projective space

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

#Contents#
* table of contents
{:toc}

## Definition

Let $T$ be an abelian [[Lawvere theory]] (one containing the theory of [[abelian groups]]). Write $\mathbb{A}^1$ for its canonical [[line object]] and $\mathbb{G}_m$ for the corresponding [[multiplicative group]]-object.

The **projective space** $\mathbb{P}_n$ of $T$ is the [[quotient]]

$$
  \mathbb{P}_n
  \coloneqq
  (\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m
$$

of the [[complement]] of the origin inside the $(n+1)$-fold [[Cartesian product]] of the line with itself by the canonical [[action]] of $\mathbb{G}_m$.

Any point $(x_0,x_1,\ldots,x_n)\in \mathbb{A}^{n+1} - \{0\}$ gives _homogeneous coordinates_ for its image under the quotient map. When considered in this fashion, one often writes $[x_0:x_1:\ldots:x_n]$. Homogeneous  coordinates were introduced in [Möbius 27](#Mobius27)

More generally, for $(X,0)$ a [[pointed topological space]] with (pointed) $\mathbb{G}_m$-[[action]], the quotient

$$
 \mathbb{P}(X)
  \;\coloneqq\;
  (X \setminus\{0\})
  /
  \mathbb{G}_m
$$

is the corresponding projective space.

If instead of forming the [[quotient]] one forms the [[quotient stack]]/[[action groupoid]], one speaks of the [[projective stack]]

$$
  \hat \mathbb{P}(X)
  \coloneqq
  (X\setminus\{0\})//\mathbb{G}_m
  \,.
$$

## Examples

### For commutative rings and algebras

For $T$ the theory of commutative [[ring]]s or more generally commutative [[associative algebra]]s over a ring $k$, $\mathbb{A}_k^1$ is the standard [[affine line]] over $k$. In this case $\mathbb{P}^n_k$ is (...) A closed sub[[scheme]] of $\mathbb{P}^n_k$ is a [[projective scheme]].

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

For $R$ a commutative $k$-algebra, there is a [[natural isomorphism]] between

* $\mathbb{Z}$-[[graded algebra|gradings]] on $R$;

* $\mathbb{G}_m$-[[action]]s on $Spec R$.

=--

The proof is spelled out at _[[affine line]]_.

### Real, complex, quaternionic, octonionic projective space
  {#RealComplexProjectiveOctonionicProjectiveSpace}

In particular, as $k$ ranges over the four finite-dimensional [[real normed division algebras]], we have

* $k =$ [[real numbers]]: [[real projective space]] $\mathbb{R}P^n$

  including the [[circle]] $\mathbb{R}P^1 \,\simeq\, S^1$

* $k =$ [[complex numbers]]: [[complex projective space]] $\mathbb{C}P^n$

  including the [[Riemann sphere]] $\mathbb{C}P^1 \,\simeq\, S^2$

* $k =$ [[quaternions]]: [[quaternionic projective space]] $\mathbb{H}P^n$

  including the [[4-sphere]] $\mathbb{H}P^1 \,\simeq\, S^4$

* $k =$ [[octonions]]: [[octonionic projective space]] $\mathbb{O}P^n$

  (for $n \leq 2$)

  including the [[8-sphere]] $\mathbb{O}P^1 \,\simeq\, S^8$

\linebreak

+-- {: .num_defn #TopologicalProjectiveSpace}
###### Definition
**(topological projective space)**

Let $n \in \mathbb{N}$. Consider the [[Euclidean space]] $k^{n+1}$ equipped with its [[metric topology]], let $k^{n+1} \setminus \{0\} \subset k^{n+1}$ be the [[topological subspace]] which is the [[complement]] of the origin, and consider on its underlying set the [[equivalence relation]] which identifies two points if they differ by [[multiplication]] with some $c \in k$ (necessarily non-zero):

$$
  (\vec x_1 \sim \vec x_2)
   \;\Leftrightarrow\;
  \left(
    \underset{c \in k}{\exists}
    ( \vec x_2 = c \vec x_1 )
  \right)
  \,.
$$

The [[equivalence class]] $[\vec x]$ is traditionally denoted

$$
  [x_1 : x_2 : \cdots : x_{n+1}]
  \,.
$$

Then the _[[projective space]]_ $k P^n$ is the corresponding [[quotient topological space]]

$$
  k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim
  \,.
$$

=--

+-- {: .num_lemma #CanonicalInclusionOfProjectiveSpaces}
###### Lemma
**(canonical inclusion of projective spaces)**

For $n \in \mathbb{N}$ the function between topological projective spaces from def. \ref{TopologicalProjectiveSpace} given by

$$
  \array{
    k P^n &\overset{}{\longrightarrow}& k P^{n+1}
    \\
    [x_1 : \cdots : x_{n+1}] &\mapsto& [ x_1 : \cdots : x_{n+1} : 0]
  }
$$

is a [[continuous function]].

=--

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

There is a [[commuting square]] of functions of underlying sets of the form

$$
  \array{
    (x_1, \cdots, x_{n+1}) &\mapsto& (x_1, \cdots, x_{n+1}, 0)
    \\
    k^{n+1} & \overset{}{\longrightarrow} & k^{n+2}
    \\
    \downarrow &\searrow& \downarrow
    \\
    k P^n &\longrightarrow& k P^{n+1}
    \\
    [x_1 : \cdots : x_{n+1}] &\mapsto& [ x_1 : \cdots : x_{n+1} : 0]
  }
  \,,
$$

where the two vertical functions are the defining quotient co-projections, which are continuous functions by nature of [[quotient spaces]]. The top function is clearly continuous ([[polynomials are continuous]]) and hence so is its composite with the right co-projection, inducated by the diagonal arrow in the above diagram.

This implies that the bottom function is continuous by the nature (the [[universal property]]) of the quotient space topology.

=--

+-- {: .num_prop #QuotientOfnSphereTopologicalProjectiveSpace}
###### Proposition
**(projective space as quotient space of an $n$-sphere)**

For $n \in \mathbb{C}$ there are [[homeomorphisms]]

1. $S^{n}/(\mathbb{Z}/2) \simeq \mathbb{R}P^n$

   between

   1. the [[quotient space]] of the [[Euclidean space|Euclidean]] [[n-sphere]] canonically regarded as a [[subspace]] of the [[Euclidean space]] $\mathbb{R}^{n+1}$ by the [[equivalence relation]] which identifies two points $\vec p \in \mathbb{R}^{n+1}$ if they differ by multiplication by $-1$

   1. [[real projective space]] (def. \ref{TopologicalProjectiveSpace})

1. $S^{2n+1}/S^1 \simeq \mathbb{C}P^{n}$

   between

   1. the [[quotient space]] of the [[Euclidean space|Euclidean]] [[n-sphere|(2n+2)-sphere]], canonically regarded as a [[subspace]] of the [[Euclidean space]] $\mathbb{R}^{2n+2} \simeq \mathbb{C}^{n+1}$ by the [[equivalence relation]] which identifies two points $\vec p \in \mathbb{C}^{n+1}$ if they differ by multiplication by a complex number of unit norm

   1. [[complex projective space]] (def. \ref{TopologicalProjectiveSpace}).

=--

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

It is clear that there is a [[bijection]] of underlying sets as claimed: Under the equivalence relation defining projective space, every element $\vec x = (x_1, \cdots, x_{n+1}) \in k^{n+1}$ is equivalent to one of unit norm, namely $\frac{1}{\vert \vec x\vert} \vec x$, hence lying on the unit sphere. Representatives of this form are unique up to further multiplication by elements in $k \setminus \{0\}$ of unit norm.

It remains to see that this bijection is a [[homeomorphism]]. For definiteness of notation, we discuss this for the case $k = \mathbb{C}$, the case $k = \mathbb{R}$ works verbatim the same, with the evident substitutions.

So we have a [[commuting diagram]] of functions of underlying sets

$$
  \array{
    S^{2n+1} &\hookrightarrow& \mathbb{C}^{n+1} \setminus \{0\}
    \\
    {}^{\mathllap{q_{S^{2n+1}}}}\downarrow &\searrow^{\mathrlap{f}}& \downarrow^{\mathrlap{q_{\mathbb{C}^{n+1}}}}
    \\
    S^{2n+1}/S^1 &\longrightarrow& \mathbb{C}P^n
  }
$$

where the top horizontal and the two vertical functions are continuous, and where the bottom function is
is a bijection.  Since the diagonal composite is also continuous, the nature of the [[quotient space topology]]
implies that the bottom function is also continuous. To see that it is a [[homeomorphism]] it hence remains to
see that it is an [[open map]] (by [this prop.](homeomorphism#HomeoContinuousOpenBijection)).

So let $U \subset S^{2n+1}/S^1$ be an open set, which means that $q_{S^{2n+1}}^{-1}(U) \subset S^{2n+1}$ is an open set.
We need to see that $f(q_{S^{2n+1}}^{-1}(U)) \subset \mathbb{C}P^{n}$ is open, hence that
$q_{\mathbb{C}^{n+1}}^{-1}(f(q_{S^{2n+1}}^{-1}(U))) \subset \mathbb{C}^{n+1}$ is open.
Now by the nature of the [[Euclidean space|Euclidean]] [[metric topology]], the open subset $q_{S^{2n+1}}^{-1}(U)$
is a union of open balls $B^\circ_x(\epsilon)$ in $\mathbb{C}^{n+1}$ intersected with $S^{2n+1}$. But then $q_{\mathbb{C}^{n+1}}^{-1}(f(B^\circ_x(\epsilon)\vert_{S^{2n+1}}))$
is their [[orbit]] under the multiplicative action by $\mathbb{C} \setminus \{0\}$,
hence is a [[cylinder]] $B^\circ_x(\epsilon)\vert_{S^{2n+1}} \times (\mathbb{C} \setminus \{0\})$. This is clearly open.

=--

+-- {: .num_prop #CellComplexStructureOnTopologicalProjectiveSpace}
###### Proposition
**([[cell structure of K-projective space]])**

There is a [[CW-complex]] structure on [[real projective space]] $\mathbb{R}P^n$ (def. \ref{TopologicalProjectiveSpace}) for $n \in \mathbb{N}$, given by [[induction]], where $\mathbb{R}P^{n+1}$ arises from $\mathbb{R}P^n$ by attaching a single cell of dimension $n+1$ with attaching map the [[projection]] $S^{n} \longrightarrow \mathbb{C}P^n$ from prop. \ref{QuotientOfnSphereTopologicalProjectiveSpace}:

$$
  \array{
     S^{n} &\longrightarrow& S^{n}/(\mathbb{Z}/2) \simeq \mathbb{R}P^n
     \\
     {}^{\mathllap{\iota_{n+1}}}\downarrow
       &(po)& \downarrow
     \\
     D^{n+1} &\underset{q}{\longrightarrow}& \mathbb{R}P^{n+1}
  }
  \,.
$$

Similarly, there is a [[CW-complex]] structure on [[complex projective space]] $\mathbb{C}P^n$ (def. \ref{TopologicalProjectiveSpace}) for $n \in \mathbb{N}$, given by [[induction]], where $\mathbb{C}P^{n+1}$ arises from $\mathbb{C}P^n$ by attaching a single cell of dimension $2(n+1)$ with attaching map the [[projection]] $S^{2n+1} \longrightarrow \mathbb{C}P^n$ from prop. \ref{QuotientOfnSphereTopologicalProjectiveSpace}:

$$
  \array{
     S^{2n+1} &\longrightarrow& S^{2n+1}/S^1 \simeq \mathbb{C}P^n
     \\
     {}^{\mathllap{\iota_{2n+2}}}\downarrow
       &(po)& \downarrow
     \\
     D^{2n+2} &\underset{q}{\longrightarrow}& \mathbb{C}P^{n+1}
  }
  \,.
$$

=--

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

we discuss the case $k = \mathbb{C}$. The case $k= \mathbb{R}$ works verbatim the same, with the evident substitutions.

Given [[homogeneous coordinates]] $(z_0 ,  z_1 ,  \cdots ,  z_n ,  z_{n+1} ,  z_{n+2}) \in \mathbb{C}^{n+2}$ for $\mathbb{C}P^{n+1}$, let

$$
  \phi \coloneqq -arg(z_{n+2})
$$

be the [[phase]] of $z_{n+2}$. Then under the equivalence relation defining $\mathbb{C}P^{n+1}$ these coordinates represent the same element as

$$
  \frac{1}{\vert \vec z\vert}(e^{i \phi} z_0, e^{i \phi}z_1,\cdots, e^{i \phi}z_{n+1}, r)
  \,,
$$

where

$$
  r = {\vert z_{n+2}\vert}\in [0,1) \subset \mathbb{C}
$$

is the [[absolute value]] of $z_{n+2}$. Representatives $\vec z'$ of this form (${\vert \vec z' \vert = 1}$ and $z'_{n+2} \in [0,1]$) parameterize the [[n-disk|2n+2-disk]] $D^{2n+2}$  with [[boundary]] the $(2n+1)$-sphere at $r = 0$.

The resulting function $q \colon D^{2n+2} \to \mathbb{C}P^{n+1}$ is continuous: It may be factored as

$$
  \array{
    q_{D^{2n+2}} \colon & D^{2n+2} &\overset{\phantom{AAA}}{\hookrightarrow}& \mathbb{C}^{n+2} \setminus \{0\} &\overset{q_{\mathbb{C}^{n+2}}}{\longrightarrow}&  \mathbb{C}P^{n+1}
    \\
    & (Re(z_1), Im(z_1),  \cdots, Re(z_{n+1}), Im(z_{n+1}), r) &\mapsto& (z_1, \cdots, z_{n+1}, r)  &\mapsto& [ z_1 : \cdots : z_{n+1} : r ]
  }
$$

and here the first map is the [[embedding of topological spaces|embedding]] of the disk $D^{2n+2}$ as a [[hemisphere]] in $\mathbb{R}^{2n+1} \hookrightarrow \mathbb{R}^{2n+2} \simeq \mathbb{C}^{2n+2}$, while the second is the defining quotient space projection.
Both of these are continuous, and hence so is their composite.

The only remaining part of the action of $\mathbb{C}-\{0\}$ which fixes the conditions ${\vert z'\vert} = 0$ and $z'_{n+2}$ is
$S^1 \subset \mathbb{C} \setminus \{0\}$ acting on the elements with $ r = \{z'_{n+2}\} = 0$ by phase shifts on the $z_0, \cdots, z_{n+1}$.
The quotient of this remaining action on $D^{2(n+1)}$ identifies
its boundary $S^{2n+1}$-sphere with $\mathbb{C}P^{n}$, by prop. \ref{QuotientOfnSphereTopologicalProjectiveSpace}.

This shows that the above square is a [[pushout]] diagram of underlying sets.

By the nature of [[colimits]] in [[Top]] ([this prop.](Top#DescriptionOfLimitsAndColimitsInTop))
it remains to see that the
[[topological space|topology]] on $\mathbb{C}P^{n+1}$ is the [[final topology]] induced by the functions
$D^{2n+2} \to \mathbb{C}P^{n+1}$ and $\mathbb{C}P^n \to \mathbb{C}P^{n+1}$, hence that a subset of
$\mathbb{C}P^{n+1}$ is open precisely if its pre-images under these two functions are open.

We saw above that $q_{D^{2n+2}}$ is continuous.
Moreover, also the function
$i_n \colon \mathbb{C}P^n \to \mathbb{C}P^{n+1}$ is continuous (by [this lemma](projective+space#CanonicalInclusionOfProjectiveSpaces)).

This shows that if a subset of $\mathbb{C}P^{n+1}$ is open, then its pre-images under these functions are open.
It remains to see that if $S \subset \mathbb{C}P^{n+1}$ is a subset with $q_{S^{2n+2}}^{-1}(S) \subset D^{2n+2}$ open and
$i_n^{-1}(S) \subset \mathbb{C}P^n$ open, then $S \subset \mathbb{C}P^{n+1}$ is open.

Notice that $q_{\mathbb{C}^{n+2}}^{-1}(S)$ contains with every point also its [[orbit]] under the [[action]]
of $\mathbb{C} \setminus \{0\}$, and that every open subset of $D^{2n+2}$ is a unions of open balls.
By the above factorization of $q_{D^{2n+2}}$ this means that if $q_{D^{2n+2}}^{-1}(S)$ is open, then
$q_{\mathbb{C}^{n+2}}^{-1}(S)$ is a union of open cyclinders, hence is open. By the nature of the
[[quotient topology]], this means that $S \subset \mathbb{C}P^n$ is open.

=--

+-- {: .num_defn #TopologicalProjectiveSpaceStandardOpenCover}
###### Definition
**(standard open cover of topological projective space)**

For $n \in \mathbb{N}$ the _standard open cover_ of the projective space $k P^n$ (def. \ref{TopologicalProjectiveSpace}) is

$$
  \left\{
    U_i \subset k P^n
  \right\}_{i \in \{1, \cdots, n+1\}}
$$

with

$$
  U_i
    \coloneqq
  \left\{
    [x_1 : \cdots : x_{n+1}] \in k P^n
    \;\vert\;
    x_i \neq 0
  \right\}
  \,.
$$

To see that this is an open cover:

1. This is a cover because with the orgin removed in $k^n \setminus \{0\}$ at every point $[x_1: \cdots : x_{n+1}]$
   at least one of the $x_i$ has to be non-vanishing.

1. These subsets are open in the [[quotient topology]] $k P^n = (k^n \setminus \{0\})/\sim$, since their [[pre-image]] under the quotient co-projection $k^{n+1} \setminus \{0\} \to k P^n$ coincides with the pre-image $(pr_i\circ\iota)^{-1}( k \setminus \{0\} )$ under the [[projection]] onto the $i$th coordinate in the [[product topological space]] $k^{n+1} = \underset{i \in \{1,\cdots, n\}}{\prod} k$ (where we write $k^n \setminus \{0\} \overset{\iota}{\hookrightarrow} k^n \overset{pr_i}{\to} k$).

=--

+-- {: .num_prop #nSphereAsCoveringSpaceOverRealProjectiveSpace}
###### Proposition
**([[n-sphere]] projecting to real projective space is [[covering space]] projection)

For $n \in \mathbb{N}$, the [[continuous function]] $p \;\colon\; S^n \to \mathbb{R}P^n$ from prop. \ref{QuotientOfnSphereTopologicalProjectiveSpace}
is a [[covering space]] projection.

=--

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

We need to produce an [[open cover]] $\{U_i \subset \mathbb{R}P^n\}_{i \in I}$ such that the restrictions of the projection to this cover are [[homeomorphism|homeomorphic]] over the base to a [[product topological space]]

$$
  \array{
     U_i \times Disc(\mathbb{Z}_2)
       && \overset{\simeq}{\longrightarrow} &&
    S^n|_{U_i}
    \\
    & \searrow && \swarrow
    \\
    && U_i
  }
  \,.
$$

Consider the standard open cover from def. \ref{TopologicalProjectiveSpaceStandardOpenCover}. Hence $i \in \{1, \cdots, n+1\}$
and $U_i$ consists of those lines through the origin in $\mathbb{R}^{n+1}$ which do not lie in the subspace defined by $x_i = 0$. The intersection of this subspace with the unit sphere $S^n \subset \mathbb{R}^{n+1}$ is an [[equator]] of the $n$-sphere, and so the [[complement]] of this equator is the [[disjoint union]] of the two open [[hemispheres]] $D_i^\pm \subset S^n$. Hence

$$
  \array{
     S^n\vert_{U_i} & \simeq D_i^+ \sqcup D_i^-
   }
  \,.
$$

Moreover, each line in $\mathbb{R}^{n+1}$ which corresponds to an element in $U_i$ intersects $D^+_i$ as well as $D^-_i$ exactly once. In particular therefore the $\mathbb{Z}_2$-action on $S^n$ restricts over $U_i$ to the interchange of these two hemispheres, and hence prop. \ref{QuotientOfnSphereTopologicalProjectiveSpace} gives the required homeomorphism as above.

=--

+-- {: .num_prop #ProjectiveSpaceOpenCoverIsAtlas}
###### Proposition
**(standard open cover is [[atlas]])**

The charts of the standard open cover of def. \ref{TopologicalProjectiveSpaceStandardOpenCover} are [[homomorphism|homeomorphic]] to [[Euclidean space]] $k^n$.

=--

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

If $x_i \neq 0$ then

$$
  [x_1 : \cdots : x_i : \cdots : x_{n+1}]
  =
  \left[
    \frac{x_1}{x_i} : \cdots : 1 : \cdots \frac{x_{n+1}}{x_i}
  \right]
$$

and the representatives of the form on the right are _unique_.

This means that

$$
  \array{
    \mathbb{R}^n &\overset{\phi_i}{\longrightarrow}& U_i
    \\
    (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1})
    &\mapsto&
    [x_1: \cdots: 1: \cdots : x_n+1]
  }
$$

is a bijection of sets.

To see that this is a [[continuous function]], notice that it is the composite

$$
  \array{
    && \mathrlap{\mathbb{R}^{n+1} \setminus \{x_i = 0\}}
    \\
    & {}^{\mathllap{\hat \phi_i}}\nearrow & \downarrow
    \\
    \mathbb{R}^n
     &
      \underset{\phi_i}{\longrightarrow}
     &
     U_i
  }
$$

of the function

$$
  \array{
    \mathbb{R}^n &\overset{\hat \phi_i}{\longrightarrow}& \mathbb{R}^{n+1} \setminus \{x_i = 0\}
    \\
    (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1})
    &\mapsto&
    (x_1, \cdots, 1, \cdots ,x_n+1)
  }
$$

with the quotient projection. Now $\hat \phi_i$ is a [[polynomial]] function and since [[polynomials are continuous]], and since the projection to a [[quotient topological space]] is continuous, and since composites of continuous functions are continuous, it follows that $\phi_i$ is continuous.

It remains to see that also the [[inverse function]] $\phi_i^{-1}$ is continuous.
Since

$$
  \array{
    \mathbb{R}^{n+1} \setminus \{x_i = 0\}
      &\overset{}{\longrightarrow}&
    U_i
      &\overset{\phi_i^{-1}}{\longrightarrow}&
    \mathbb{R}^n
    \\
    (x_1, \cdots, x_{n+1})
    && \mapsto &&
    ( \frac{x_1}{x_i}, \cdots, \frac{x_{i-1}}{x_i}, \frac{x_{i+1}}{x_i}, \cdots , \frac{x_{n+1}}{x_i})
  }
$$

is a [[rational function]], and since [[rational functions are continuous]], it follows, by nature of the [[quotient topology]], that $\phi_i$ takes open subsets to open subsets, hence that $\phi_i^{-1}$ is continuous.

=--

+-- {: .num_prop #SmoothManifoldRealComplexProjectiveSpace}
###### Proposition
**(real/complex projective space is [[smooth manifold]])**

For $k \in \{\mathbb{R}, \mathbb{C}\}$ the topological projective space $k P^n$ (def. \ref{TopologicalProjectiveSpace}) is a [[topological manifold]].

Equipped with the standard open cover of def. \ref{TopologicalProjectiveSpaceStandardOpenCover} regarded as an [[atlas]] by prop. \ref{ProjectiveSpaceOpenCoverIsAtlas}, it is a [[differentiable manifold]], in fact a [[smooth manifold]].

=--

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

By prop. \ref{ProjectiveSpaceOpenCoverIsAtlas} $k P^n$ is a [[locally Euclidean space]]. Moreover, $kP^n$ admits the structure of a [[CW-complex]] (by prop. \ref{CellComplexStructureOnTopologicalProjectiveSpace})
and therefore it is a [[paracompact Hausdorff space]] since [[CW-complexes are paracompact Hausdorff spaces]]. This means that it is a [[topological manifold]].

It remains to see that the [[gluing functions]] of this atlas are [[differentiable functions]] and in fact [[smooth functions]]. But by prop. \ref{ProjectiveSpaceOpenCoverIsAtlas} they are even [[rational functions]].

=--

## Related concepts

* [[projective plane]]

* [[projective variety]]

* [[Grassmannian]]

* [[projective G-space]]

  [[infinite complex projective G-space]]

* [[projective geometry]], [[synthetic projective geometry]]

* [[projective linear group]]

* [[tautological line bundle]]

* [[projective bundle]]

* [[affine space]], [[conical space]]

* [[direction of a vector]]

## References

* {#Mobius27} [[August Ferdinand Möbius]], _Der barycentrische Calcül_ (1827)

Textbook account:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §9 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

See also:

* Miles Reid, _Graded rings and varieties in weighted projective space_ ([pdf](http://www.warwick.ac.uk/~masda/surf/more/grad.pdf))

* [[Aurelio Carboni]], [[Marco Grandis]] , _Categories of projective spaces_ , JPAA **110** (1996) pp.241-258.

[[!redirects projective spaces]]

[[!redirects homogeneous coordinates]]


projective special linear group

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

#Contents#
* table of contents
{:toc}

## Definition

The _projective general special linear group_ $PSL(n)$ (in some [[dimension]] $n$ and over some [[coefficients]]) is the [[quotient group]] of the [[special linear group]] $SL(n)$ by its [[center of a group|center]].

## Examples

* there is an [[exceptional isomorphism]] between the [[rotational icosahedral group]] $I$ and the projectiver special linear group in dimension 2 with [[coefficients]] in a [[prime field]]:  $I \simeq PSL_2(\mathbb{F}_5) $ ([this Prop.](icosahedral+group#IsomorphismToSL25))

* the [[automorphism group]] of the [[Fano plane]] is $\simeq PSL(2,\mathbb{F}_7)$ (...)

## Related concepts

* [[projective general linear group]]

* [[projective space]]

## References

* Wikipedia, _[Projective linear group](https://en.wikipedia.org/wiki/Projective_linear_group)_

[[!redirects special linear groups]]


projective stack

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

#Contents#
* table of contents
{:toc}

## Definition

A _[[projective space]]_ is a [[quotient]]

$$
 \mathbb{P}(X) :=  (X-\{0\})/\mathbb{G}_m
$$

by the action of a [[multiplicative group]] object (for a given [[line object]]).

If instead of forming the [[quotient]] one forms the weak quotient/[[action groupoid]], one speaks of the **projective stack**

$$
  \hat \mathbb{P}(X) := (X-\{0\})//\mathbb{G}_m
  \,.
$$

## References

The definition appears for instance as def 2.1.1 in

* Dan Abramovich and Brendan Hasset, _Stable varieties with a twist_ ([ps](http://math.rice.edu/~hassett/papers/projective/slcmodEMS2.ps))

[[!redirects projective spaces]]


projective structure

This is a disambiguation page.

Word _projective structure_ may mean a [[projective model structure]] where model structure is in the sense of Quillen model structure and it may refer to such a "projective" model structures in a variety of contexts, most classically on a category of 1-sided modules over a [[ring]] and also in the context of [[model structure on functors]].

It may also mean one of the [[Cartan geometry|Cartan geometries]] (locally homogeneous geometric structures), which is locally modeled on a projective space. This subject or its underlying object is sometimes called projective differential geometry, projective Cartan geometry, locally projective geometry, locally projective manifold, or a manifold with a (locally) projective structure.

It may also refer to one of the - realizations of/formalisms for - the projective structures, for example a projective atlas on a manifold or a projective Cartan connection. See also [[projectively flat connection]].

* William Goldman, _What is a... projective structure_, Notices Amer. Math. Soc., November 2008, [pdf](http://www.ams.org/notices/200811/tx081101418p.pdf)


projective unitary group

> Some overlap with [[PU(ℋ)]]. Needs to be disentangled.

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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

For $\mathcal{H}$ a [[Hilbert space]], the _projective unitary group_ $P U(\mathcal{H})$ is the [[quotient]] of the [[unitary group]] $U(\mathcal{H})$ by its [[center]] $Z U(\mathcal{H}) \simeq U(1)$, the [[circle group]]

$$
 P U(\mathcal{H}) \coloneqq U(\mathcal{H})/U(1)
  \,.
$$

=--

This is naturally a [[topological group]]. For $\mathcal{H}$ of finite-[[dimension]] $n$ $P U(n) := P U(\mathcal{H})$ is also naturally a [[Lie group]].

## Properties

### Homotopy type
 {#HomotopyType}

\begin{proposition}\label{KZ2}
If $\mathcal{H}$ is an infinite-[[dimension]]al [[separable Hilbert space]] the underlying [[topological space]] of its projective unitary group has the [[homotopy type]] of an [[Eilenberg-MacLane space]] $K(\mathbb{Z}, 2)$.
\end{proposition}

\begin{proof}
The [[unitary group]] $U(\mathcal{H})$ in this case is [[contractible]] (by [[Kuiper's theorem]]) and the [[circle group]] $U(1)$ [[action|acts]] free and faithfully on it. Therefore the quotient map $U(\mathcal{H}) \to P U(\mathcal{H})$ is a model for the [[circle group]]-[[universal principal bundle]] and in particular the topological space underlying $P U(\mathcal{H})$ is equivalent to the [[classifying space]] $B U(1) \simeq B^2 \mathbb{Z} \simeq K(\mathbb{Z},2)$.
\end{proof}

\begin{proposition}\label{ClassesOfPUHBundles}
For $\mathcal{H}$ an infinite-dimensional separable Hilbert space $P U(\mathcal{H})$-[[principal bundle]]s over a [[topological space]] $X$ are classidied by third [[integral cohomology]] of $X$

$$
  P U(\mathcal{H}) Bund(X) \simeq H^3(X, \mathbb{Z})
  \,.
$$
\end{proposition}

\begin{proof}
By prop. \ref{KZ2} we have that the [[classifying space]] of $P U(\mathcal{H})$ itself is an [[Eilenberg-MacLane space]]

$$
 B P U(\mathcal{H})
  \simeq
  B B U(1)
  \simeq
  B^3 \mathbb{Z}
  \simeq
  K(\mathbb{Z}, 3)
  \,.
$$

This is the classifying space for degree-3 integral cohomology (see [[Eilenberg-MacLane spectrum]] for more on this).
\end{proof}

\begin{prop}\label{ClassesOfPUHBundles}
Every [[circle n-bundle with connection|circle 2-bundle]]/[[bundle gerbe]] on $X$ is equivalent to the [[lifting gerbe]] of some $P U(\mathcal{H})$-principal bundle to a $U(\mathcal{H})$-bundle, and the equivalence classes of these structures correspond uniquely.
\end{prop}

\begin{proof}
The [[twisted bundles]] of a given bundle gerbe are given by the [[twisted cohomology]] relative to the morphism $\mathbf{B} P U(\mathcal{H}) \to \mathbf{B}^2 U(1)$ that is part of the long [[fiber sequence]]

$$
  \mathbf{B} U(1) \to \mathbf{B} U(\mathcal{H}) \to B \mathbf{P} U(\mathcal{H}) \to \mathbf{B}^2 U(1)
  \,.
$$

Since the topological space underlying $U(\mathcal{H})$ is contractible, on the underlying topological spaces this is

$$
  K(\mathbb{Z},2)
   \to
  *
   \to
  K(\mathbb{Z},3)
    \stackrel{\simeq}{\to}
  K(\mathbb{Z}, 3)
  \,.
$$

This means that the morphism that sends $P U(\mathcal{H})$-bundles to the twist that they induce is an isomorphism.
\end{proof}

> (Somebody should force me to say this in more detail).

For more on this see also [[twisted K-theory]].

### Action on Fredholm operators

Let $\mathcal{H}$ be an infinite-dimensional separable Hilbert space.

Since by the above $PU(\mathcal{H}) \simeq B U(1)$ and since there is a canonical action of [[line bundles]] on complex vector bundles, hence on the [[topological K-theory]] of a manifold $X$, there must also be a natural action of $PU(\mathcal{H}) \times Fred \to Fred$ of $PU(\mathcal{H})$ on the space of [[Fredholm operators]] (on connected components).

This is given by letting a projective unitary act by conjugation on a Fredholm operator: $(g, F) \mapsto g F g^{-1}$.

## Related concepts

* [[PU(ℋ)]], [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]

* [[projective linear group]]

* [[unitary group]]

  * [[special unitary group]], **projective unitary group**

* [[orthogonal group]]

  * [[special orthogonal group]]

  * [[spin group]]

## References

On the [[cohomology]] of the [[classifying spaces]] $B PU(n)$:

* Xing Gu, _On the Cohomology of the Classifying Spaces of Projective Unitary Groups_,  Journal of Topology and AnalysisVol. **13** 02 (2021)  535-573 ([arXiv:1612.00506](https://arxiv.org/abs/1612.00506), [doi:10.1142/S1793525320500211](https://doi.org/10.1142/S1793525320500211))

On the spaces of [[group homomorphisms]] into $PU(\mathcal{H})$ (with an eye towards the [[universal equivariant PU(H)-bundle|universal equivariant $PU(\mathcal{H})$-bundle]]):

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], Section 1 of: _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

[[!redirects projective unitary groups]]

[[!redirects PU(n)]]


projective variety

#Contents#
* table of contents
{:toc}

## Idea

A _projective [[algebraic variety]]_ (over an [[algebraically closed field]] $k$) is the 0-locus of a [[homogeneous polynomial|homogeneous]] ideal of [[polynomial]]s in $(n+1)$ variables over $k$ in the [[projective space|projective n-space]] $\mathbb{P}^n$.

## Examples

The archetypical example is [[projective space]] itself. In direct generalization of this but less evident: every [[Grassmannian]] is naturally a projective variety, even a [[smooth variety|smooth variety]].

This generalizes to [[quiver representations]]. Every Grassmannian of a Quiver representation is a projective variety. In fact, every projective variety arises this way ([Reineke, '12](#Reineke)).

## Related concepts

* [[projective manifold]]

## References

* Markus Reineke, _Every projective variety is a quiver Grassmannian_ ([arXiv:1204.5730](http://arxiv.org/abs/1204.5730), [blog discussion](http://www.neverendingbooks.org/index.php/quiver-grassmannians-can-be-anything.html))
  {#Reineke}

[[!redirects projective varieties]]
[[!redirects projective algebraic variety]]

[[!redirects projective algebraic varieties]]


projectively cofibrant diagram

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $\mathcal{C}$ be a ([[cofibrantly generated model category|cofibrantly generated]]) [[model category]] and let $\mathcal{D}$ be any [[category]], regarded as a [[diagram]]-shape in the following.

Write $[\mathcal{D}, \mathcal{C}]_{proj}$ for the [[projective model structure]] on the [[functor category]] of [[functors]] from $\mathcal{D}$ to $\mathcal{C}$, hence of $\mathcal{D}$-[[diagrams]] in $\mathcal{C}$.

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

A functor/diagram $X : \mathcal{D} \to \mathcal{C}$ is a **projectively cofibrant diagram** in $\mathcal{C}$ if it is a [[cofibrant object]] in the [[projective model structure]] $[\mathcal{D}, \mathcal{C}]_{proj}$.

=--

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

We unwind the condition in def. \ref{ProjectivelyCofibrantDiagram}.

 First of all it says of course that a [[diagram]] $X \colon \mathcal{D}\to \mathcal{C}$ is projectively cofibrant precisely if the inclusion $\emptyset \to X$ of the [[initial object|initial]] diagram has the [[left lifting property]] with respect to [[natural transformations]] of diagrams

$$
  (A \stackrel{p}{\to} B) : \mathcal{C} \to \mathcal{D}
$$

which are projective acyclic fibrations, hence which are such that for each $c \in \mathcal{C}$ the component $\eta_c : A(c) \to B(c)$ is an acyclic fibration in $\mathcal{C}$.

This in turn means that $F$ is projectively cofibrant precisely if for every diagram of natural transformations

$$
 \array{
  &&A
  \\
  &
  &\downarrow^{\mathrlap{p}}
  \\
  X
  &\to&
  B
}
$$

with $p$ as above, there exists a lift $\sigma$ in

$$
 \array{
  &&A
  \\
  &
  {}^{\mathllap{\sigma}}\nearrow &\downarrow^{\mathrlap{p}}
  \\
  X
  &\to&
  B
}
 \;\;\;\;\;\;
 \in [\mathcal{D}, \mathcal{C}]
  \,.
$$

making the triangle commute.

=--

## Properties

The main point of projectively cofibrant diagrams is that the ordinary [[colimit]] over them is a presentation of the [[homotopy colimit]]:

because the ([[colimit]] $\dashv$ constant diagram)-[[adjunction]]

$$
 (\underset{\longrightarrow}{\lim} \dashv const)
  :
  \mathcal{C} \stackrel{\overset{\underset{\longrightarrow}{\lim}}{\leftarrow}}{\underset{const}{\to}}
  [\mathcal{C}, \mathcal{D}]
$$

is a [[Quillen adjunction]] (because $const$ is by the very definition of the [[projective model structure]] a [[right Quillen functor]]), the [[homotopy colimit]], being the left [[derived functor]] $\mathbb{L}\underset{\longrightarrow}{\lim}$ of the [[colimit]], is computed as the ordinary colimit evaluated on a cofibrant resolution $Q X$ of a diagram $X : \mathcal{D} \to \mathcal{C}$:

$$
  (\mathbb{L} \underset{\longrightarrow}{\lim})(X)
  \simeq
  \underset{\longrightarrow}{\lim})(Q X)
  \,.
$$

## Examples

### For specific diagram shapes

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

A _[[span]]_ diagram $X_1 \leftarrow X_0 \to X_2$ is projectively cofibrant precisely if the two morphisms are cofibrations in $\mathcal{D}$ and $X_0$, hence all three objects, are cofibrant.

The colimit over such a diagram is the [[homotopy pushout]] of the span.

=--

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

A _cotower_ diagram

$$
  X_0 \to X_1 \to X_2 \to \cdots
$$

is projectively cofibrant precisely if every morphism is a cofibration and if the first object $X_0$, and hence all objects, are cofibrant in $\mathcal{D}$.

The colimit over such a diagram is a homotopy [[sequential colimit]].

=--

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

A _[[parallel morphisms]]_ diagram

$$
  X_0 \stackrel{\overset{f}{\to}}{\underset{g}{\to}} X_1
$$

is projectively cofibrant precisely if $X_0$ is cofibrant, and if the morphism
$(f,g) : X_0 \coprod X_0 \to X_1$ is a cofibration.

This implies that also $f$ and $g$ are cofibrations and hence that $X_1$ is cofibrant.

=--

The colimit over such a diagram is a homotopy [[coequalizer]].

### For specific ambient model categories

Let $\mathcal{C} = $ [[sSet]]${}_{Quillen}$ be the standard [[model structure on simplicial sets]]. Then $[\mathcal{D}, \mathcal{C}]_{proj}$ is the projective [[model structure on simplicial presheaves]].

For the following see at _[[model structure on simplicial presheaves]]_ the section _[Cofibrant objects](model+structure%20on%20simplicial%20presheaves#CofibrantObjects)_ for more details (due to [[Dan Dugger]], see also [Richard Garner's answer on
MathOverflow](https://mathoverflow.net/questions/97690/necessary-conditions-for-cofibrancy-in-global-projective-model-structure-on-simp/127187#127187)).

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

A necessary and sufficient condition for a diagram $X : \mathcal{D} \to sSet$ to be projectively cofibrant is:

1. $X$ is degreewise a coproduct of [[retracts]] of [[representable functor|representables]]
   $$
    X_n = \coprod_{i} U^n_i \;\;\;\;  \{U^n_i \in \mathcal{C} \hookrightarrow [\mathcal{C}, Set]\}
   $$

1. Each simplicial level, as a presheaf of sets, can be presented as the [[coproduct]] of two presheaves of sets, one of which is the presheaf of degenerate simplices:
   $$
      X_n = NonDegenerate_n \coprod Degenerate_n
      \,.
   $$

=--

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

A _[[split hypercover]]_ is of this form.

=--

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

For $X : \mathcal{D} \to sSet$ any simplicial presheaf, a cofibrant [[resolution]] is given by
$$
  (Q X)_n : \coprod_{U_0 \to \cdots \to U_n \to X_n} U_0
  \,,
$$
where the coproduct runs over all sequences of morphisms between representables $U_i$, as indicated.

=--

## References

See the references at _[[homotopy colimit]]_ and generally at _[[model category]]_.

Related discussion is at

* MathOverflow _[When are "diagrams of cofibrations" projectively cofibrant](http://mathoverflow.net/questions/70612/when-are-diagrams-of-cofibrations-projectively-cofibrant)_

Related discussion is for instance also in

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

where cofibrant cotowers are mentioned as example 2.3.15.

[[!redirects projectively cofibrant diagrams]]


projectively flat connection

## Idea

A [[connection on a bundle|connection]] that is [[flat connection|flat]] up to a central term.

In [[Cartan geometry|Cartan geometry]], a torsion-free affine connection $\nabla$  on a manifold is said to be projectively flat if the corresponding normal projective Cartan connection induced by the projective class of $\nabla$ is flat. For example, the Levi-Civita connection of a hyperbolic manifolds are projectively flat in this sense, because the corresponding normal standard tractor connection is flat.

## Related concepts

* [[projective representation]]

[[!redirects projectively flat connections]]


projector

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### Context
#### Homological algebra
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[[!include homological algebra - contents]]
=--
#### Idempotents
+-- {: .hide}
[[!include idempotents - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[linear algebra]] a _projector_ is a [[linear map]] $e \colon V \to V$ that "squares to itself" in that its [[composition]] with itself is again itself: $e \circ e = e$.

A projector $e$ leads to a decomposition of the [[vector space]] $V$ that it acts on into a [[direct sum]] of its [[kernel]] and its [[image]]:

$$
  V \simeq ker(e) \oplus im(e)
  \,.
$$

The notion of projector is the special case of that of [[idempotent]] [[morphism]].

In [[functional analysis]], one sometimes requires additionally that this idempotent is in fact [[self-adjoint operator|self-adjoint]]; or one can use the slightly different terminology __projection operator__.

## Properties

Projectors relate to the notion of [[projections]] in [[category theory]] as follows: the existence of the projector $P \colon V \to V$ canonically induces a decomposition of $V$ as a [[direct sum]] $V \simeq ker(V) \oplus im(V)$ and in terms of this $P$ is the [[composition]]

$$
  P \colon V \simeq im(v) \oplus ker(V)\to im(V) \hookrightarrow V
$$

of the [[projection]] (in the sense of maps out of [[products]]) out of the [[direct sum]] $im(V) \oplus ker(V) \simeq im(V) \times ker(V)$ followed by the [[subobject]] inclusion of $im(V)$. Hence:

_A projector is a projection followed by an inclusion_.

## Related concepts

* [[projection]]

[[!redirects projector]]
[[!redirects projectors]]

[[!redirects projection operator]]
[[!redirects projection operators]]


prometric space

# Prometric spaces
* table of contents
{: toc}

## Definitions

Let $X$ be an abstract [[set]].  For purposes of this definition, let a _distance function_ on $X$ be a [[positive number|nonnegative]]-[[extended real number|extended]]-[[real number|real]]-valued [[binary function|binary]] [[function]] on $X$; that is, a function $d\colon X \times X \to [0,\infty]$.

(For most purposes, we may assume that these distance functions are _pointwise-bounded_: taking only finite values.  On the other hand, for full generality in [[constructive mathematics]], we must allow the distance functions to take nonnegative extended *[[upper real|upper]]* real values, although again we may assume them to be pointwise-bounded and pointwise-[[located real number|located]] for many purposes.)

Let $G$ be a collection of such distance functions.

### Elementary axioms {#elementary}

Consider the following potential properties of $G$:

1. _Reflexivity_:  For every $d \in G$ and $x \in X$, we have $d(x,x) = 0$.

2. _Transitivity_ (a version of the [[triangle identity]]):  For any $d \in G$ there exists an $e \in G$ with
   $$ d(x,z) \leq e(x,y) + e(y,z) $$
   for all $x,y,z \in X$.

3. _Symmetry_:  For every $d \in G$, there is an $e \in G$ with $d(x,y) \leq e(y,x)$ for all $x,y \in X$.  (In light of Isotony below, we may require $d(x,y) = e(y,x)$.)

4. _Nontriviality_:  There exists a $d \in G$.  (In light of Isotony below, we may require $d(x,y) = 0$ for all $x,y \in X$.)

5. _Filtration_:  For all $d,e \in G$, there is an $f \in G$ with $d(x,y) \leq f(x,y)$ and $e(x,y) \leq f(x,y)$ for all $x,y \in X$.  (In light of Isotony below, we may require $f(x,y) = \max(d(x,y), e(x,y))$ for all $x,y \in X$.)

6. _Isotony_:  If $d \in G$ and $e$ is a distance function with $e(x,y) \leq d(x,y)$ for all $x,y \in X$, then $e \in G$.

### Sophisticated axioms {#sophisticated}

With the aid of [[abstract algebra]] (but nothing too fancy), we may view these same 6 axioms in another light, as follows.

First, the [[function set]] $[0,\infty]^{X \times X}$ of all distance functions $d\colon X \times X \to [0,\infty]$ on $X$ is a [[lattice group|lattice]] $*$-[[star-monoid|monoid]] under these operations:

* The order $\leq$ is given pointwise:
  $$ d \leq e \;\iff\; \forall(x,y\colon X),\; d(x,y) \leq e(x,y) .$$

* The corresponding [[join]] operation $\vee$ is also pointwise:
  $$ (d \vee e)(x,y) \coloneqq \max(d(x,y), e(x,y)) .$$

* The corresponding [[bottom element]] is the [[zero function]] $0$:
  $$ 0(x,y) \coloneqq 0 .$$

* The monoid operation $\circ$ is defined so:
  $$ (d \circ e)(x,z) \coloneqq \inf_{y \in X} (d(x,y) + e(y,z)) .$$

* The corresponding [[identity element]] is the infinite [[Kronecker delta]] $\delta$:
  $$ \delta(x,y) \coloneqq \begin{cases}
                           0      & if\; x = y,\\
                           \infty & if\; x \ne y.
                           \end{cases} $$
  (This works in [[constructive mathematics]] because we are using extended *[[upper real number|upper]]* reals; $\delta(x,y)$ is the [[infimum]] of the set $\{ t\colon \mathbb{R} \;|\; t = 0 \;\wedge\; x = y \}$.)

* The [[involution]] $d \mapsto d^{\op}$ is defined so:
  $$ d^{\op}(x,y) \coloneqq d(y,x) .$$

Then the same 6 axioms may be expressed as follows:

1. Reflexivity:  For every $d \in G$, we have $d \leq \delta$.

2. Transitivity:  For every $d \in G$, there exists $e \in G$ such that $d \leq e \circ e$.

3. Symmetry:  For every $d \in G$, there exists $e \in G$ such that $d \leq e^{\op}$.  (In light of Isotony, we may require $d = e^op$; in other words, $d^op \in G$.)

4. Nontriviality:  There exists a $d \in G$.  (In light of Isotony, we may require $d = 0$.)

5. Filtration:  For all $d,e \in G$, there is an $f \in G$ with $d \leq f$ and $e \leq f$.  (In light of Isotony, we may require $f = d \vee e$.)

6. Isotony:  If $d \in G$ and $e$ is a distance function with $e \leq d$, then $e \in G$.

It is possible to generate these even more systematically by specifying a [[proarrow equipment]] and considering the (possibly symmetric) [[pro-object|pro]]-[[monads]] in it; see the [Categorial interpretation](#proarrows) below.

### Unbiased versions

Reflexivity and Transitivity are a binary--nullary pair whose [[unbiased]] combination is as follows:

* [Elementary](#elementary) version:  For every [[natural number]] $n = 0,1,2,\ldots$ and every $d \in G$, there exists $e \in G$ with
   $$ d(x_0,x_n) \leq \sum_{i \lt n} e(x_i,x_{i+}) $$
   for each [[list]] $x_0,\ldots,x_n \in X$.

* [Sophisticated](#sophisticated) version:  For every [[natural number]] $n = 0,1,2,\ldots$ and every $d \in G$, there exists $e \in G$ such that $d \leq e^{\circ n}$.

We can combine these with Symmetry by generalizing $n$ from a natural number to a [[list]] $\epsilon$ of [[bits]] and replacing $e(x_i,x_{i+})$ with $e(x_{i+},x_i)$ (that is, replacing $e$ with $e^{\op}$ in the $i$th position) when $\epsilon_i = 1$.

Similarly, Nontriviality and Filtration are a binary--nullary pair whose unbiased version states (in light of Isotony) closure under finitary joins; but this is properly discussed at [[ideal]].

### Structures and spaces

A collection $G$ of distance functions that satisfies all of (1--6) is a __prometric__; if we drop (3), then we still have a __quasi-prometric__.  If we are working with quasi-prometrics generally, then one that happens to satisfy (3) is called __symmetric__; in other words, a prometric is precisely a symmetric quasi-prometric.

Finally, a __prometric space__ is a set equipped with a prometric, and likewise a __quasi-prometric space__ is a set equipped with a quasi-prometric.

### Generating (quasi)-prometrics

We sometimes wish to consider collections of distance functions that generate (quasi)-prometrics.  In the following table, a collection $G$ satisfying the conditions listed on the left has the name on the right:

| Conditions | Name |
| - | - |
| 1,2 | __pre-quasi-prometric__ |
| 1,2,3 | __pre-prometric__ |
|       4,5 | [[ideal base]] |
| 1,2,  4,5 | quasi-prometric __[[base]]__ |
| 1,2,3,4,5 | prometric __base__ |
|       4,5,6 | [[ideal]] |
| 1,2,  4,5,6 | quasi-prometric |
| 1,2,3,4,5,6 | prometric |

Here we always use the original (either [elementary](#elementary) or [sophisticated](#sophisticated)) formulation of (3--5); one might consider what it means if these satisfy the stronger versions rewritten in light of Isotony, but it\'s getting a bit far along into [[centipede mathematics]] to actually give these things names.

A (quasi)-prometric base is precisely an ideal base whose generated ideal is a (quasi)-prometric.  We may also speak of a (quasi)-prometric __subbase__ as an [[ideal subbase]] (that is, an arbitrary collection) whose generated ideal base is a (quasi)-prometric base, or equivalently whose generated ideal is a (quasi)-prometric.  A pre-(quasi)-prometric is always a (quasi)-prometric subbase, but not conversely; but thinking too hard about subbases risks more centipedes.

A pre-quasi-prometric is __symmetric__ if its generated quasi-prometric is symmetric (hence a prometric); a quasi-prometric base is symmetric iff it is a prometric base (and the analogous result holds for subbases), but a symmetric pre-quasi-prometric need not be a pre-prometric.  Some authors may require a strong version of Symmetry in which the distance functions $d$ are all individually required to be [[symmetric binary function|symmetric]]; that is, $d(x,y) = d(y,x)$ (or simply $d = d^{\op}$).  Every prometric has a base with this property; in particular, if $G$ is a prometric, then
$$ \{ d \in G \;|\; d = d^{\op} \} $$
is a prometric base that generates $G$, and it is this base that some authors may refer to as the prometric itself.  (I write 'some authors may', but there are few authors on this subject; still, it\'s the sort of thing that somebody might do.)

A pre-quasi-prometric $G$ is __pointwise-bounded__ if every $d \in G$ is pointwise-bounded; that is, $d(x,y) \lt \infty$ for every $x,y \in X$ (or simply $d \lt \infty$).  If $G$ is a (quasi)-prometric, then
$$ G_b \coloneqq \{ d \in G \;|\; d \lt \infty \} $$
is a (quasi)-prometric that is very similar to $G$ (in particular, they are [[uniformly equivalent]]), and some authors may prefer to work with $G_b$.  Indeed, when pointwise boundedness is *not* required, some authors may call the structure __extended__, as an instance of the [[red herring principle]].  However, requiring pointwise boundedness interferes with the more sophisticated approaches to (quasi)-prometrics; in particular, $\delta$ is not pointwise-bounded.  (It is possible that prometric spaces would work better with a closure condition that would make $G_b$ generate $G$.)

Most definitions are no more complicated when phrased in terms of (quasi)-prometric bases or even pre-(quasi)-prometrics, and some constructions give one of these more naturally than the generated (quasi)-prometric.  When working in [[predicative mathematics]], it is preferable to work exclusively with (quasi)-prometric bases, as the generated (quasi)-prometric will typically be a [[proper class]].  However, it is ultimately the generated (quasi)-prometric (even if referred to only obliquely) that matters.

Any (quasi)-[[gauge space|gauge]] is a (quasi)-prometric base; similarly, given any (quasi)-[[pseudometric]] $d$, the [[singleton subset|singleton]] $\{d\}$ is a (quasi)-prometric base (and its generated (quasi)-gauge in turn generates the same (quasi)-prometric).  One might call a (quasi)-prometric (space) _simple_ if it is generated in this way by a (quasi)-pseudometric; this term is used analogously in the theory of [[syntopogenous spaces]], but the term _(quasi)-pseudometrizable_ is more likely to be understood.  (See [Subcategories](#subcategories) below.)

## Morphisms

A **[[short map]]** between (quasi)-prometric spaces $X$ and $Y$ is a function $f:X\to Y$ such that for every $d\in G_Y$, we have $d\circ (f\times f) \in G_X$.  We write $ProMet$ for the category of prometric spaces and short maps, and similarly $QProMet$ for the category of quasi-prometric spaces and short maps.

If the (quasi)-prometrics of $X$ and $Y$ are presented by bases, then this is equivalent to saying that for any basic distance function $d$ on $Y$, there is a basic $e$ on $X$ such that $d(f(x),f(x'))\le e(x,x')$ for all $x,x'\in X$.  Thus, for (quasi)-pseudometric spaces and (quasi)-gauge spaces considered as (quasi)-prometric spaces, this reduces to the usual notion of short map (i.e., distance-decreasing map).  Hence the category $Gau$ of gauge spaces and short maps is included as a full subcategory of $ProMet$, and similarly with $QGau$ in $QProMet$.

## Subcategories {#subcategories}

Since $Gau$ includes the categories of metric spaces and uniform spaces (disjointly), so does $ProMet$.  Likewise, since $QGau$ includes the category of topological spaces (disjointly from metric and uniform spaces), so does $QProMet$.

There is also another embedding of $Unif$ into $ProMet$, however, which is notably simpler than its embedding into $Gau$ (both easier to state and not using [[dependent choice]]).  Given a [[uniform space]] $X$, we define for each [[entourage]] $U\subseteq X\times X$ a distance function
$$
d_U(x,y) =
\begin{cases}
  0 & (x,y)\in U\\
  \infty & (x,y)\notin U.
\end{cases}
$$
(Again, this may be defined constructively as an infimum.)  The collection of such $d_U$ is a base for a prometric on $X$.  The short maps between such prometric spaces are precisely the uniformly continuous ones, so this defines another embedding of $Unif$ into $ProMet$.  The full image of this embedding consists precisely of those prometric spaces generated by a base of $\{0,\infty\}$-valued functions.  Note that replacing $\infty$ by any positive real number also defines an embedding of $Unif$ into $ProMet$, but a yet different one.

Conversely, every prometric induces a uniformity, where the entourages are the sets
$$ U_{d,\epsilon} = \{ (x,y) \;|\; d(x,y) \lt \epsilon \} .$$
In this way every short map induces a uniformly continuous map as well.  This operation is compatible with the above inclusions of $Unif$, as well as with the inclusion via $Gau$.

## Categorical interpretation {#proarrows}

As observed by Lawvere, an extended quasi-pseudo-metric space is a [[category enriched]] over the [[monoidal category]] $([0,\infty],\geq,+,0)$.  In other words, it is a monoid (or [[monad]]) in the [[bicategory]] $[0,\infty] Mat$ of [[matrices]] with values in this monoidal category.  Analogously, an extended quasi-prometric space is a monad in the bicategory $Pro [0,\infty] Mat$ whose hom-categories are the categories of [[pro-object]]s in the hom-categories of $[0,\infty] Mat$.

Note that if $Rel = \{0,1\} Mat$ denotes the bicategory of [[relation]]s in $Set$, then a monad in $Rel$ is a [[preorder]], while a monad in $Pro Rel$ is a quasi-[[uniform space]].

In all these cases, in order to recover the correct notion of morphism abstractly, we must consider monads in a [[double category]] or [[equipment]] rather than merely a bicategory.

[[!include generalized uniform structures - table]]

## References

* Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen.  "One Setting for All: Metric, Topology, Uniformity, Approach Structure." ([pdf](http://www.math.yorku.ca/~tholen/ProMat_V_.pdf))

[[!redirects prometric]]
[[!redirects prometrics]]
[[!redirects prometric space]]
[[!redirects prometric spaces]]

[[!redirects quasiprometric]]
[[!redirects quasiprometrics]]
[[!redirects quasiprometric space]]
[[!redirects quasiprometric spaces]]


promonoidal category

+-- {: .rightHandSide}
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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

# Promonoidal categories
* table of contents
{: toc}

## Idea

A **promonoidal category** is like a [[monoidal category]] in whose structure (namely, [[tensor product]] and [[unit object]]) we have replaced [[functors]] by [[profunctors]].

## Definition

A **promonoidal category** is a [[pseudomonoid]] in the [[monoidal bicategory]] [[Prof]].  This means that it is a [[category]] $A$ together with

* A [[profunctor]] $P \colon A\times A ⇸ A$.
* A profunctor $J\colon 1$ ⇸ $A$.
* [[associator|Associativity]] and [[unitor|unit isomorphisms]] $P \odot (P\times 1) \cong P\odot (1\times P)$, $P\odot (J\times 1) \cong 1$, and $P\odot (1\times J) \cong 1$.
* The usual [[pentagon identity|pentagon]] and unit conditions hold, as in a [[monoidal category]].

Recalling that a profunctor $A$ ⇸ $B$ is defined to be a functor of the form $B^{op}\times A \to Set$, we can make this more explicit.  We can also generalize it by replacing [[Set]] by a [[Benabou cosmos]] $V$ and $A$ by a $V$-[[enriched category]]; then a profunctor is a $V$-[[enriched functor]] $B^{op}\times A \to V$.

Thus, we obtain the following as an explicit definition of **promonoidal $V$-category**:

We have the following data

1. A $V$-[[enriched category]] $A$.

1. A $3$-ary [[enriched functor]] $P:A^\op \otimes A \otimes A\to V$.  For notational clarity, we may write $P(a,b,c)$ as $P(a,b \diamond c)$.

1. A $V$-functor $J:A^{op}\to V$.

and [[enriched natural isomorphisms]]

1. $\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)$

1. $\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)$

1. $\alpha_{abcd}: \int^x (P(x,a\diamond b)\otimes P(d,x\diamond c)) \to \int^x(P(x,b\diamond c)\otimes P(d,a\diamond x))$

satisfying the pentagon and unit axioms for promonoidal categories. Explicitly, writting $P^{A}_{B,C}$ for $P(A,B;C)$, $\mathsf{h}^{A}_{B}$ for $\mathrm{Hom}_{A}(A,B)$, $J^X$ for $J(X)$, and $\diamond$ for composition of profunctors, we require the following conditions to hold:

1. **The triangle identity for promonoidal categories.** For each $A,B,C\in\mathrm{Obj}(A)$, the [[diagram]]

   \begin{imagefromfile}
       "file_name": "pro-triangle-corrected.svg"
   \end{imagefromfile}

   [[commuting diagram|commutes]];

1. **The pentagon identity for promonoidal categories.** For each $A,B,C,D,E\in\mathrm{Obj}(A)$, the [[diagram]]

   \begin{imagefromfile}
       "file_name": "pro-pentagon.svg"
   \end{imagefromfile}

   [[commuting diagram|commutes]].

## Properties

### Versus monoidal categories

Since any functor induces a representable profunctor, any monoidal category can be regarded as a promonoidal category.  A given promonoidal category arises in this way if and only if the profunctors $P$ and $J$ are representable.

### Day convolution

A promonoidal structure on $A$ suffices to induce a monoidal structure on $V^{A^{op}}$ by [[Day convolution]].  In fact, given a small $V$-category $A$, there is an equivalence of categories between

1. the category of pro-monoidal structures on $A$, with strong pro-monoidal functors between them, and

1. the category of biclosed monoidal structures on $V^{A^{op}}$, with strong monoidal functors between them.

### Versus multicategories

A promonoidal structure on $A$ can be identified with a particular sort of [[multicategory]] structure on $A^{op}$, i.e. with a co-multicategory structure on $A$.  The set $P(x, y, z)$ is regarded as the set of co-multimorphisms $x \to (y,z)$.

More generally, we define a co-multicategory $\bar A$ as follows.  The objects of $\bar A$ are the objects of $A$.  The co-multimorphisms $b\to a_1\dots a_n$ in $\bar A$ are defined by induction on $n$ as follows: $\bar A(b;)=Jb$, and $\bar A(b;a_1,\dots,a_{n+1})=\int^x\bar A(x;a_1,\dots,a_n)\otimes P(b,x\diamond a_{n+1})$.

Not every co-multicategory arises from a promonoidal one in this way.  Roughly, a promonoidal category is a co-multicategory whose $n$-ary co-multimorphisms are determined by the binary, unary, and nullary morphisms.  In general, co-multicategories can be identified with a certain sort of "lax promonoidal category".

## Notes

[[Brian Day]] introduced the notion of a "premonoidal" category in [(Day 1970)](#Day70), and later renamed this to a "promonoidal" category in [(Day 1974)](#Day74) while reformulating the identity and associativity isomorphisms $\lambda,\rho,\alpha$ explicitly in terms of profunctor composition.  However, note that his definition is op'd from the definition used in this article, in the sense that a Day-promonoidal structure on a category $C$ corresponds to a pseudomonoid structure on $C^{op}$ in [[Prof]].  In particular, one example Day considers is that of a [[closed category]], which is actually a co-promonoidal category in the sense used here (analogous to the co-promonoidal structure on a multicategory described above).

Regarding monoidal categories as promonoidal is useful in order to express extra structure on them, such as [[closed monoidal category|closedness]], [[star-autonomous category|$\ast$-autonomy]], or [[compact closed category|compact closedness]], in abstract bicategorical terms: these notions can be defined by adding extra structure to a [[pseudomonoid]] in the [[monoidal bicategory]] [[Prof]] (i.e. a promonoidal category), but the extra structure does not lie inside the sub-monoidal bicategory [[Cat]].

## Related pages

* [[monoidal category]], [[multicategory]]
* [[Day convolution]]
* A (co-)promonoidal [[poset]] is called a [[ternary frame]], when its Day convolution monoidal category is used to model [[substructural logic]].

## References

* {#Day70} [[Brian Day]], On closed categories of functors, _Lecture Notes in Mathematics_ 137 (1970), 1-38.

* {#Day74} [[Brian Day]], An embedding theorem for closed categories, _Lecture Notes in Mathematics_ 420 (1974), 55-64.

* Day, Panchadcharam and Street, *On centres and lax centres for promonoidal categories*.

The relationship between [[multicategories]], [[promonoidal categories]], [[lax monoidal categories]], and [[monoidal categories]] is exposited in:

* [[Brian Day]] and [[Ross Street]], _Lax monoids, pseudo-operads, and convolution_, Contemporary Mathematics 318 (2003): 75-96.

[[!redirects promonoidal categories]]
[[!redirects pro-monoidal category]]
[[!redirects pro-monoidal categories]]
[[!redirects promonoidal structure]]
[[!redirects pro-monoidal structure]]
[[!redirects promonoidal structures]]
[[!redirects pro-monoidal structures]]


proof

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

> The interesting conception of the [[propositions-as-types]] principle is what I call _Brouwer's Dictum_, which states that all of mathematics, including the concept of a proof, is to be derived from the concept of a *[[constructive mathematics|construction]]*, a [[computation]] classified by a [[type]].  In [[intuitionistic mathematics]] proofs are themselves "first-class" mathematical objects that inhabit [[type|types]] that may as well be identified with the [[proposition]] that they prove.  Proving a proposition is no different than constructing a [[program]] of a type.  In this sense [[logic]] is a branch of [[mathematics]], the branch concerned with those constructions that are proofs.  And mathematics is itself a branch of [[computer science]], since according to Brouwer's Dictum all of mathematics is to be based on the concept of computation.  But notice as well that there are many more constructions than those that correspond to proofs.  [[natural number|Numbers]], for example, are perhaps the most basic ones, as would be any [[inductive type|inductive]] or [[coinductive type|coinductive]] types, or even more exotic objects such as Brouwer's own [[choice sequence|choice sequences]].  From this point of view the [[judgement]] $t\in A$ stating that $t$ is a [[term|construction]] of type $A$ is of fundamental importance, since it encompasses not only the formation of "ordinary" mathematical constructions, but also those that are distinctively intuitionistic, namely mathematical proofs.

> An often misunderstood point that must be clarified before we continue is that the concept of proof in [[intuitionistic logic|intuitionism]] is *not* to be identified with the concept of a *formal proof* in a fixed formal system.  What constitutes a proof of a [[proposition]] is a *[[judgement]]*, and there is no reason to suppose _a priori_ that this judgement ought to be [[decidable judgment|decidable]].  It should be possible to recognize a proof when we see one, but it is not required that we be able to *rule out* what is a proof in all cases.  In contrast formal proofs are inductively defined and hence fully circumscribed, and we expect it to be decidable whether or not a purported formal proof is in fact a formal proof, that is whether it is well-formed according to the given inductively defined rules.  But the upshot of [[incompleteness theorem|Gödel's theorem]] is that as soon as we fix the concept of formal proof, it is immediate that it is not an adequate conception of proof _simpliciter_, because there are propositions that are true, which is to say have a proof, but have no *formal proof* according to the given rules.  The concept of truth, even in the intuitionistic setting, eludes formalization, and it will ever be thus.  Putting all this another way, according to the intuitionistic viewpoint (and the mathematical practices that it codifies), there is no truth other than that given by proof.  Yet the rules of proof cannot be given in decidable form without missing the point. ([Harper](#Harper))

## Definition

* [In type theory](#Proof)

* [Formal proofs](#FormalProof)

### In type theory
 {#Proof}

In [[type theory]], a [[proposition]] is identified with the [[type]] of all its proofs (the _[[propositions as types]]_-aspect of [[computational trinitarianism]]).
Here a proof consists of exhibiting a [[term]] of the corresponding type (showing that it is [[inhabited]]), hence a proof is a typing [[judgement]] for a term of the type representing the proposition.

See also _[[proofs as programs]]_.

### Formal proofs
 {#FormalProof}

A *[[formal proof]]* is whatever is called a 'proof' in a [[formal system]]; a formal system for [[mathematics]] then gives rules for producing a proof in the above sense.

Typically, a formal system is [[induction|inductively]] defined, and hence its proofs are fully circumscribed; this is the case for [[deductive systems]] such as [[natural deduction]], [[sequent calculus]], and [[Hilbert systems]].  [[incompleteness theorem|Gödel's theorem]] suggests, however, that no such system can encapsulate all of mathematics.

\begin{remark}\label{MeaningOfTheWordFormal}
**(meaning of "formal")**
\linebreak
  Generally, people tend to speak of *formal* arguments when referring to arguments by *formalism*, hence by purely *symbolic manipulations* following [[syntax|syntactic]] *[[rewrite rules]]*.

Beware that the quality of such arguments depends drastically on how good the formalism is --- with the result that the term "formal" appears in practice in two different and almost opposite meanings:

1. If it is granted that the syntactic rules properly reflect the intended logic in its most fine detail, then a *formal argument* via these rules is the most rigorous hence the **strongest form of argument**. This is the sense in which *formal proof* is understood in mathematics.

1. But if the syntactic rules employed actually *fail* to reflect all intended aspects of a given problem, then the *formal arguments* based on these rules are superficial arguments, may not account for the actual detail in question, may give conclusions that are considered wrong and hence constitute a **weak form of argument**.

The second sense is, incidentally, the sense in which reference to "formal argument" is typically used in *[[physics]]* (for instance when manipulating the [[integrand]] of a [[path integral]] as if one were dealing with an actual [[integral]]).

But also mathematicians sometimes speak of "formal computations" in this second sense of "superficial argument by overly naïve formalism" (for instance when disregarding issues of [[convergence of a sequence|convergence]] in [[analysis]]).

See also:

* Wikipedia: *[Formal computation](https://en.wikipedia.org/wiki/Formal_calculation)*

* MathOverflow: *[Why do people use "formal calculation"
to describe informal calculations?
](https://mathoverflow.net/q/206544/381)*

\end{remark}

## Relation to observation in physics
 {#RelationToObservationInPhysics}

In ([Jaffe-Quinn 93, p. 2](JaffeQuinn93)) it was claimed that

> {#JaffeQuinnProofIsObservation} the role of rigorous [[proof]] in mathematics is functionally [[analogy|analogous]] to the role of [[experiment]] in the natural sciences

+-- {: .un_remark}
###### Remarks
After making this statement about proof as observation, the article by Jaffe-Quinn goes further to suggest that the analogue in physics of speculation and conjecture in mathematics is the realm of theoretical physics, and then maybe even further in suggesting that mathematics could reasonably be purely "theoretical" in this sense. These further claims were considered faulty by several authors ([math/9404229](http://arxiv.org/abs/math/9404229)).

([[Saunders MacLane]] in particular argued for theorems and proofs as the final goal of a piece of developed mathematics, and emphasized the explanatory function of proofs. This in contrast with experiment in physics, which is neither considered as the final goal nor as playing an explanatory role.)

While all this does contradict some of Jaffe-Quinn's claims, it may still harmonize with the statement [above](#JaffeQuinnProofIsObservation). Indeed, another perspective on the claim is that finding a proof is an observational or witnessing act, which resonates with the nature of proof in [[constructive mathematics]], such as made manifest in the [[propositions as types]]-paradigm. This perspective sees proof as something more than merely establishing the truth of a proposition.

> ... to show that a proposition is true in type theory corresponds to exhibiting an element $[$ [[term]] $]$ of the type corresponding to that proposition. We regard the elements of this type as _evidence_ or _witnesses_ that the proposition is true. (They are sometimes even called _proofs_... (from [[Homotopy Type Theory -- Univalent Foundations of Mathematics]], section 1.11)

Thus a proof *qua* witness is a construction, and in more elaborated developments (for instance in [[intensional type theory]]) a formal proof is itself a mathematical object, with internal mathematical structure.
=--

## Proof methods
 {#ProofMethods}

* [[refutation by contradiction]]

* [[proof by contradiction]],

## Libraries of formal proofs
 {#LibrariesOfFormalProofs}

Libraries of [[formal proofs]] formalized in some [[proof assistant]]:

* [[UniMath project]]

* [[Xena project]]

* [[ForMath project]]

* [[Archive of Formal Proofs]]

* [[The QED Project]]

## Related concepts

* [[computational trinitarianism]]

* [[proof theory]]

\linebreak

[[!include mathematical statements --- contents]]

\linebreak

[[!include proof assistants and formalization projects -- list]]

## References

General introduction to the notion of proof in modern rigorous mathematics:

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

* {#Quinn2009} [[Frank Quinn]]: *Proof Projects for Teachers* (2009) [[pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf), [[Quinn-ProofProjects.pdf:file]]]

A brief exposition of the notion of proof and formal proof in [[constructive mathematics]]/[[type theory]] is in

* {#Harper} [[Robert Harper]], _Extensionality, Intensionality, and Brouwer's Dictum_, August 2012 ([web](http://existentialtype.wordpress.com/2012/08/11/extensionality-intensionality-and-brouwers-dictum/))

* [[Robert Harper]], _Constructive Mathematics is not Meta-Mathematics_, July 2013 ([web](http://existentialtype.wordpress.com/2013/07/10/constructive-mathematics-is-not-meta-mathematics/))

Discussion of formal proof in view of [[proof assistants]]:

* [[Thomas Hales]], _Formal proof_, Notices of the AMS **55** 11 (2008) 1370-1379 [[pdf](http://www.ams.org/notices/200811/tx081101370p.pdf)]

* John Harrison, _Formal proof -- theory and practice_ ([pdf](http://www.ams.org/notices/200811/tx081101395p.pdf))

* [[Jeremy Avigad]], Kevin Donnelly, David Gray, Paul Raff, _A formally verified proof of the prime number theorem_ ([arXiv:cs/0509025](http://arxiv.org/abs/cs/0509025))

* [[Jeremy Avigad]], John Harrison, _Formally Verified Mathematics_, Communications of the ACM, Vol. 57 No. 4, Pages 66-75 ([web](http://cacm.acm.org/magazines/2014/4/173219-formally-verified-mathematics/fulltext))

* [[Jeremy Avigad]], _Formal verication, interactive theorem proving, and automated reasoning_ (2014) ([pdf](http://www.andrew.cmu.edu/user/avigad/Talks/baltimore.pdf))

A discussion of the relation of mathematical proof to [[phenomenology]] of [[theories]] of [[physics]] is in

* {#Miquel} [[Alexandre Miquel]], _The experimental effectiveness of mathematical proof_ ([pdf](http://perso.ens-lyon.fr/alexandre.miquel/publis/effectiveness.pdf))

Discussion of the statistics and *reception* of formal proof:

* {#ViteriDeDeo22} Scott Viteri, Simon DeDeo, *Epistemic Phase Transitions in Mathematical Proofs*, Cognition **225** (2022) 105120 [[arxiv:2004.00055](https://arxiv.org/abs/2004.00055), [doi:10.1016/j.cognition.2022.105120](https://doi.org/10.1016/j.cognition.2022.105120)]

Texts on genuine [[proof theory]] include

*  [[Richard Statman]], _[[Structural Complexity of Proofs]]_, 1974

Projects aiming to formalize parts of mathematics include

* [[ForMath]]

* [[The QED Project]]

* [Xena](https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/) (by [[Kevin Buzzard]])

Consideration of the relation of mathematical proof to physics icludes

* {#JaffeQuinn93} [[Arthur Jaffe]], [[Frank Quinn]], _"Theoretical Mathematics": Towards a cultural synthesis of mathematics and theoretical physics_, Bulletin of the AMS, Volume 29,Number 1, July 1993 ([arXiv:math/9307227](http://arxiv.org/abs/math/9307227))

[[!redirects proof]]
[[!redirects proofs]]

[[!redirects formal proof]]
[[!redirects formal proofs]]

[[!redirects rigorous proof]]
[[!redirects rigorous proofs]]


proof assistant

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

# Contents
* table of contents
{: toc}

## Idea

A _proof assistant_ or _proof management system_ is a kind of software designed to help with [[proofs]] in formalized [[mathematics]]. Many proof assistants resemble and/or include a [[programming language]].

There are arguably two threads of current development in proof systems, which may be called "foundational" and "coverage".

The "foundational" work tries to find the best foundational theory to formalize mathematics (see also at _[[foundations of mathematics]]_). Out of that work first came [[dependent type theory|dependent types]] ([[Automath]], in the late 60s), then the [[calculus of constructions]] (early [[Coq]]), and the [[calculus of inductive constructions]] (current [[Coq]]). More recently a new wave of such work is being done in [[homotopy type theory]] as another step in this direction. Coq's library is not that large, except in the area of [[group theory]] where the results of the work on [[Feit-Thompson theorem]] has produced something larger.

The "coverage" work tries to formalize as much as possible of mathematics in existing theories.  For instance, for decades people have been building [[Mizar]]'s library (Mizar is based on [[Tarski–Grothendieck set theory]] rather than [[type theory]]). Its library is a couple of orders of magnitude larger than anyone else's. On the other hand, despite this quantity, it remains an issue to attack problems of contemporary research interest in these systems, see also at _[Mizar -- problem of pertinence](Mizar#ProblemOfPertinence)_.

Similar to Mizar is [[NuPRL]], [[HOL light]] and [[Isabelle]], which all have decently sized libraries.  (Isabelle can be used with either [[material set theory]], like Mizar, or [[higher-order type theory]], like the others.)

## Examples

[[!include proof assistants and formalization projects -- list]]

## References

Gentle exposition in

* [[Théo Winterhalter]], §1 of: *Formalisation and Meta-Theory of Type Theory*, Nantes (2020) [[pdf](https://github.com/TheoWinterhalter/phd-thesis/releases/download/v1.2.1/TheoWinterhalter-PhD-v1.2.1.pdf), [github](https://github.com/TheoWinterhalter/phd-thesis)]

Further exposition:

* {#Avigad12} [[Jeremy Avigad]], _Interactive Theorem Proving, Automated Reasoning, and Mathematical Computation_, (2012) [pdf slides](http://www.andrew.cmu.edu/user/avigad/Talks/icerm.pdf)

On computer assisted proofs in [[analysis]]:

* [[Oscar E. Lanford]], *Computer assisted proofs*, in: *Computational Methods in Field Theory*, Lecture Notes in Physics, **409**, Springer (1992) [[doi:10.1007/3-540-55997-3_30](https://doi.org/10.1007/3-540-55997-3_30)]

See also:

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

List of web resources:

* [[Freek Wiedijk]], _Digital math by alphabet_ ([web](http://www.cs.ru.nl/~freek/digimath/index.html))

See also

* _[Conference Series on Intelligent Computer Mathematics](http://www.cicm-conference.org/)_

  ([2014](http://cicm-conference.org/2014/cicm.php))

* Conference series on Interactive theorem proving

  ([2014](http://www.cs.uwyo.edu/~ruben/itp-2014/))

Parts of the above text are taken from [this MO comment](http://mathoverflow.net/questions/133572/at-which-level-is-it-currently-possible-to-write-formal-proofs/134009#134009) by [[Jacques Carette]].

Proof assistants specifically for [[homotopy type theory]]:

* [HoTT wiki](https://ncatlab.org/homotopytypetheory/show/HomePage), *[[homotopytypetheory:Proof Assistants]]*

See also

* [[Kevin Buzzard]], _[Xena project](https://xenaproject.wordpress.com/2018/10/07/what-is-the-xena-project/)_

  on [[formal proof]] and proof assistants in undergraduate mathematics.

[[!redirects proof assistant]]
[[!redirects proof assistants]]

[[!redirects proof management system]]
[[!redirects proof management systems]]

[[!redirects computer proof assistant]]
[[!redirects computer proof assistants]]

[[!redirects theorem prover]]


proof assistants and formalization projects -- list

**[[proof assistants]]:**

based on plain [[type theory]]/[[set theory]]:

* [[Metamath]]

* [[Mizar]], [[NuPRL]], [[Isabelle]], [[HOL]]

based on [[dependent type theory]]/[[homotopy type theory]]:

* [[Coq]], [[Agda]], [[Lean]], [[Arend]]

{#BasedOnCubicalTypeTheory} based on [[cubical type theory]]:

* [cubicaltt](https://github.com/mortberg/cubicaltt)

* [redtt](http://github.com/redprl/redtt)

* [yacctt](http://github.com/mortberg/yacctt)

* [RedPRL](http://www.redprl.org/en/latest/)

* [Cubical Agda](Agda#CubicalAgda)

  * [library and demos](https://github.com/agda/cubical)

  * [[1lab]] (cross-linked reference resource)

based on [[modal type theory]]:

* [[Agda-flat]]: [installation instructions](http://www.cl.cam.ac.uk/~amp12/agda/internal-universes/code/agda-flat/install.txt)

based on [[simplicial type theory]]:

* [[Rzk]]

For [[monoidal category|monoidal]] [[category theory]]:

* [[DisCoPy]]

For [[higher category theory]]:

* [[opetopic type theory]]

* [[Globular]], [[homotopy.io]]

**projects for [[formal proof|formalization]] of [[mathematics]] with proof assistants:

* [[Archive of Formal Proofs]] (using [[Isabelle]])

* [[ForMath project]] (using [[Coq]])

* [[MathScheme]]

* [[UniMath project]] (using [[Coq]] and [[Agda]])

* [[Xena project]] (using [[Lean]])

Other proof assistants

* [Andromeda](http://www.andromeda-prover.org/)

Historical projects that died out:

* [[The QED Project]]


proof by contradiction

# Contents
* table of contents
{: toc}

## Idea

The method called _proof by contradiction_ is a method of [[formal proof]] using the [[principle of excluded middle]].

This principle says that for $P$ any [[proposition]], then it or its [[negation]] is [[true]]. (The 'or' here is meant internally, as a formal disjunction $P \vee \neg P$.) One also speaks of _[[classical logic]]_ if this principle is taken to hold.

But this principle implies that in order to prove a [[proposition]] $P$ it is sufficient to show that its [[negation]] is [[false]], hence that assuming that $P$ were not true would lead to a "contradiction". This is _proof by contradiction_.

Proof by contradiction is used frequently in [[classical mathematics]].

While this method of proof is classical, it has some peculiar consequences. Notably when $P$ is an [[existential quantification|existential statement]] of the form "there exists an object $X$ with the property $Y$", then a proof of such existence by contradiction alone gives no means to actually construct an example.

Therefore one also says that proof by contradiction is "non-constructive" and that an alternative proof not invoking the [[principle of excluded middle]] (if it exists) is a "constructive" proof, see at _[[constructive logic]]_ and _[[constructive mathematics]]_.

## Relation to refutation by contradiction

One should take care to distinguish proof by contradiction from _[[refutation by contradiction]]_ (also called _proof of negation_), which nevertheless involves the derivation of a contradiction: assuming the truth of $P$, a contradiction is found, and one concludes that $\neg P$. This remains [[constructive mathematics|constructively]] valid, since it simply uses the constructive _meaning_ of [[negation]], i.e. the [[introduction rule]] for the connective of [[negation]] (see [Bauer 2010](#bauer)).

Since this distinction is sometimes hard for those who have learned classical logic to the point it becomes hard to unlearn, here is another way of putting it (also mentioned by Bauer):

* A proof by contradiction is where you argue "suppose $P$ is false; then [argue blah blah blah]..., contradiction. Therefore $P$ is true.

* A refutation by contradiction is where you argue "suppose $P$ is true; then [blah blah blah]..., contradiction. Therefore $P$ is false.

For example, the standard proofs of the [[irrational number|irrationality]] of $\sqrt{2}$ are sometimes called proofs by contradiction, but they are actually refutations by contradiction (showing that "$\sqrt{2}$ is rational" is false).  (The usual meaning of 'irrational' in [[constructive analysis]] is actually something stronger than 'not rational', so such refutations by contradiction are not enough as they stand to prove irrationality; however, they usually can be easily strengthened to prove irrationality in the strong sense.)

## Related concepts

* [[formal proof]]

## References

* {#bauer} [[Andrej Bauer]], _[Proof of negation and proof by contradiction](http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/)_ 2010

* Wikipedia, _[Proof by contradiction](https://en.wikipedia.org/wiki/Proof_by_contradiction)_

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