mid \rho_1^2 = \rho_0\rho_1\rho_2 = 1\rangle$$

is called the **oriented cartographic group**.  Specifically, this terminology comes from [[Grothendieck]]'s [[Esquisse d'un programme]], and is motivated by the fact that [[transitive action|transitive]] [[permutation representations]] (or equivalently, [[conjugacy class]]es of subgroups) of $\mathcal{C}_2^+$ can be identified with [[topological maps]] on connected, oriented surfaces without boundary, while more generally, transitive permutation representations of $\mathcal{C}_2$ can be identified with maps on connected surfaces which may or may not be orientable or have a boundary.

## Higher dimensions

The $n$-dimensional analogue of the cartographic group is

$$\mathcal{C}_n = \langle\sigma_0,\dots,\sigma_n \mid \sigma_i^2 = (\sigma_i\sigma_j)^2 = 1, (|i-j|\gt 1)\rangle,$$

which is a [[Coxeter group]].  For related references, see the last section of [Jones and Singerman](#JonesSingerman94).

## Related concepts

* [[topological map]]
* [[combinatorial map]]
* [[child's drawing]]

## References

* [[Alexander Grothendieck]]. [[Esquisse d'un programme]] (section 3)
* [[Christine Voisin]] and [[Jean Malgoire]]. Cartes cellulaires, Cahiers Mathématiques, 12, Montpellier, 1977. ([sudoc](http://www.sudoc.fr/009461914))
* [[Gareth A. Jones]] and [[David Singerman]]. Theory of Maps on Orientable Surfaces. Proceedings of the London Mathematical Society, 37:273-307, 1978. ([doi](https://doi.org/10.1112/plms/s3-37.2.273))
* Robin P. Bryant and [[David Singerman]]. Foundation of the Theory of Maps on Surfaces With Boundary. Quarterly Journal of Mathematics, 2(36):17-41, 1985. ([doi](https://doi.org/10.1093/qmath/36.1.17))
* {#JonesSingerman94} [[Gareth Jones]] and [[David Singerman]]. Maps, hypermaps, and triangle groups.  In _The Grothendieck Theory of Dessins d'Enfants_, [[L. Schneps]] (ed.), London Mathematical Society Lecture Note Series 200, Cambridge University Press, 1994. ([doi](https://doi.org/10.1017/CBO9780511569302.006))
* See also the Wikipedia page on [generalized maps](https://en.wikipedia.org/wiki/Generalized_map), which correspond to permutation representations of $\mathcal{C}_n$.


CartSp

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

# Contents
* table of contents
{: toc}

## Definition

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

Write $CartSp$ for the [[category]] whose

* [[objects]] are [[Cartesian space]]s $\mathbb{R}^n$ for $n \in \mathbb{N}$;

* [[morphisms]] are suitable structure-preserving [[functions]] between these spaces.

For definiteness we write

$CartSp_{lin}$ for the category whose objects are [[Cartesian space]]s regarded as [[real number|real]] [[vector spaces]] and whose morphisms are [[linear function]]s between these;

* $CartSp_{top}$ for the category whose objects are Cartesian spaces regarded as [[topological spaces]] equipped with their [[Euclidean topology]] and morphisms are [[continuous maps]] between them.

* $CartSp_{smth}$ for the category whose objects are Cartesian spaces regarded as [[smooth manifolds]] with their standard [[smooth structure]] and morphisms are [[smooth function]]s.

=--

## Properties

### As a small category of objects with a basis

A [[Cartesian space]] carries a lot of structure, for instance [[CartSp]] may be naturally regarded as a [[full subcategory]] of the category $C$, for $C$ (any one of) the category of

*  [[vector spaces]],
*  [[affine spaces]],
*  [[normed vector spaces]],
*  [[inner product spaces]],
*  [[Euclidean spaces]].

In all these cases, the inclusion $CartSp \hookrightarrow C$ is an [[equivalence of categories]]: choosing an [[isomorphism]] from any of these objects to a [[Cartesian space]] amounts to choosing a [[basis]] of a [[vector space]], a [[coordinate system]].

### As a site

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

Write

* [[CartSp]]${}_{top}$ for the category whose [[object]]s are Cartesian spaces and whose [[morphism]]s are all [[continuous maps]] between these.

* [[CartSp]]${}_{smooth}$ for the category whose [[object]]s are Cartesian spaces and whose [[morphism]]s are all [[smooth f# Refefunctions]] between these.

* [[CartSp]]${}_{synthdiff}$ for the [[full subcategory]] of the category of [[smooth loci]] on those of the form $\mathbb{R}^n \times D$ for $D$ an [[infinitesimal space]] (the formal dual of a Weil algebra).

=--

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

In all three cases there is the [[good open cover]] [[coverage]] that makes [[CartSp]] a [[site]].

=--

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

For [[CartSp]]${}_{top}$ this is obvious. For [[CartSp]]${}_{smooth}$ this is somewhat more subtle. It is a folk theorem (see the references at [[open ball]]). A detailed proof is at [[good open cover]]. This directly carries over to $CartSp_{synthdiff}$.

=--

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

* The site $CartSp_{top}$ is a [[dense subsite]] of the site of [[paracompact topological space|paracompact]] [[topological manifold]]s with the [[open cover]] [[coverage]].

* The site $CartSp_{smooth}$ is a [[dense subsite]] of the [[site]] [[Diff]] of [[paracompact topological space|paracompact]] [[smooth manifold]]s equipped with the [[open cover]] [[coverage]].

=--

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

Equipped with this structure of a site, [[CartSp]] is an
[[∞-cohesive site]].

=--

The corresponding [[cohesive topos]] [[sheaf topos|of sheaves]] is

* $Sh_{(1,1)}(CartSp_{smooth})$, discussed at [[diffeological space]].

* $Sh_{(1,1)}(CartSp_{synthdiff})$, discussed at [[Cahiers topos]].

The corresponding [[cohesive (∞,1)-topos]] [[(∞,1)-category of (∞,1)-sheaves|of (∞,1)-sheaves]] is

* $Sh_{(\infty,1)}(CartSp_{top}) =$ [[ETop∞Grpd]];

* $Sh_{(\infty,1)}(CartSp_{smooth}) =$ [[Smooth∞Grpd]];

* $Sh_{(\infty,1)}(CartSp_{synthdiff}) =$ [[SynthDiff∞Grpd]];

+-- {: .num_cor}
###### Corollary

We have [[equivalences of categories]]

* $Sh(CartSp_{top}) \simeq Sh(TopMfd)$

* $Sh(CartSp_{smooth}) \simeq Sh(Diff)$

and [[equivalences of (∞,1)-categories]]

* $Sh_{(\infty,1)}(CartSp_{top}) \simeq Sh_{(\infty,1)}(TopMfd)$;

* $Sh_{(\infty,1)}(CartSp_{smooth}) \simeq Sh_{(\infty,1)}(Diff)$.

=--

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

The first two statements follow by the [above proposition](#AsDenseSubsite) with the _comparison lemma_ discussed at [[dense sub-site]].

For the second condition notice that since an [[∞-cohesive site]] is in particular an [[∞-local site]] we have that $Sh_{(\infty,1)}(CartSp)$ is a [[local (∞,1)-topos]]. As discussed there, this implies that it is a [[hypercomplete (∞,1)-topos]]. By the discussion at [[model structure on simplicial presheaves]] this means that it is [[presentable (∞,1)-category|presented]] by the Joyal-Jardine-[[model structure on simplicial sheaves]] $Sh(CartSp)^{\Delta^{op}}_{loc}$. The claim then follows with the first two statements.

=--

### As a category with open maps

There is a canonical structure of a category with [[open map]]s on $CartSp$ (...)

### As an algebraic theory

The category $CartSp$ is (the [[syntactic category]] of ) a [[Lawvere theory]]: the theory for [[smooth algebra]]s.

### As a pre-geometry

Equipped with the above [[coverage]]-structure,  [[open map]]-structure and [[Lawvere theory]]-property, $CartSp$ is essentially a [[pregeometry (for structured (∞,1)-toposes)]].

(Except that the pullback stability of the open maps holds only in the weaker sense of [[coverage]]s).

(...)

$\,$

## Related concepts

[[!include geometries of physics -- table]]

$\,$

## References
 {#References}

$CartSp_{smth}$ as an example of a "cartesian [[differential category]]":

* [[R. Blute]], [[J.R.B. Cockett]], [[Robert Seely]], section 2 of _Cartesian differential categories_, Theory and Applications of Categories **22** 23 (2009) 622-672 [[tac:22-23](http://www.tac.mta.ca/tac/volumes/22/23/22-23abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/22/23/22-23.pdf)]

{#CartSpAsSiteReference} $CartSp_{smth}$ as a convenient [[site]] for [[diffeological spaces]], [[smooth sets]], [[smooth groupoids]], ... [[smooth infinity-groupoids|smooth $\infty$-groupoids]] (and the term "CartSp", or similar, for it) was first considered in:

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], p. 22 in: *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012) 169-213 [[arXiv:0910.4001](http://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3](https://doi.org/10.1007/s00220-012-1510-3)]

* [[Urs Schreiber]], [[Zoran Škoda]], [§6.2](https://arxiv.org/pdf/1004.2472.pdf#page=15) of: *Categorified symmetries* [[arXiv:1004.2472](https://arxiv.org/abs/1004.2472)]

* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], Appendix of: _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]

* [[Urs Schreiber]], §3.2.1 & §4.4 in: *[[schreiber:differential cohomology in a cohesive topos]]* [[arXiv:1310.7930](https://arxiv.org/abs/1310.7930)]

following the [[site]] [[ThCartSp|$CartSp_{synthdiff}$]] of [[formal smooth manifold|infinitesimally thickened]] Cartesian spaces, previously claimed (then without proof, it seems) to be a site for the [[Cahiers topos]] in:

* [[Anders Kock]], Section 5 of: *Convenient vector spaces embed into the Cahiers topos*, 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)]

* {#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)]

The idea is also implicit in

* [[Denis-Charles Cisinski]], Exp. 6.1.2 in: *Faisceaux localement asphériques* (2003) [[pdf](https://cisinski.app.uni-regensburg.de/mtest2.pdf), [[Cisinski-FaisceauxLocAsph.pdf:file]]]

With an eye towards [[Frölicher spaces]], $CartSp_{synthdiff}$ also briefly appears in:

* Hirokazu Nishimura, Section 5 of: _Microlinearity in Frölicher -- Beyond the Regnant Philosophy of Manifolds_ [[arXiv:0912.0827](http://arxiv.org/abs/0912.0827)]

category: category

[[!redirects Cart]]
[[!redirects site of cartesian spaces]]
[[!redirects sites of cartesian spaces]]
[[!redirects site of cartesian manifolds]]
[[!redirects sites of cartesian manifolds]]

[[!redirects CartSp]]
[[!redirects Cart Sp]]

[[!redirects CartesianSpace]]
[[!redirects CartesianSpaces]]

[[!redirects SmoothCartSp]]


Cary Malkiewich

* [webpage](http://math.uiuc.edu/~cmalkiew/)

## Selected writings

On the [[stable homotopy category]] ([[stable homotopy theory]]):

* {#Malkiewich14} [[Cary Malkiewich]], _The stable homotopy category_, 2014 ([pdf](https://people.math.binghamton.edu/malkiewich/stable.pdf), [[MalkiewichStableHomotopyCategory.pdf:file]])

On the [[parameterized stable homotopy theory]] of [[parameterized spectra]]:

* [[Cary Malkiewich]], *Parametrized spectra, a low-tech approach* [[arXiv:1906.04773](https://arxiv.org/abs/1906.04773), user guide: [pdf](https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf), [[Malkiewich-ParametrizedSpectraUserGuide.pdf:file]]]

* {#Malkiewich23} [[Cary Malkiewich]], *A convenient category of parametrized spectra* [[arXiv:2305.15327](https://arxiv.org/abs/2305.15327)]

On the [[Dennis trace]] on [[algebraic K-theory]] and relating to [[bicategorical trace]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

## Related $n$Lab entries

* [[stable homotopy theory]]

* [[stable homotopy category]]

* [[Becker-Gottlieb transfer]]

category: people


Casimir effect

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum electrodynamics]] there is a net [[force]] between two conducting but uncharged parallel plates, which is absent in the [[classical field theory]] of [[electromagnetism]]. This quantum effect is called the _Casimir effect_.

## Related concepts

* [[Lamb shift]]

## References

Named after [[Hendrik Casimir]].

Discussion in the context of [[causal perturbation theory]] is in

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], pages 45-47 of _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))

Popular accounts often attribute the Casimir effect to [[vacuum energy]]. Discussion that this is false includes

* R. L. Jaffe, _The Casimir Effect and the Quantum Vacuum_, Phys.Rev. D72 (2005) 021301 ([arXiv:hep-th/0503158](https://arxiv.org/abs/hep-th/0503158))

* {#Nikolic16} [[Hrvoje Nikolić]], _Proof that Casimir force does not originate from vacuum energy_, Phys. Lett. B 761, 197 (2016) ([arXiv:1605.04143](https://arxiv.org/abs/1605.04143))

See also

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

[[!redirects Casimir effects]]


Casimir function

Casimir function is an element in the center of a [[Poisson algebra]].

Quantum analogues are the [[Casimir operator]]s.

[[!redirects Casimir functions]]


Casimir operator

#Contents#
* table of contents
{:toc}

## Definition

Given a [[Lie algebra]] $\mathfrak{g}$, then a _Casimir element_ or _Casimir [[invariant]]_ is an element in the [[center]] of its [[universal enveloping algebra]] $U(\mathfrak{g})$.

Often this is considered with some [[Lie algebra representation]] in mind, in which case one speaks of _Casimir operators_.

## References

Named after [[Hendrik Casimir]], who discovered the concept in the context of [[rigid body dynamics]].

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

[[!redirects Casimir operators]]
[[!redirects Casimir invariant]]
[[!redirects Casimir invariants]]
[[!redirects Casimir element]]
[[!redirects Casimir elements]]
[[!redirects Casimir]]


Casson invariant

Let $M$ be an integral [[homology sphere|homology 3-sphere]]. Then one can consider the moduli space of equivalence classes of $SU(2)$-valued representations of $\pi_1(M)$, i.e. $SU(2)$-local systems. Casson invariant counts them algebraically to yields an integral invariant of $M$.

There is a generalization to rational homology 3-spheres, so called **Casson-Walker invariant**. A holomorphic counterpart for (Calabi-Yau 3-folds) of the Casson invariant are essentially the [[Donaldson-Thomas invariants]], or physically counting [[BPS states]].

Floer homology of 3-manifolds, which uses the Chern-Simons functionals as Morse functions yields some related invariants as well.

* wikipedia [Casson invariant](http://en.wikipedia.org/wiki/Casson_invariant)

* Kevin Walker, _An extension of Casson invariant to rational homology spheres_, Bull. AMS, __22__, 2 (1990) 261--267; _A generalization of Casson invariant_, Annals of Math. Studies __126__, Princeton Univ. Press 1992.; and the book review by W. Goldman: [ps](http://www-users.math.umd.edu/~wmg/walker.ps), [dvi](http://www-users.math.umd.edu/~wmg/walker.dvi)

* C. H. Taubes, _Casson's invariant and gauge theory_,  J. Diff. Geom __31__ (1990) 547--599, [euclid](http://projecteuclid.org/euclid.jdg/1214444327), [MR1037415](http://www.ams.org/mathscinet-getitem?mr=1037415)

* [[eom]]: [Chern–Simons functional](http://eom.springer.de/C/c120140.htm)
[[!redirects Casson-Walker invariant]]

[[!redirects Casson's invariant]]


CAT

Recall that [[Cat]] is generally used to denote the (or a) [[category]] (or [[2-category]]) of [[categories]] (and [[functors]] and [[natural transformations]]).  Of course, we cannot usually expect to have a category of *all* categories, for the same reasons we cannot have a set of all sets, so $Cat$ usually denotes the category of [[small categories]], whatever that means in your chosen [[foundations]].  Of course, $Cat$ is then not a small category but a [[large category]].

In some foundations, such as [[Grothendieck universes]], it nevertheless makes sense to talk about the "category of [[large categories]]," which is itself not a large category (for the same reason) but a "very large" category (some older work calls it a [[metacategory]]).  It is fairly common to denote this "[[very large category|very large]]" category (or 2-category) by $CAT$.

#References#

* [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

category: category