shrinking lemma

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#### Topology
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[[!include topology - contents]]
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#Contents#
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## Idea

In [[topology]], the "shrinking lemma" (lemma \ref{PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained} below) states that on a [[normal topological space]] the patches of every [[locally finite cover]] may be replaced by smaller patches which still cover the space, but such that their [[topological closures]] are contained in the original patches.

If there is more than a [[countable set]] of elements in the original cover, then this conclusion requires [[excluded middle]] and [[Zorn's lemma]], hence the [[axiom of choice]].

The shrinking lemma is needed in the proof that [[ paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]].

## Statement

+-- {: .num_lemma #PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained}
###### Lemma
**(shrinking lemma for locally finite covers)**

Assuming the [[axiom of choice]] then:

Let $X$ be a [[topological space]] which is [[normal topological space|normal]] and let $\{U_i \subset X\}_{i \in I}$ be a [[locally finite cover|locally finite]] [[open cover]].

Then there exists another open cover $\{V_i \subset X\}_{i \in I}$ such that the [[topological closure]] $Cl(V_i)$ of its elements is contained in the original patches:

$$
  \underset{i \in I}{\forall}
  \left(
     V_i \subset Cl(V_i) \subset U_i
  \right)
  \,.
$$

=--

We now **prove** this in  increasing generality, first for  binary open covers (lemma \ref{ShrinkingLemmaForBinaryCover} below), then for finite covers (lemma \ref{ShrinkinglemmaForFiniteCovers}), then for locally finite countable covers (lemma \ref{ShrinkingLemmaForLocallyFiniteCountableCovers}), and finally for general locally finite covers (lemma \ref{PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained}, proof [below](#PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained)). It is only the last statement that needs the [[axiom of choice]].

+-- {: .num_lemma #ShrinkingLemmaForBinaryCover}
###### Lemma
**(shrinking lemma for binary covers)**

Let $(X,\tau)$ be a [[normal topological space]] and let $\{U_i \subset X\}_{i \in \{1,2\}}$ an [[open cover]] by two [[open subsets]].

Then there exists an open set $V_1 \subset X$ whose [[topological closure]] is contained in $U_1$

$$
  V_1 \subset Cl(V_1) \subset U_1
$$

and such that $\{V_1,U_2\}$ is still an open cover of $X$.

=--

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

Since $U_1 \cup U_2 = X$ it follows (by [[de Morgan's law]]) that their [[complements]] $X \backslash U_i$ are [[disjoint subset|disjoint]] [[closed subsets]]. Hence by normality of $(X,\tau)$ there exist disjoint open subsets

$$
  V_1 \supset X \backslash U_2
  \phantom{AAA}
  V_2 \supset X \backslash U_1
  \,.
$$

By their disjointness, we have the following inclusions:

$$
  V_1 \subset X \backslash V_2 \subset U_1
  \,.
$$

In particular, since $X \backslash V_2$ is closed, this means that $Cl(V_1) \subset X \backslash V_2$.

Hence it only remains to observe that $V_1 \cup U_2 = X$, by definition of $V_1$.

=--

+-- {: .num_lemma #ShrinkinglemmaForFiniteCovers}
###### Lemma
**(shrinking lemma for finite covers)**

Let $(X,\tau)$ be a [[normal topological space]], and let $\{U_i \subset X\}_{i \in \{1, \cdots, n\}}$ be an [[open cover]] with a [[finite number]] $n \in \mathbb{N}$ of patches. Then there exists another open cover $\{V_i \subset X\}_{i \in I}$ such that $Cl(V_i) \subset U_i$ for all $i \in I$.

=--

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

By [[induction]] using lemma \ref{ShrinkingLemmaForBinaryCover}.

To begin with, consider $\{ U_1, \underoverset{i = 2}{n}{\cup} U_i\}$. This is a binary open cover, and hence lemma \ref{ShrinkingLemmaForBinaryCover} gives an open subset $V_1 \subset X$ with $V_1 \subset Cl(V_1) \subset U_1$ such that $\{V_1, \underoverset{i = 2}{n}{\cup} U_i\}$ is still an open cover, and accordingly so is

$$
  \{ V_1  \} \cup \left\{ U_i \right\}_{i \in \{2, \cdots, n\}}
  \,.
$$

Similarly we next find an open subset $V_2 \subset X$ with $V_2 \subset Cl(V_2) \subset U_2$ and such that

$$
  \{ V_1, ,V_2  \} \cup \left\{ U_i \right\}_{i \in \{3, \cdots, n\}}
$$

is an open cover. After $n$ such steps we are left with an open cover $\{V_i \subset X\}_{i \in \{1, \cdots, n\}}$ as required.

=--

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

Beware that the [[induction]] in lemma \ref{ShrinkinglemmaForFiniteCovers} does _not_ give the statement for infinite [[countable covers]]. The issue is that it is not guaranteed that $\underset{i \in \mathbb{N}}{\cup} V_i$ is a cover.

And in fact, assuming the [[axiom of choice]], then there exists a  counter-example of a countable cover on a normal spaces for which the shrinking lemma fails (a [[Dowker space]] due to [Beslagic 85](#Beslagic85)).

=--

This issue is evaded if we consider [[locally finite covers]]:

+-- {: .num_lemma #ShrinkingLemmaForLocallyFiniteCountableCovers}
###### Lemma
**(shrinking lemma for locally finite countable covers)**

Let $(X,\tau)$ be a [[normal topological space]] and $\{U_i \subset X\}_{i \in \mathbb{N}}$ a [[locally finite cover|locally finite]] [[countable cover]]. Then there exists [[open subsets]] $V_i \subset X$ for $i \in \mathbb{N}$ such that $V_i \subset Cl(V_i) \subset U_i$ and such that $\{V_i \subset X\}_{i \in \mathbb{N}}$ is still a cover.

=--

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

As in the proof of lemma \ref{ShrinkinglemmaForFiniteCovers}, there exist $V_i$ for $i \in \mathbb{N}$ such that $V_i \subset Cl(V_i) \subset U_i$ and such that for every finite number, hence every $n \in \mathbb{N}$, then

$$
  \underoverset{i = 0}{n}{\cup} V_i \cup
  \underoverset{i = n+1}{\infty}{\cup} U_i
  \;=\; X\,.
$$

Now the extra assumption that $\{U_i \subset X\}_{i \in I}$ is [[locally finite cover|locally finite]] implies that every $x \in X$ is contained in only finitely many of the  $U_i$, hence that for every $x \in X$ there exists $n_x \in \mathbb{N}$ such that

$$
  x \notin \underoverset{i = n_x+1}{\infty}{\cup} U_i
  \,.
$$

This implies that for each $x$

$$
  x \in \underoverset{i = 0}{n_x}{\cup} V_i
  \subset \underset{i \in \mathbb{N}}{\cup} V_i
$$

hence that $\{V_i \subset X\}_{i \in \mathbb{N}}$ is indeed a cover of $X$.

=--

We now invoke [[Zorn's lemma]] to generalize the shrinking lemma for finitely many patches (lemma \ref{ShrinkinglemmaForFiniteCovers}) to arbitrary sets of patches:

+-- {: .proof}
###### Proof
of the general shrinking lemma \ref{PatchesOfOpenCoverOfNormalSpaceMayBeMadeSmallerSoThatTheirClosuresAreContained}

Let $\{U_i \subset X\}_{i \in I}$ be the given locally finite cover of the normal space $(X,\tau)$. Consider the set $S$ of [[pairs]] $(J, \mathcal{V})$ consisting of

1. a [[subset]] $J \subset I$;

1. an $I$-indexed set of open subsets $\mathcal{V} = \{V_i \subset X\}_{i \in I}$

with the property that

1. $(i \in J \subset I) \Rightarrow ( Cl(V_i) \subset U_i )$;

1. $(i \in I \backslash J) \Rightarrow ( V_i = U_i )$.

1. $\{V_i \subset X\}_{i \in I}$ is an open cover of $X$.

Equip the set $S$ with a [[partial order]] by setting

$$
  \left(
    (J_1, \mathcal{V})
    \leq
    (J_2, \mathcal{W})
  \right)
    \Leftrightarrow
  \left(
    \left(
      J_1 \subset J_2
    \right)
    \,\text{and}\,
    \left(
      \underset{i \in J_1}{\forall}
      \left(
         V_i = W_i
      \right)
    \right)
  \right)
  \,.
$$

By definition, an element of $S$ with $J = I$ is an open cover of the required form.

We claim now that a [[maximal element]] $(J, \mathcal{V})$ of $(S,\leq)$ has $J = I$.

For assume on the contrary that there were $i \in I \backslash J$. Then we could apply the construction in lemma \ref{ShrinkingLemmaForBinaryCover} to replace that single $V_i$ with a smaller open subset $V'_i$ to obtain $\mathcal{V}'$ such that $Cl(V'_i) \subset V_i$ and such that $\mathcal{V}'$ is still an open cover. But that would mean that $(J,\mathcal{V}) \lt (J \cup \{i\}, \mathcal{V}')$, contradicting the assumption that $(J,\mathcal{V})$ is maximal. This [[proof by contradiction|proves by contradiction]] that a maximal element of $(S,\leq)$ has $J = I$ and hence is an open cover as required.

We are reduced now to showing that a maximal element of $(S,\leq)$ exists. To achieve this we invoke [[Zorn's lemma]]. Hence we have to check that every [[chain]] in $(S,\leq)$, hence every [[total order|totally ordered]] [[subset]] has an [[upper bound]].

So let $T \subset S$ be a [[total order|totally ordered]] subset. Consider the union of all the index sets appearing in the pairs in this subset:

$$
  K
    \;\coloneqq\;
  \underset{(J,\mathcal{V}) \in T  }{\cup} J
  \,.
$$

Now define open subsets $W_i$ for $i \in K$ picking any $(J,\mathcal{V})$ in $T$ with $i \in J$ and setting

$$
  W_i \coloneqq V_i \phantom{AAA} i \in K
  \,.
$$

This is independent of the choice of $(J,\mathcal{V})$, hence well defined, by the assumption that $(T,\leq)$ is totally ordered.

Moreover, for $i \in I\backslash K$ define

$$
  W_i \coloneqq U_i  \phantom{AAA} i \in I \backslash K
  \,.
$$

We claim now that $\{W_i \subset X\}_{i \in I}$ thus defined is a cover of $X$. Take an arbitrary point $x \in X$. If $x \in U_i$ for some $i \notin K$, we have $U_i = W_i$ and therefore $x$ is in $\underset{i \in I}{\cup} W_i$. Otherwise, combining with the assumption that $\{U_i \subset X\}_{i \in I}$ is locally finite, the set $J_x$ of indices $i \in I$ such that $x \in U_i$ is finite and $J_x \subset K$. Since $(T,\leq)$ is a total order, it must contain an element $(J, \mathcal{V})$ such that $J_x \subset J$.  And since that $\mathcal{V}$ is a cover and $x$ cannot belong to any $U_i$ with $i$ outside of $J_x$, it must be that $x \in \underset{i \in J_x}{\cup} V_i \subset \underset{i \in J}{\cup} V_i$, and hence $x$ is in $\underset{i \in I}{\cup} W_i$.

This shows that $(K,\mathcal{W})$ is indeed an element of $S$. It is clear by construction that it is an upper bound for $(T ,\leq )$. Hence we have shown that every [[chain]] in $(S,\leq)$ has an upper bound, and so Zorn's lemma implies the claim.

=--

## References

The above account follows

* [[Matt Rosenzweig]], _[General shrinking lemma for normal spaces](https://matthewhr.wordpress.com/2014/06/11/general-shrinking-lemma-for-normal-spaces/)_

The example (a [[Dowker space]]) of a normal space with a (not locally-finite) countable cover to which the shrinking lemma does not apply is given in

* {#Beslagic85} Amer Beslagic, _A Dowker product_, Transactions of the AMS, vol 292, number 2 (1985) ([pdf](http://www.ams.org/journals/tran/1985-292-02/S0002-9947-1985-0808735-X/S0002-9947-1985-0808735-X.pdf))


shtuka

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#### Arithmetic geometry
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[[!include arithmetic geometry - contents]]
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#Contents#
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{:toc}

## Idea

In [[arithmetic geometry]] over a [[finite field]] a _shtuka_ on an [[arithmetic scheme]] is essentially an [[equivariant bundle|equivariant]] [[algebraic vector bundle]] on the [[product]] of the scheme with a given [[arithmetic curve]], where equivariance is with respect to the [[action]] of the [[Frobenius endomorphism]] (e.g. [Scholze-Weinstein, def. 1](#ScholzeWeinstein14)).

(*Shtuka* is a Russian word colloquially meaning "thing".)

## Definition
\begin{definition} (Definition 7.1 in [#Scholze17](#Scholze17))

Let $X$ be a smooth projective curve over $\mathbb{F}_{p}$ and let $S$ be a scheme over $\mathbb{F}_{p}$. A shtuka over $S$ relative to $X$ with legs at $x_{1},\ldots,x_{n}:S\to X$ is a [[vector bundle]] $\mathcal{E}$ over $S\times_{\mathbb{F}_{p}}X$ together with an isomorphism

$$\mathrm{Frob}_{S}^{\ast}\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}\cong\mathcal{E}\vert_{S\times_{\mathbb{F}_{p}} X\setminus \bigcup_{i=1}^{n}\Gamma_{x_{i}}}$$

\end{definition}

## Related concepts

* [[Rapoport-Zink space]]

##References

Reviews of the basic definition include

* [[David Goss]], *What is ... a Shtuka?*, Notices of the AMS **50** 1 (2003) 36-37 [[pdf](http://www.ams.org/notices/200301/what-is.pdf), full issue:[pdf](http://www.ams.org/notices/200301/200301FullIssue.pdf)]

* Wikipedia, _[Drinfeld module -- Shtuka](http://en.wikipedia.org/wiki/Drinfeld_module#Shtukas)_

Definition 1 in

* {#ScholzeWeinstein14} [[Jared Weinstein]], notes from lecture by [[Peter Scholze]], _Peter Scholze's lectures on $p$-adic geometry_, MSRI 2014 ([pdf](http://math.bu.edu/people/jsweinst/Math274/ScholzeLectures.pdf))

More conceptual discussion, in the context of the [[function field analogy]], is in

* {#Hartl06} [[Urs Hartl]], _A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic_ ([arXiv:math/0607182](http://arxiv.org/abs/math/0607182))

* {#Scholze17} [[Peter Scholze]], _p-adic geometry_, [arXiv:1712.03708](https://arxiv.org/abs/1712.03708)

[[!redirects shtukas]]


Shu-Heng Shao

* [Institute page](https://www.stonybrook.edu/commcms/physics/people/_profiles/shaos)

* [InSpire page](https://inspirehep.net/authors/1064881)

* [ResearchGate page](https://www.researchgate.net/profile/Shu-Heng-Shao)

## Selected writings

On [[generalized global symmetries]]:

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], [[Shu-Heng Shao]], *Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond* [[arXiv:2205.09545](https://arxiv.org/abs/2205.09545)]

category: people


shuffle

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### Context
#### Combinatorics
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[[!include combinatorics - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[mathematics]], by "shuffles" or "un-shuffles" one means  certain [[permutations]] , namely those which (when thought of as permuting the [[natural numbers]] $\{1, 2, \cdots, n \}$) are [[linear order]]-preserving along the first $p$ steps and then again along the remaining $q = n - p$ steps.

The term 'shuffle' conjures up the idea of shuffling a pack of cards. In fact the mathematical idea is nearer to shuffling two packs of cards one through the other. Suppose we have a pack of $p$ cards and a pack of $q$ cards and we build a pack of $p+q$ cards, whilst retaining the order on the two 'sub-packs'.  The result is a $(p,q)$-shuffle.

Or rather, the permutation in question may be seen as describing the un-doing of such a shuffling, whence some authors say "un-shuffle" for the same concept.

{#Illustration} In this vein, the following graphics shows an illustration of an example of a $(4,3)$-(un-)shuffle of an ordered list of 7 elements:

\begin{tikzcd}[
  row sep=0pt
  ]
  7 &[-20pt]
  {} \ar[rr, -, line width=5pt, blue, opacity=.3] &[-10pt] {}  &[+25pt] {} &[-10pt] {}
  &[+30pt]
  5 &[-20pt] {} &[-10pt] {} \ar[rr, -, line width=5pt, blue, opacity=.3] &[+25pt] {} &[-10pt] {}
  \\
  6 & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {}  & {} & {}
  &
  3 & {} & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {}
  \\
  5 & {} & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {}
  &
  2 & {} & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {}
  \\
  4 & {}  \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {} & {}
  &
  7 &
  {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {}  & {} & {}
  \\
  3 & {} & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {}
  &
  6 & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {}  & {} & {}
  \\
  2 & {} & {} \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {}
  &
  4 & {}  \ar[rr, -, line width=5pt, blue, opacity=.3] & {} & {} & {}
  \\
  1 & {} \ar[rr, -, line width=5pt, blue, opacity=.3]  & {} & {} & {}
  &
  1 & {} \ar[rr, -, line width=5pt, blue, opacity=.3]  & {} & {} & {}
\end{tikzcd}

## Definitions

\begin{definition}

For $p,q \in \mathbb{N}$ two [[natural numbers]], a **$(p,q)$-shuffle** is a [[permutation]]

$$
  (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)
$$

of $(1,2, \cdots, p+q)$ subject to the condition that

$$
  \mu_1 \lt \mu_2 \lt \cdots \lt \mu_p
$$

and

$$
  \nu_1 \lt \nu_2 \lt \cdots \lt \nu_q
  \,.
$$

In more formal terms, a $(p,q)$-shuffle is a [[permutation]] $\sigma \,\in\, \Sigma_{p + q}$ (an [[element]] of the [[symmetric group]] [[group action|acting]] on the [[finite set]] $\{1, \cdots, p + q\}$ of $p + q$ [[elements]]), such that

$$
  \sigma(1)
  \lt
  \sigma(2)
  \lt
  \cdots
  \lt
  \sigma(p)
  \;\;\;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;\;\;
  \sigma(p+1)
  \lt
  \sigma(p+2)
  \lt
  \cdots
  \lt
  \sigma(p+q)
  \,;
$$

hence, equivalently, such that

$$
  \underset{1 \leq i \lt p+q}{\forall}
  \;\;\;\;\;\;\;\;\;\;
  i \neq p
    \;\;\;\;
      \Rightarrow
    \;\;\;\;
  \sigma(i) \lt \sigma(i + 1)
  \,.
$$

The **signature** of a $(p,q)$-shuffle is the [[signature of a permutation|signature]] of the corresponding permutation.

\end{definition}

Two other equivalent (and dual) ways of defining the notion of $(p,q)$-shuffle are as follows (e.g. [Hoffbeck-Moerdijk 17, section 1.1](#HoffbeckMoerdijk17)), naturally understood as characterizing the non-degenerate [[simplices]] in a [[product of simplices]] (e.g. [Kerodon 2.5.7.2](#Kerodon): [00RH](https://kerodon.net/tag/00RH)):

* Consider $p$ and $q$ as the [[linear orders]] $(p) = \{ 1 \lt \dots \lt p \}$ and $(q) = \{ 1 \lt \dots \lt q \}$. Then a $(p,q)$-shuffle is a way of extending the [[partial order]] on the [[coproduct]] $(p) + (q)$ to a linear order, or equivalently, a [[surjective]] [[monotone function]]

   $$
     (p) + (q) \to (p+q)
   $$

* Consider $p$ and $q$ as [[non-empty]] [[linear orders]] $[p] = \{ 0 \lt \dots \lt p \}$ and $[q] = \{ 0 \lt \dots \lt q \}$. Then a $(p,q)$-shuffle is a [[maximum|maximal]] [[chain]] within the [[product]] [[partial order]] $[p] \times [q]$, or equivalently, an [[injective]] [[monotone function]]:

  $$
    [p + q] \to [p] \times [q]
    \,.
  $$

## Applications

### Products of simplices

Shuffles control the combinatorics of [[product]]s of [[simplices]]. See [[products of simplices]] for details.

### Eilenberg-Zilber map

Related to the product of simplices: shuffles control the [[Eilenberg-Zilber map]]. See there for details.

###Differential graded coalgebras

Shuffles are used in defining the pre-cgc structure on $\bigwedge V$ in the theory of [[differential graded coalgebras]]

###Differential graded Hopf algebras
Shuffles are also used for defining the shuffle product on $T(V)$, see [[differential graded Hopf algebra]].

### In $L_\infty$-algebras ###

In the definition of [[L-∞ algebras]]
the _unshuffle_ side of shuffles is used.

## Properties
 {#Properties}

### Basic properties

\begin{remark}\label{NumberOfShuffles}
**(number of shuffles)**
\linebreak
  The number of $(p,q)$-shuffles is given by the [[binomial coefficient]]
$$
  #
  Sh(p,q)
  \;\;=\;\;
  \left(
    {p + q} \atop p
  \right)
  \,.
$$
Because, due to the preservation of order in the two "decks", a shuffle $\sigma \in Sh(p,q)  \subset Aut\big(\{1, \cdots, p + q\}\big)$ is fully determined by which $p$ elements of $\{1, \cdots, p + q\}$ are in the  image of $\{1, \cdots p\}$, which is the number of choices of $p$ out of $p + q$ elements.
\end{remark}

### The poset of $(p,q)$-shuffles

Any pair $(p,q)$ yields a [[poset]] relating the various $(p,q)$-[[shuffles]].

For example, consider this $(2,1)$-shuffle:
$$
  \begin{matrix}
    (0\lt 1) &-& (0\lt 2) &-& (1\lt 2)
  \end{matrix}
$$

(This [[Hasse diagram]] has been laid out horizontally to save space.  The bottom is to the left. The vertex $(0\lt 1)$ corresponds to the shuffle with $\mu_1 = 0, \mu_1 = 1$, and so on.)

We need here to explain the partial order.  We take the '$\mu$-signature' of the shuffle, that is, the ordered set $\mu_1\lt\ldots \lt \mu_p$. (Of course, this determines the $\nu$-signature as that is the complement of $\mu$.)

+--{: .num_defn}
######Definition######
Given two $(p,q)$-shuffles, represented by $(\mu, \nu)$ and $(\mu\prime,\nu\prime)$, we set

$$(\mu, \nu) \leq (\mu\prime,\nu\prime)$$

if, for each $i$ in the range $1\leq i\leq p$, $\mu_i \leq \mu_i\prime.$  We refer to this poset as $(Shuff(p,q),\leq)$.
=--

Going to  $(Shuff(3,1),\leq)$, the poset for $(3,1)$-shuffles,  the fact that $q = 1$ will mean that this poset is again [[linear order|linear]]:
$$\begin{matrix}
(0 \lt 1 \lt 2) &-& (0 \lt 1 \lt 3)& - &(0 \lt 2 \lt 3)& - &(1 \lt 2 \lt 3)
\end{matrix}$$

This is true in general:

+--{: .num_lemma}
######Lemma
If $p = 1$ or $q = 1$, then  $(Shuff(p,q),\leq)$ is a [[linear order|linear poset]].
=--

+-- {: .proof}
######Proof######
If $p = 1$, $\mu = (\mu_1)$ is a singleton, and the poset will be:

$$\begin{matrix}
(0) & - & (1) & - & \ldots & - & (q).
\end{matrix}$$

For $q = 1$, $\nu = (\nu_1)$, and the poset is

$$\begin{matrix}
(0 \lt 1 \lt \ldots \lt p-1) & - & (0 \lt 1 \lt \ldots \lt p-2 \lt p) & - & \ldots & - & (1 \lt \ldots \lt p),
\end{matrix}$$
where at each stage one misses out the single $\nu$-value.
=--

In all cases, each position is obtained from the one immediately to its 'left' by increasing _one_ value, yet remaining with a shuffle.  This is more clearly seen in the (2,2) example, which is no longer [[linear order|linear]].  First we display the grid in which things are happening.
$$\begin{matrix}
(0,2) & - & (1,2) & - & (2,2) \\
| & & | & & | \\
(0,1) & - & (1,1) & - & (2,1) \\
| & & | & & | \\
(0,0) & - & (1,0) & - & (2,0)
\end{matrix}$$

We can then look at the shuffle poset, noting again that it is not [[linear order|linear]]:

\begin{xymatrix@=1.5em}&&(1<2)\ar@{-}[dr]&&\\
(0<1)\ar@{-}[r]&(0<2)\ar@{-}[ur]\ar@{-}[dr]&&(1<3)\ar@{-}[r]&(2<3)\\
&&(0<3)\ar@{-}[ur]&&\end{xymatrix}

The left hand shuffle, labelled $(0\lt1)$, corresponds to $\left(\begin{array}{ccccc}0&1&2&2&2\\0&0&0&1&2\end{array}\right)$, so gives the path along the bottom of the square and then up the right hand side.

\begin{xymatrix@=1.5em}&&(2,2)\ar@{-}[d]\\
&&(2,1)\ar@{-}[d]\\
(0,0)\ar@{-}[r]&(1,0)\ar@{-}[r]&(2,0)\end{xymatrix}

The first change goes to $(0\lt 2)$ and gives a path with 2 steps,

\begin{xymatrix@=1.5em}&&(2,2)\ar@{-}[d]\\
&(1,1)\ar@{-}[r]\ar@{-}[d]&(2,1)\\
(0,0)\ar@{-}[r]&(1,0)&\end{xymatrix}

and corresponds to the shuffle, $\left(\begin{array}{ccccc}0&1&1&2&2\\0&0&1&1&2\end{array}\right)$,
so at this position, $(0\lt2)$, there is a choice, either increase 0 by 1 to get $(1\lt 2)$ or increase 2 by 1 to get $(0\lt3)$. In the first case, we get the path

\begin{xymatrix@=1.5em}&&(2,2)\ar@{-}[d]\\
(0,1)\ar@{-}[d]\ar@{-}[r]&(1,1)\ar@{-}[r]&(2,1)\\
(0,0)&&\end{xymatrix}

going horizontally across the square,
and the shuffle, $\left(\begin{array}{ccccc}0&0&1&2&2\\0&1&1&1&2\end{array}\right)$,
and in the second case, we get the shuffle:
 $\left(\begin{array}{ccccc}0&1&1&1&2\\0&0&1&2&2\end{array}\right)$, and the obvious path up the middle of the square:

\begin{xymatrix@=1.5em}&(1,2)\ar@{-}[r]\ar@{-}[d]&(2,2)\\
&(1,1)\ar@{-}[d]&\\
(0,0)\ar@{-}[r]&(1,0)&\end{xymatrix}

From $(1 \lt 2)$ or $(0\lt3)$, there is only one way to go, namely to $(1 \lt 3)$ and a 2-step path (left to you), and, finally it is just one step from  $(1 \lt 3)$ to $(2 \lt 3)$ and the other extremal path.

\linebreak

In summary, each path corresponds to a simplex of maximal dimension in the product. The poset encodes the simplest relationships between those paths with the links in the Hasse diagram corresponding to simple changes in the path, and adjacency of the two simplices in the product, but note that the poset is usually not linear.

### The Anti-Lex order

There is a total order on $Shuff(p,q)$ related to the above partial order and which is useful when checking cancellation of terms in non-commutative contexts, as occurs in the Whitehead product formula given by Curtis and attributed to Kan.  This is the **anti-lexicographic order.

We  represent   a $(p,q)$-shuffle $(\mu,\nu)$ by an increasing $\mu$-sequence $\mu_1\lt \ldots \lt \mu_p$, (so with complementary $\nu$-sequence $\nu_1\lt \ldots \lt \nu_q$).

We order the $(\mu,\nu)$ using just the $\mu$-sequence as, of course, that determines the $\nu$-sequence completely.  Inductively in $p$, we set

$$\mu = (\mu_1,\ldots, \mu_p)\prec \mu\prime = (\mu\prime_1,\ldots, \mu\prime_p),$$

(i) if $\mu_p\lt \mu\prime_p$,

or

(ii) if $\mu_p = \mu\prime_p$ and $\mu_p$ is odd,

$$(\mu_1,\ldots, \mu_{p-1})\prec (\mu\prime_1,\ldots, \mu\prime_{p-1}),$$

whilst if $\mu_p = \mu\prime_p$ and $\mu_p$ is even,

$$(\mu_1,\ldots, \mu_{p-1})\succ (\mu\prime_1,\ldots, \mu\prime_{p-1}),$$

where we adopt the notation $(\mu_1,\ldots, \mu_p)$ instead of $(\mu_1\lt\ldots \lt \mu_p)$.

For example, on our (2,2)-shuffles, the total order is

$$(0,1) - (1,2) - (0,2) - (0,3) - (1,3) - (2,3). $$

We note the lexicographic order on the sector with $\mu_2 = 2$ is reversed.

For illustrative purposes, we will look at two other examples.

 First a generic $(p,1)$ case:  the shuffle poset for $(p,1)$ is, more or less,

$$(0,\ldots , p-2) - (0,\ldots, p-4,p-3,p-1) - (0, \ldots, p-4,p-2,p-1) -
 \ldots  -  (1, \ldots , p-1)$$

as, at each position, there is only one transposition that can be applied.  The anti-lex total order will correspond _exactly_ to this if $p-1$ is odd, but, if $p-1$ is even, it will reverse the order on the last $p-1$ positions giving

$$(0,\ldots , p-2) - (1, \ldots , p-1) - (0,2, \ldots, p-1) -  \ldots  - (0,\ldots, p-3,p-1).$$

There are various points to note.  Firstly that if $p$ is even, the permutation corresponding to $(1, \ldots , p-1)$ is odd.  Secondly, the geometric picture is simply a prism, $\Delta[p]\times \Delta[1]$, and we can easily interpret the above order as a filling scheme for the simplices of $(\Delta[p]\times \Delta[1])\times \Delta[1]$, starting with the empty shell of

$$(\Delta[p]\times \Delta[1])\times \{0\} \cup \partial(\Delta[p]\times \Delta[1])\times \Delta[1],$$

together with part of the top of the 'canister'.  (In general the total order seems to correspond to some sort of filling scheme although this is not always as clear as here.)

Our second case will be (3,2).  Again we will write $(i,j,k)$ instead of $i\lt j\lt k$ in order to save confusion over the various different orders being considered.  The shuffle poset, $\mathrm{Shuff}(3,2)$, has Hasse diagram given below:

\begin{xymatrix@=1.5em}&&(2,3,4)\ar@{-}[d]&\\
&&(1,3,4)\ar@{-}[dl]\ar@{-}[dr]&\\
&(1,2,4)\ar@{-}[dl]\ar@{-}[dr]&&(0,3,4)\ar@{-}[dl]\\
(1,2,3)\ar@{-}[dr]&&(0,2,4)\ar@{-}[dl]\ar@{-}[dr]&\\
&(0,2,3)\ar@{-}[dr]&&(0,1,4)\ar@{-}[dl]\\
&&(0,1,3)\ar@{-}[d]&\\
&&(0,1,2)&\end{xymatrix}

The subgraph of those with $\mu_3 = 4$, of course, is a copy of our earlier $(2,2)$-example, whilst that with $\mu_3 = 3$, is a copy of $Shuff(3,1)$.  Within that we have, on deleting the 3s from the final positions, a copy of $Shuff(2,1)$ and so on.  Taking $Shuff(2,1) \times \{3\}$, we can match it with a $Shuff(2,1) \times \{4\}$ within the $\mu_3 = 4$ block, the matching being given by, for instance, $(0,1,3)\leftrightarrow(0,1,4)$, etc.  The anti-lex order gives
\[
(0,1,2)-(0,1,3) - (1,2,3)-(0,2,3)-(2,3,4)-(1,3,4)-(0,3,4) - (0,2,4) - (1,2,4) - (0,1,4),\]

which is
$$Shuff(3,1)\oplus Shuff(2,2)^{op}\times \{4\},$$
the ordinal sum, or join, of the anti-lex ordered shuffle sets.  This sort of decomposition is quite general.

## Related concepts

* [[shuffles of trees]]

* [[Eilenberg-Zilber map]]

* [[Eilenberg-Zilber theorem]]

## References

Original articles:

* [[Samuel Eilenberg]], [[Saunders MacLane]], On the groups $H(\Pi,n)$, I, Ann. of Math. (2) 58, (1953), 55–106. ([jstor](https://www.jstor.org/stable/1969820))

Review:

* {#AguiarMahajan10} [[Marcelo Aguiar]], [[Swapneel Mahajan]], Section 2.2.3 in: _[[Monoidal Functors, Species and Hopf Algebras]]_, With forewords by [[Kenneth Brown]], [[Stephen Chase]], [[André Joyal]]. CRM Monograph Series __29__ Amer. Math. Soc. 2010. lii+784 pp. ([author pdf](http://www.math.cornell.edu/~maguiar/a.pdf))

* [[Kerodon]], 2.5.7.2 *$(p,q)$-Shuffles* ([00RH](https://kerodon.net/tag/00RH))

The two dual equivalent characterizations of $(p,q)$-shuffles (called _shuffles of linear trees_ or _shuffles of linear orders_) are discussed in section 1.1 of

* {#HoffbeckMoerdijk17} [[Eric Hoffbeck]], [[Ieke Moerdijk]], _Shuffles of trees_, ([arXiv:1705.03638](https://arxiv.org/abs/1705.03638)).

[[!redirects shuffles]]
[[!redirects unshuffle]]
[[!redirects unshuffles]]


shuffle of trees

## Idea

The generalization of the concept of [[shuffles]] to [[trees]].

## References

* {#HoffbeckMoerdijk} [[Eric Hoffbeck]], [[Ieke Moerdijk]], _Shuffles of trees_, ([arXiv:1705.03638](https://arxiv.org/abs/1705.03638)).

[[!redirects shuffles of trees]]


Shun-ichi Amari

__Shun-ichi Amari__ is, along with N. N. Chentsov, a pioneer of [[information geometry]] and its applications to [[machine learning]].

* Shun-ichi Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, C. R. Rao, _Differential geometry in statistical inference_ ([project Euclid](http://projecteuclid.org/euclid.lnms/1215467056))
* Shun-ichi Amari, _Information geometry and its applications_, Applied Mathematical Sciences 194, Springer 2016
* Shun-ichi Amari, Hiroshi Nagaoka, _Methods of information geometry_, Transactions of mathematical monographs __191__, American Mathematical Society, 2000.

* Shun-ichi Amari, T. Ozeki, H. Park, _Learning and inference in hierarchical models with singularities_, Syst. Comput. Japan 34:7 (2003) 34--42

category: people


Shuwei Liu

## Selected writings

On [[KK-compactification]] of [[D=6 N=(2,0) SCFT]] on [[Riemann surfaces]] [[4d superconformal gauge field theory]]:

* [[Jin Chen]], [[Babak Haghighat]], [[Shuwei Liu]], [[Marcus Sperling]], _4d $\mathcal{N}=1$ from 6d D-type $\mathcal{N}=(1,0)$_,  J. High Energ. Phys. **2020** 152 (2020) [[arXiv:1907.00536](https://arxiv.org/abs/1907.00536), doi:10.1007/JHEP01(2020)152]

category: people


Shôrô Araki

* [IAS Past Members page](https://www.ias.edu/scholars/sh%C3%B4r%C3%B4-araki)

## Selected writings

Discussion of $\mathbb{Z}_2$-[[equivariant stable homotopy groups]]/[[equivariant stable cohomotopy]] of [[spheres]]:

* {#ArakiIriye82} [[Shôrô Araki]], [[Kouyemon Iriye]], _Equivariant stable homotopy groups of spheres with involutions. I_, Osaka J. Math. Volume 19, Number 1 (1982), 1-55. ([Euclid:1200774828](http://projecteuclid.org/euclid.ojm/1200774828))

On [[real oriented cohomology theory]] in general and in particular on [[Real cobordism cohomology]] and [[Real K-theory]]:

* {#Araki78} [[Shôrô Araki]], *$\tau$-Cohomology Theories*, Japanese Journal of Mathematics **4** 2 (1978) 363-416 [[doi:10.4099/math1924.4.363](https://doi.org/10.4099/math1924.4.363)]

* {#Araki79} [[Shôrô Araki]], *Forgetful spectral sequences*, Osaka Journal of Mathematics **16** 1 (1979) 173-199 [[Euclid](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-16/issue-1/Forgetful-spectral-sequences/ojm/1200771837.full)]

  > (the relevant [[spectral sequences]])

* [[Shôrô Araki]], *Orientations in $\tau$-cohomology theories*, Japan Journal of Mathematics (N.S.) **5** 2 (1979) 403-430 [[doi:10.4099/math1924.5.403](https://doi.org/10.4099/math1924.5.403)]

On generalization of the [[h-cobordism theorem]] to [[equivariant cobordism]]:

* [[Shôrô Araki]], *Equivariant $s$-cobordism theorems*, J. Math. Soc.  Japan **40** 2 (1988) [[pdf](file:///C:/Users/us13/Downloads/04020349-2.pdf)]

category: people

[[!redirects Shoro Araki]]


Si Li

* [webpage](https://sili-math.github.io/)

## Selected writings

On [[BCOV theory]]:

* [[Kevin Costello]], [[Si Li]], _Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model_ ([arXiv:1201.4501](http://arxiv.org/abs/1201.4501))

* [[Si Li]], _BCOV theory on the elliptic curve and higher genus mirror symmetry_ ([arXiv:1112.4063](http://arxiv.org/abs/1112.4063))

* [[Si Li]], _Variation of Hodge structures, Frobenius manifolds and Gauge theory_ ([arXiv:1303.2782](http://arxiv.org/abs/1303.2782))

* {#CostelloLi15} [[Kevin Costello]], Si Li, _Quantization of open-closed BCOV theory, I_ ([arXiv:1505.06703](http://arxiv.org/abs/1505.06703))

On [[vertex operator algebras]] and [[mirror symmetry]]:

* [[Si Li]], _Vertex algebras and quantum master equation_([arxiv/1612.01292](https://arxiv.org/abs/1612.01292))

On [[topologically twisted supergravity]]:

* [[Kevin Costello]], [[Si Li]], _Twisted supergravity and its quantization_ ([arXiv:1606.00365](https://arxiv.org/abs/1606.00365), [spire:1466640](http://inspirehep.net/record/1466640))

category: people


Sibe Mardesic

__Sibe Mardešić__ was a Croatian [[topology|topologist]], retired Professor at the University of Zagreb and a full member of Croatian Academy of Arts and Sciences (HAZU). His son Pavao Mardešić is also a mathematician, a specialist in the theory of (low dimensional) dynamical systems.

A brief biography can be found in the article [HAPPY 80-th BIRTHDAY PROFESSOR MARDEŠIĆ](http://web.math.hr/glasnik/vol_42/mardesic.html), in Glasnik Matematički, the journal that he helped to found.

## Selected writings

Most of Sibe Mardešić's important works are related to [[shape theory]] and geometric topology, mainly in the approach via inverse systems, which he pioneered (there was also a parallel development by [[Tim Porter]]). Sibe Mardešić wrote several monographs and edited several proceedings volumes in geometric topology and shape theory.

* S. Mardešić, [[Jack Segal]], _Shape theory_, North Holland 1982.

* S. Mardešić, Strong shape and homology, Springer monographs in mathematics, 2000. xii+489 pp.

* HAZU pages: [Akademik Sibe Mardešić](http://www.hazu.hr/Akademici/SMardesic.html), in Croatian; and [bio data in English](http://www.hazu.hr/Akademici/SMardesic_bio.html).

* Ju. T. Lisica and S. Mardesic, Steenrod--Sitnikov homology for arbitrary spaces, Bull. Amer. Math. Soc. 9 (1983), 207--210;

* S. Mardešić and L. R. Rubin, Cell-like mappings and and non-metrizable compacta of finite cohomological dimension, Trans. Amer. Math. Soc. 311 (1989), 53--79;

* S. Mardešić, Nonvanishing derived limits in shape theory, Topology 35(1996), 521--532.

* S. Mardešić, Comparison of singular and Čech homology i
locally connected spaces, Michigan Math. J. 6 (1959), 151--166.
* S. Mardešić, On covering dimension and inverse limits
for compact spaces, Illinois J. Math. 4 (1960), 278--291.
* S. Mardešić, On the Hahn-Mazurkiewicz theorem in non-metric
spaces, Proc. Amer. Math. Soc. 11 (1960), 929--937.
* S. Mardešić, J. Segal, epsilon-mappings   onto polyhedra,
Trans. Amer. Math. Soc. 109 (1963), 146--164.
* S. Mardešić, Images of ordered compacta are locally
peripherally metric, Pacific J. Math. 23 (1967), 557--568.
* S. Mardešić, J. Segal, Shapes of compacta and ANR-systems,
Fund. Math. 72 (1971), 41--59.
* S. Mardešić, Shapes for topological spaces,  General
Topology and its Appl. 3 (1973), 265--282.
* S. Mardešić, On the Whitehead theorem in shape theory I,
Fund. Math. 91 (1976), 51--64.
* S. Mardešić, Approximate polyhedra, resolutions of maps
and shape fibrations, Fund. Math. 114  (1981), 53--78.
* Ju. T. Lisica, S. Mardešić, Coherent prohomotopy and
strong shape theory, Glasnik Mat. 19 (39) (1984), 335--399.
* Ju. T. Lisica, S. Mardešić, Strong homology of inverse
systems of spaces, I, II, Topology and its Appl. 19 (1985), 29--43. 45--64.
* S. Mardešić, A. V. Prasolov, Strong homology is not additive,
Trans. Amer. Math. Soc. 307 (1988), 725--744.
* S. Mardešić, L.R. Rubin, Approximate inverse systems
of compacta and covering dimension, Pacific J. Math. 138 (1989), 129--144.
* S. Mardešić, T. Watanabe, Approximate resolutions of
spaces and mappings, Glasnik Mat. 24 (1989), 587--637.
* S. Mardešić, Strong expansions and strong shape theory,
Topology and its Appl. 38 (1991), 275--291.
* S. Mardešić, A. V. Prasolov, On strong homology of
compact spaces, Topology and its Appl. 82 (1998), 327--354.
* S. Mardešić, Coherent homotopy and localization,
Topology and its Appl. 94 (1999), 253--274.

category: people

[[!redirects Sibe Mardešić]]
[[!redirects Сибе Мардешић]]
[[!redirects S. Mardešić]]
[[!redirects Mardešić]]
[[!redirects Mardesic]]


side effect

#Contents#
* table of contents
{:toc}

##Idea

In [[computer science]], a **side effect** refers simply to the modification of some kind of state, such as changing the value of a variable, writing some data to disk or raising an exception.

## Relation to Monads and Algebras and Multicategories

The presence of side effects in a programming language usually means the language cannot be given a [[denotational semantics]] in the category of [[sets]], or in any [[bicartesian closed category]]. Instead, they can be interpreted in the category of [[algebras]] of a [[monad]] on a [[bicartesian closed category]], often a monad presented by an [[algebraic theory]]. There is a direct relationship between properties of the monad or theory and the type theoretic syntax that can be interpreted in its [[Eilenberg-Moore category|category of algebras]].

| [[monads]]  | [[algebraic theories]] |  flavor of [[multicategory]] | typical model |
|---|---|---|---|
| [[strong monad]] | arbitrary theory | [[call-by-push-value]]/[[Freyd multicategory]]  | [[enriched category]] with [[powers]] and [[copowers]] |
| [[monoidal monad]] | [[commutative algebraic theory]] | [[linear logic]]/[[linear-non-linear logic]] | [[symmetric monoidal closed category]]/linear-non-linear adjunction
| [[affine monad]]  | [[affine algebraic theory]] | [[affine logic]] | [[semicartesian monoidal category]] |
| [[relevant monad]] | [[relevant algebraic theory]] | [[relevance logic]] | [[relevance monoidal category]] |
| preserves [[cartesian products]] | affine and relevant theory | [[cartesian multicategory]]/[[S4]] | monoidal adjunction between [[bicartesian closed]] categories

So in typical models, you have a category $V$ of "value types" and pure morphisms and a category $C$ of "comptuation types" or "linear types" and homomorphisms, where $C$ is the category of algebras of a monad on $V$. Then, how the types and terms of a language are interpreted in the model depends on the [[evaluation order]] of the language. In [[call-by-value]] languages, types are interpreted as objects of $V$, but terms $\Gamma \vdash M : A$ are interpreted as Kleisli morphisms $\Gamma \to T A$, or equivalently homomorphisms of free algebras in $C$: $F \Gamma \to F A$. In [[call-by-name]] languages, types are dually interpreted as objects of $C$, but terms $\Gamma \vdash M : A$ are interpreted as co-Kleisli morphisms $U\Gamma \to U A$.

A dual approach would be to start with a category of linear morphisms and construct a category of "value types" as the category of coalgebras of some well-behaved [[comonad]]. This approach is advocated by Girard in his original work on [[linear logic]].

## Related concepts

* [[algebraic effect]]

* [[monad (in computer science)]]

* [[effect handler]]

## References

The conditions on monads above and their relationship to structural rules are described in the following (though the notions themselves are originally due to [[Anders Kock]]).

* {#Jacobs94} [[Bart Jacobs]], _Semantics of Weakening and Contraction_, Annals of Pure and Applied Logic, volume 69, Issue 1, (1994) pp.73-106, doi:[10.1016/0168-0072(94)90020-5](https://doi.org/10.1016/0168-0072%2894%2990020-5)

On side effects in [[dependent type theory|dependent]] ([[dependent linear type theory|linear]]) [[type theory]]:

* [[Matthijs Vákár]], *In Search of Effectful Dependent Types* ([arXiv:1706.07997](https://arxiv.org/abs/1706.07997), [uuid:e91e19b3-7e10-4fda-9433-f23b469e4049](https://ora.ox.ac.uk/objects/uuid:e91e19b3-7e10-4fda-9433-f23b469e4049))

[[!redirects side effects]]


Sidney A. Morris

* [personal page](https://www.sidneymorris.net/)

## Selected writings

On [[Pontrjagin duality]]:

* {#Morris77} [[Sidney A. Morris]], _Pontryagin Duality and the Structure of Locally Compact Abelian Groups_, London Math. Soc. Lecture Notes 29, Cambridge U. Press, 1977 ([doi:10.1017/CBO9780511600722](https://doi.org/10.1017/CBO9780511600722))

category: people

[[!redirects Sidney Morris]]


Sidney Coleman

* [Wikipedia entry](http://de.wikipedia.org/wiki/Sidney_Coleman)

## Selected writings

Introducing the [[Coleman-Mandula theorem]]:

* [[Sidney Coleman]], [[Jeffrey Ellis Mandula]], _All Possible Symmetries of the S Matrix_, Physical Review __159__ (5): 1251--1256 (1967) ([doi:10.1103/PhysRev.159.1251](https://doi.org/10.1103/PhysRev.159.1251))

On [[chiral perturbation theory]]:

* [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. I_, Phys. Rev. 177, 2239 (1969) ([doi:10.1103/PhysRev.177.2239](https://doi.org/10.1103/PhysRev.177.2239))

* [[Curtis Callan]], Jr., [[Sidney Coleman]], [[Julius Wess]], [[Bruno Zumino]], _Structure of Phenomenological Lagrangians. II_, Phys. Rev. 177, 2247 (1969) ([doi:10.1103/PhysRev.177.2247](https://doi.org/10.1103/PhysRev.177.2247))

On [[vacuum decay]]:

* [[Sidney Coleman]], _Fate of the false vacuum: Semiclassical theory_ , Phys. Rev. D 15, 2929–2936 (1977) ([doi:10.1103/PhysRevD.15.2929](https://doi.org/10.1103/PhysRevD.15.2929))

* [[Sidney Coleman]], [[Frank De Luccia]], _Gravitational effects on and of vacuum decay_ , Phys. Rev. D 21, 3305–3315 (1980) ([doi:10.1103/PhysRevD.21.3305](https://doi.org/10.1103/PhysRevD.21.3305))

category: people


Siegel modular form

[[!redirects Siegel modular forms]]

## Idea

_Siegel modular forms_ are one generalization of [[modular forms]] to functions in more than one complex variable (another such generalization are [[Hilbert modular forms]]).

Where ordinary modular forms are locally functions on the [[moduli stack of elliptic curves]] over the complex numbers, so Siegel modular forms are locally functions on something like the moduli stack of higher-dimensional complex [[abelian varieties]].

## Definition

Let $\mathcal{H}_{g}$ be the [[Siegel upper half-space]] of degree $g$ and let $\Sp_{2g}(\mathbb{Z})$ the subgroup of elements in the [[symplectic group]] $\Sp_{2g}(\mathbb{R})$ with integral entries. Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and fix a representation

$$\rho:\GL_{g}(\mathbb{C})\to \GL(V).$$

A _Siegel modular form of weight $\rho$_ is a [[holomorphic function]] $f:\mathcal{H}_{g}\to V$ such that

$$f(\gamma(\tau))=\rho(C\tau+D)f(\tau)$$

for all $\tau=\begin{pmatrix}A & B\\C & D\end{pmatrix}\in\Sp_{2g}(\mathbb{Z})$ and all $\tau\in\mathcal{H}_{g}$. If $g=1$, we additionally require that $f$ is holomorphic at $\infty$. A _classical Siegel modular form of weight $k$_ is the special case when the representation $\rho$ is given by taking $k$-th powers of the determinant. Further specializing to $g=1$, this gives classical (also known as elliptic) [[modular forms]].

## Related concepts

* [[Siegel upper half space]]

## References

Gerard van der Greer, _Siegel Modular Forms_, [arxiv:0605346](https://arxiv.org/abs/math/0605346)


Siegel upper half-space

#Contents#
* table of contents
{:toc}

## Idea

The _Siegel upper half space_ is the generalization of the [[upper half-plane]] as one passes from the description of complex [[elliptic curves]] to that of [[Riemann surfaces]] of higher [[genus]].

## Definition

The _Siegel upper half space of degree $g$ (or genus $g$)_ is the set of $g\times g$ [[symmetric matrices]] whose entries are complex numbers with positive-definite imaginary part.

## Relation to the Symplectic Group

The Siegel upper half space of degree $g$ may be obtained as the quotient $\mathcal{H}_{g}=\Sp_{2g}(\mathbb{R})/\U(g)$ of the [[symplectic group]] $\Sp_{2g}(\mathbb{R})$ by its maximal compact subgroup (and the stabilizer of $i\cdot\mathbb{1}_{g}$), the [[unitary group]] $\U(g)$, embedded into $\Sp_{2g}(\mathbb{R})$ as follows:

$$\U(g)=\lbrace \begin{pmatrix} A & B \\ -B & A\end{pmatrix}\in\Sp_{2g}(\mathbb{R}):A\cdot A^{t}+B\cdot B^{t}=\mathbb{1}_{g}\rbrace.$$

## Related concepts

* [[Siegel modular form]]

* [[symplectic group]]

## References

* Wikipedia, _[Siegel upper half-space](http://en.wikipedia.org/wiki/Siegel_upper_half-space)_

[[!redirects Siegel upper half space]]


Siegfried Bosch

* [webpage](http://wwwmath.uni-muenster.de/persdb/show_perspage.php?id=73)

## Selected writings

On [[algebraic geometry]]:

* [[Siegfried Bosch]], *Algebraic Geometry and Commutative Algebra*, Universitext, Springer (2017) [[doi:10.1007/978-1-4471-4829-6](https://doi.org/10.1007/978-1-4471-4829-6)]

## Related enties

* [[rigid analytic geometry]]

category: people


Sierpinski space

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

# Contents
* table of contents
{: toc}

## Definition

### In impredicative mathematics

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

The __Sierpiński space__ $\Sigma$ is the [[topological space]]

1. whose underlying set has two elements, say $\{0,1\}$,

1. whose set of [[open subsets]] is $\left\{ \emptyset, \{1\}, \{0,1\}  \right\}$.

(We could exchange "0" and "1" here, the result would of course be [[homeomorphism|homeomorphic]]).

Equivalently we may think of the underlying [[set]] as the set of [[classical mathematics|classical]] [[truth values]]  $\{\bot, \top\}$, equipped with the [[specialization topology]], in which $\{\bot\}$ is [[closed subset|closed]] and $\{\top\}$ is an [[open subset|open]] but not conversely.

=--

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

In [[constructive mathematics]], it is important that $\{\top\}$ be open (and $\{\bot\}$ closed), rather than the other way around.  Indeed, the general definition (since we can\'t assume that every element is either $\top$ or $\bot$) is that a subset $P$ of $\Sigma$ is open as long as it is upward closed: $p \Rightarrow q$ and $p \in P$ imply that $q \in P$.  The ability to place a topology on $\Top(X,\Sigma)$ is fundamental to [[abstract Stone duality]], a constructive approach to general topology.

=--

### In predicative constructive mathematics

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

The __Sierpiński space__ $\Sigma$ is the [[initial object|initial]] [[sigma-frame|$\sigma$-frame]].

=--

### As a higher inductive type

In type theory, as a [[higher inductive type]] __Sierpinski space__ $\Sigma$ is generated by the following constructors:

* A type $\Sigma$,
* A top term $1:\Sigma$
* A meet operation $(-)\wedge(-): \Sigma \times \Sigma \to \Sigma$
* A left unit identity for meet
$$m_\lambda:\prod_{a:\Sigma} 1 \wedge a = a$$
* A right unit identity for meet
$$m_\rho:\prod_{a:\Sigma} a \wedge 1 = a$$
* A associative identity for meet
$$m_\alpha:\prod_{a:\Sigma} \prod_{b:\Sigma} \prod_{c:\Sigma} (a \wedge b) \wedge c = a \wedge (b \wedge c)$$
* A commutative identity for meet
$$m_\kappa:\prod_{a:\Sigma} \prod_{b:\Sigma} a \wedge b = b \wedge a$$
* An idempotent identity for meet
$$m_\iota:\prod_{a:\Sigma} a \wedge a = a$$
* A bottom term $0:\Sigma$
* A join operation $(-)\vee(-): \Sigma \times \Sigma \to \Sigma$
* A left unit identity for join
$$j_\lambda:\prod_{a:\Sigma} 0 \vee a = a$$
* A right unit identity for join
$$j_\rho:\prod_{a:\Sigma} a \vee 0 = a$$
* A associative identity for join
$$j_\alpha:\prod_{a:\Sigma} \prod_{b:\Sigma} \prod_{c:\Sigma} (a \vee b) \vee c = a \vee (b \vee c)$$
* A commutative identity for join
$$j_\kappa:\prod_{a:\Sigma} \prod_{b:\Sigma} a \vee b = b \vee a$$
* An idempotent identity for join
$$j_\iota:\prod_{a:\Sigma} a \vee a = a$$
* A meet-join absorption identity
$$m_j:\prod_{a:\Sigma} \prod_{b:\Sigma} a \wedge (a \vee b) = a
* A join-meet absorption identity
$$j_m:\prod_{a:\Sigma} \prod_{b:\Sigma} a \vee (a \wedge b) = a
* A sequential join function:
$$\Vee_{n:\mathbb{N}} (-)(n): (\mathbb{N} \to \Sigma) \to \Sigma$$
* An ascending sequence condition:
$$s: \prod_{n:\mathbb{N}} \prod_{s:\mathbb{N} \to \Sigma} s(n) \wedge \left(\Vee_{i:\mathbb{N}} s(i)\right) = s(n)$$
* A sequential least upper bound condition:
$$l: \prod_{x:\Sigma} \prod_{s:\mathbb{N} \to \Sigma} \left(\prod_{n:\mathbb{N}} s(n) \wedge x = s(n)\right) \to \left(\left(\Vee_{n:\mathbb{N}} s(n)\right) \wedge x = \Vee_{n:\mathbb{N}} s(n)\right)$$
* A function
$$\mathcal{t}:(\Sigma \times (\mathbb{N} \to \Sigma)) \to \mathbb{N} \to \Sigma$$
* A dependent function
$$p:\prod_{x:\Sigma} \prod_{s:\mathbb{N} \to \Sigma} \mathcal{t}(x,s)(n) = x \wedge s(n)$$
* A sequentially distributive condition:
$$d: \prod_{x:\Sigma} \prod_{s:\mathbb{N} \to \Sigma} x \wedge \left(\Vee_{n:\mathbb{N}} s(n)\right) = \Vee_{n:\mathbb{N}} \mathcal{t}(x,s)(n)$$
* A 0-truncator
$$\tau_0: \prod_{a:\Sigma} \prod_{b:\Sigma} isProp(a=b)$$

### As a higher inductive-inductive type ###

The __Sierpinski space__ $\Sigma$ is inductively generated by

* a term $\bot:\Sigma$
* a binary operation $(-)\vee(-):\Sigma \times \Sigma \to \Sigma$
* a function $\Vee_{n:\mathbb{N}} (-)(n): (\mathbb{N} \to \Sigma) \to \Sigma$
* a term $\top:\Sigma$
* a binary operation $(-)\wedge(-):\Sigma \times \Sigma \to \Sigma$

and the [[partial order]] [[type family]] $\leq$ is simultaneously inductively generated by

* a family of dependent terms
$$a:\Sigma, b:\Sigma \vdash \pi(a,b):isProp(a \leq b)$$
representing that each type $a \leq b$ is a [[proposition]].

* a family of dependent terms
$$a:\Sigma \vdash \rho(a):a \leq a$$
representing the reflexive property of $\leq$.

* a family of dependent terms
$$a:\Sigma, b:\Sigma, c:\Sigma \vdash \tau(a, b, c):(a \leq b) \times (b \leq c) \to (a \leq c)$$
representing the transitive property of $\leq$.

* a family of dependent terms
$$a:\Sigma, b:\Sigma \vdash \sigma(a, b):(a \leq b) \times (b \leq a) \to (a = b)$$
representing the anti-symmetric property of $\leq$.

* a family of dependent terms
$$a:\Sigma \vdash t(a):\bot \leq a$$
representing that $\bot$ is initial in the poset.

* three families of dependent terms
$$a:\Sigma, b:\Sigma \vdash i_a(a, b):a \leq a \vee b$$
$$a:\Sigma, b:\Sigma \vdash i_b(a, b):b \leq a \vee b$$
$$a:\Sigma, b:\Sigma, (-)\oplus(-):\Sigma \times \Sigma \to \Sigma, i_a(a, b):a \leq a \oplus b, i_b(a, b):b \leq a \oplus b \vdash i(a,b):(a \vee b) \leq (a \oplus b)$$
representing that $\vee$ is a coproduct in the poset.

* two families of dependent terms
$$n:\mathbb{N}, s:\mathbb{N} \to \Sigma \vdash i(n, s):s(n) \leq \Vee_{n:\mathbb{N}} s(n)$$
$$x:\Sigma, s:\mathbb{N} \to \Sigma \vdash i(s, x):\prod_{n:\mathbb{N}}(s(n) \leq x) \to  \left(\Vee_{n:\mathbb{N}} s(n) \leq x\right)$$
representing that $\Vee$ is a denumerable/countable coproduct in the poset.

* a family of dependent terms
$$a:\Sigma \vdash t(a):a \leq \top$$
representing that $\top$ is terminal in the poset.

* three families of dependent terms
$$a:\Sigma, b:\Sigma \vdash p_a(a, b):a \wedge b \leq a$$
$$a:\Sigma, b:\Sigma \vdash p_b(a, b):a \wedge b \leq b$$
$$a:\Sigma, b:\Sigma, (-)\otimes(-):\Sigma \times \Sigma \to \Sigma, p_a(a, b):a \leq a \otimes b, p_b(a, b):b \leq a \otimes b \vdash p(a,b):(a \otimes b) \leq (a \wedge b)$$
representing that $\wedge$ is a product in the poset.

* a family of dependent terms
$$x:\Sigma, s:\mathbb{N} \to \Sigma \vdash i(s, x):\left(\prod_{n:\mathbb{N}}(s(n) \leq x)\right) \to  \left(\Vee_{n:\mathbb{N}} s(n) \leq x\right)$$
representing the countably infinitary distributive property.

## Properties

### As a topological space

This Sierpinski space

* is a [[contractible space]];

* is a [[locally contractible space]];

* has a [[focal point]];

* is a [[sober topological space]] (but not [[separation axiom|T1]]).

According properties are inherited by the [[Sierpinski topos]] and the [[Sierpinski (∞,1)-topos]] over $Sierp$.

### As a classifer for open/closed subspaces

The Sierpinski space $S$ is a classifier for [[open subspaces]] of a [[topological space]] $X$ in that for any open subspace $A$ of $X$ there is a unique [[continuous function]] $\chi_A: X \to S$ such that $A = \chi_A^{-1}(\top)$.

Dually, it classifies [[closed subsets]] in that any closed subspace $A$ is $\chi_A^{-1}(\bot)$.  Note that the closed subsets and open subsets of $X$ are related by a [[bijection]] through [[complement|complementation]]; one gets a topology on the set of either by identifying the set with $\Top(X,\Sigma)$ for a suitable [[function space]] topology.  (This part does not work as well in [[constructive mathematics]].)

## Related concepts

* [[empty space]], [[point space]]

* [[Cantor space]]

* [[Sierpinski topos]], [[Sierpinski (∞,1)-topos]]

* [[Stone duality]]

* [[separation axioms in terms of lifting properties]]

* In [[synthetic topology]], an analogue of the Sierpinski space is called a [[dominance]].

* [[sigma-frame|$\sigma$-frame]]

## References

* Wikipedia, _[Sierpinski space](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_space)_

* [[Paul Taylor]],  _Foundations for computable topology -- 7 The Sierpinski space_ ([html](http://www.paultaylor.eu/ASD/foufct/sierpinski.html))

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type ([abs:1610.09254](https://arxiv.org/abs/1610.09254))

* Martin Escardó, Cory Knapp, Partial Elements and Recursion via Dominances in Univalent Type Theory ([pdf](https://drops.dagstuhl.de/opus/volltexte/2017/7682/))

* Andrej Bauer, Davorin Lešnik, Metric spaces in synthetic topology ([pdf](https://www.sciencedirect.com/science/article/pii/S0168007211000935))

[[!redirects Sierpinski space]]
[[!redirects Sierpi?ski space]]
[[!redirects Sierpinski topology]]
[[!redirects Sierpi?ski topology]]


Sierpinski topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

The **Sierpinski [[topos]]** is the [[arrow category]] of [[Set]].

Equivalently, this is the [[category of presheaves]] over the [[interval category]] $\Delta[1] := \mathbf{2} = \{0 \to 1\}$, or equivalently the [[category of sheaves]] over the [[Sierpinski space]] $Sierp$

$$
  Sh(Sierp) \simeq PSh(\Delta[1]) \simeq Set^{\Delta[1]}
  \,.
$$

Yet another description is that it is the [[Freyd cover]] of **Set**.

=--

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

Similarly, the **Sierpinski [[(∞,1)-topos]]** is the [[arrow (∞,1)-category]] $\infty Grpd^{\Delta[1]}$ of [[∞Grpd]].

Equivalently this is the [[(∞,1)-category of (∞,1)-presheaves]] on $\Delta[1]$ and equivalently the [[(∞,1)-category of (∞,1)-sheaves]] on $Sierp$:

$$
  Sh_{(\infty,1)}(Sierp)
  \simeq
  PSh_{(\infty,1)}(\Delta[1])
  \simeq
  \infty Grpd^{\Delta[1]}
  \,.
$$

=--

## Properties

### Presentation and Homotopy type theory

Being a [[(∞,1)-category of (∞,1)-functors]], the Sierpinski [[(∞,1)-topos]] is [[presentable (∞,1)-category|presented]] by any of the [[model structure on simplicial presheaves]]  $[\Delta[1], sSet]$.

Specifically the [[Reedy model structure]] of [[simplicial presheaves]] on the [[interval category]] $[\Delta[1], sSet]_{Reedy}$ provides a [[univalence|univalent]] model for [[homotopy type theory]] in the Sierpinski $(\infty,1)$-topos ([Shulman](#Shulman))

### Connectedness, locality, cohesion
 {#Cohesion}

We discuss the connectedness, locality and cohesion of the Sierpinski topos. We do so relative to an arbitrary [[base topos]]/[[base (∞,1)-topos]] $\mathbf{H}$, hence regard the [[global section geometric morphism]]

$$
  \mathbf{H}^I \to \mathbf{H}
  \,.
$$

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

The Sierpinski topos is a _[[cohesive topos]].

The Sierpinski $(\infty,1)$-topos is a [[cohesive (∞,1)-topos]].

$$
  (\Pi \dashv \Disc \dashv \Gamma \dashv coDisc)
  :
  \mathbf{H}^I
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \mathbf{H}
  \,.
$$

=--

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

For the first statement, see the detailed discussion at _[[cohesive topos]]_ _[here](cohesive+topos#FamiliesOfSets)_.

For the second statement, see the discussion at _[[cohesive (∞,1)-topos]]_  _[here](cohesive+%28infinity%2C1%29-topos#CohesiveDiagramToposes)_.

=--

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

The fact that the Sierpienski $(\infty,1)$-topos is, therefore, in particular

1. a [[locally ∞-connected (∞,1)-topos]];

1. an [[∞-connected (∞,1)-topos]];

1. a [[local (∞,1)-topos]]

all follow directly from the fact that it is the image, under [[localic reflection]], of the [[Sierpinski space]] (hence that it is [[n-localic (∞,1)-topos|0-localic]], its [[n-truncated|(-1)-truncation]] being the [[frame]] of opens of the Sierpinski space).

That space $Sierp$, in turn,

1. is a [[contractible topological space]];

1. a [[locally contractible topological space]].

1. has a [[focal point]]

which implies the corresponding three properties of the Sierpinski $\infty$-topos above.

=--

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

By the discussion at _[[cohesive (∞,1)-topos]]_ every such may be thought of as a _fat point_, the abstract _cohesive blob_. In this case, this fat point _is_ the Sierpinski space. This space can be thought of as being the abstract "point with open neighbourhood".

=--

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

Accordingly, the objects of the Sierpinski $(\infty,1)$-topos may be thought of as [[∞-groupoids]] (relative to $\mathbf{H}$) equipped with the notion of [[cohesive (∞,1)-topos|cohesion]] modeled on this: they are [[bundles]] $[P \to X]$ of [[∞-groupoids]] whose [[fibers]] are regarded as being [geometrically contractible](cohesive+%28infinity%2C1%29-topos+--+structures#Homotopy), in that

$$
  \Pi([P \to X]) \simeq X
$$

and so in particular

$$
  \Pi([Q \to *]) \simeq *
  \,.
$$

Hence these objects are [[discrete ∞-groupoids]] $X$, to each of whose points $ x : * \to X$ may be attached a contractible cohesive blob with inner structure given by the $\infty$-groupoid $P_x := P \times_X \{x\}$.

Accordingly, the _underlying_ $\infty$-groupoid of such a bundle $[P \to X]$ is the union

$$
  \Gamma([P \to X]) \simeq P
$$

of the discrete base space and the inner structure of the fibers.

The [[discrete object]] in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is the bundle

$$
  Disc(X) \simeq [X \stackrel{id}{\to} X]
$$

which is $X$ with "no cohesive blobs attached".

Finally the [[codiscrete object]] in the Sierpinski $(\infty,1)$-topos on an object $X \in \mathbf{H}$ is

$$
  coDisc(X) \simeq [X \to *]
  \,,
$$

the structure where all of $X$ is regarded as one single contractible cohesive ball.

The $(\Pi \dashv Disc)$-[[unit |adjunction unit]]

$$
  i :  id \to Disc \Pi
$$

on $[P \to X]$ is

$$
  \array{
    \mathllap{[}P &\to& X\mathrlap{]}
    \\
    \downarrow && \downarrow
    \\
    \mathllap{[}X &\to& X\mathrlap{]}
  }
  \,.
$$

The $(Disc  \dashv \Gamma)$-counit $ Disc \Gamma  \to id$ on $[P \to X]$ is

$$
  \array{
    \mathllap{[}P &\to& P\mathrlap{]}
    \\
    \downarrow && \downarrow
    \\
    \mathllap{[}P &\to& X\mathrlap{]}
  }
  \,.
$$

Hence the canonical [[natural transformation]]

$$
  \array{
    \Gamma && \to && \Pi
    \\
     & {}_{\mathllap{\Gamma(i)}}\searrow & & \nearrow_{\mathrlap{\simeq}}
     \\
     && \Gamma Disc \Pi
  }
$$

from "points to pieces" is on $[P \to X]$ simply the morphism $P \to X$ itself

$$
  (\Gamma \to \Pi)([P \to X]) = (P \to X)
  \,.
$$

Therefore

1. the [[full sub-(∞,1)-category]] on those objects in $\mathbf{H}^I$ for which "[pieces have points](cohesive%20%28infinity,1%29-topos#PiecesHavePoints)", hence those for which $\Gamma \to \Pi$ is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]], is the $(\infty,1)$-category of effective epimorphisms in the ambient $(\infty,1)$-topos, hence the $(\infty,1)$-category of [[groupoid object in an (∞,1)-category|groupoid objects]] in the ambient $(\infty,1)$-topos;

1. the full sub-$(\infty,1)$-category on the objects with "[one point per piece](cohesive+topos#ObjectsWithOnePointPerCohesivePiece)" is the ambient $(\infty,1)$-topos itself.

=--

### Cohesive structures

We unwind what some of the canonical  [[cohesive (infinity,1)-topos -- structures|structures in a cohesive (∞,1)-topos]] are when realized in the Sierpinski $(\infty,1)$-topos.

A group object $\mathbf{B}[\hat G \to G]$ in $\mathbf{H}^I$ is a morphism in $\mathbf{H}$ of the form $ = \mathbf{B}\hat G \to \mathbf{B}G$.

The corresponding flat coefficient object $\mathbf{\flat} \mathbf{B}[\hat G \to G] \to \mathbf{B}[\hat G \to G]$ is

$$
  \array{
    \mathbf{\hat G} &\to& \mathbf{B} \hat G
    \\
    \downarrow && \downarrow
    \\
    \mathbf{\hat G} &\to& \mathbf{G}
  }
  \,.
$$

Hence the corresponding de Rham coefficient object is

$$
  \mathbf{\flat}_{dR} \mathbf{B}[\hat G \to G]
  =
  [* \to \mathbf{B}A]
  \,,
$$

where $A \to \hat G \to G$ exhibits $\hat G$ has an $\infty$-group extension of $G$ by $A$ in $\mathbf{H}$.

The corresponding Maurer-Cartan form

$$
  [\hat G \to G] \to \mathbf{\flat}_{dR}\mathbf{B}[\hat G \to G]
$$

is

$$
  \array{
    \hat G &\to& G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}A
  }
$$

exhibiting the $A$-cocycle that classifies the extension $\hat G \to G$.

### Differential cohesion
 {#DifferentialCohesion}

The leftmost adjoint, $\Pi$, of the string of four adjoints exhibiting cohesion [above](#Cohesion) itself possesses a left adjoint, $X \mapsto [0 \to X]$. Taking this functor along with the leftmost three adjoints of this previous string yields a second quadruple adjunction which exhibits [[differential cohesion]].

For $\mathbf{H}$ any [[cohesive (∞,1)-topos]], we have the "Sierpinski $(\infty,1)$-topos relative to $\mathbf{H}$" given by the [[arrow category]] $\mathbf{H}^{\Delta[1]}$, whose [[geometric morphism]] to the [[base topos]] is the [[domain cofibration]]

$$
  \mathbf{H}^{\Delta[1]}
   \stackrel{\overset{const}{\hookleftarrow}}{\underset{dom}{\to}}
   \mathbf{H}
  \,.
$$

Conversely, we may think of $\mathbf{H}^{\Delta[1]}$ as being an "infinitesimal thickening" of $\mathbf{H}$, as formalized at _[[differential cohesion]]_, where we regard

$$
  (i_! \dashv i^* \dashv i_* \dashv i^!)
  :
  \mathbf{H}
    \stackrel{\overset{\top_!}{\hookrightarrow}}{\stackrel{\overset{\top^*}{\leftarrow}}{\stackrel{\overset{const}{\hookrightarrow}}{\underset{\bot^*}{\leftarrow}}}}
  \mathbf{H}^{\Delta[1]}
$$

as exhibiting $\mathbf{H}^{\Delta[1]}$ as an infinitesimal cohesive neighbourhood of $\mathbf{H}$ (here $(\bot, \top) : \Delta[0] \coprod \Delta[0] \to \Delta[1]$ denotes the endpoint inclusions, following the notation [here](http://ncatlab.org/nlab/show/cohesive+%28infinity,1%29-topos#CohesiveDiagramToposes)).

(See also the corresponding examples at _[[Q-category]]_.)

+-- {: .num_prop #iInclusion}
###### Observation

We have for all $X \in \mathbf{H}$ that

$$
  i_!(X) \simeq [\emptyset \to X]
  \,.
$$

=--

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

For all $[A \to B]$ in $\mathbf{H}^{\Delta[1]}$ we have

$$
  \mathbf{H}(X, i^*[A \to B])
  \simeq
  \mathbf{H}(X, cod(A \to B))
  \simeq
  \mathbf{H}(X, B)
  \,,
$$

which is indeed naturally equivalent to

$$
  \mathbf{H}^{\Delta[1]}([\emptyset \to X], [A \to B])
  \,.
$$

=--

Therefore an object of $\mathbf{H}^{\Delta[1]}$ given by a morphism $[P \to X]$ in $\mathbf{H}$ is regarded by the differential cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$ as being an infinitesimal thickening of $X$ by the fibers of $P$: where before we just had that the fibers of $P$ are "contractible cohesive thickenings" of the discrete object $X$, now $X$ is "discrete relative to $\mathbf{H}$" (hence not necessarily discrete in $\mathbf{H}$) and  the fibers are in addition regarded as being infinitesimal.

This is of course a very crude notion of infinitesimal extension. Notice for instance the following

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

With respect to the above differential cohesion $i : \mathbf{H} \hookrightarrow \mathbf{H}^{\Delta[1]}$, every morphism in $\mathbf{H}$ is a [[formally étale morphism]].

=--

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

By definition, given a morphism $f : X \to Y$, it is formally étale precisely if

$$
  \array{
    i_! X &\stackrel{i_! f}{\to}& i_! Y
    \\
    \downarrow && \downarrow
    \\
    i_* X &\stackrel{i_*}{\to}& i_* Y
  }
$$

is an [[(∞,1)-pullback]].

By prop. \ref{iInclusion} the above square diagram in $\mathbf{H}^{\Delta[1]}$ is

$$
  \array{
    [\emptyset \to X] &\to& [\emptyset \to Y]
    \\
    \downarrow && \downarrow
    \\
    [X \stackrel{id}{\to} X]
    &\to&
    [Y \stackrel{id}{\to} Y]
  }
  \,.
$$

Since $(\infty,1)$-pullbacks of $(\infty,1)$-presheaves are computed objectwise, this is an $(\infty,1)$-pullback in $\mathbf{H}^{\Delta[1]}$ precisely if the "back and front sides"

$$
  \array{
    \emptyset &\to& \emptyset
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& Y
  }
$$

and

$$
  \array{
    X &\stackrel{f}{\to}& Y
    \\
    \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}}
    \\
    X &\stackrel{f}{\to}& Y
  }
$$

are $(\infty,1)$-pullbacks in $\mathbf{H}$. This is clearly always the case.

The adjunction $i_{!} \circ i^{\ast} \vdash i_{\ast} \circ i^{\ast}$ forms the [[Aufhebung]] of $\emptyset \vdash \ast$ for the Sierpinski topos.

=--

### As a classifying topos
 {#AsAClassifyingTopos}

The Sierpinski topos is the [[classifying topos]] for [[subterminal objects]] in [[toposes]] (see e.g. [Johnstone 77, p. 117](#Johnstone77)).

The generic subterminal inclusion in the Sierpinski topos is the unique inclusion of $[\emptyset \to \ast]$ into $[\ast  \to \ast]$.

The (non-full) inclusion

$$
  \Delta^1 = \{\ast \to S^0\} \longrightarrow FinSet^{\ast/} \hookrightarrow \infty Grpd_{fin}^{\ast/}
$$

induces via restriction and right [[Kan extension]] an [[essential geometric morphism|essential]] [[geometric morphism]]

$$
  \mathbf{H}^{\Delta^1} \stackrel{\longleftarrow}{\hookrightarrow}
  \mathbf{H}[X_\ast]
$$

of the Sierpinski topos into the [[classifying topos]] for [[pointed objects]], $\mathbf{H}[X_\ast]$. (This becomes a [[geometric embedding]] if we refined the Sierpinski topos of bundles to the topos of bundles with sections).

The pointed object in $\mathbf{H}^{\Delta^1}$ which is classified by this [[geometric morphism]] is $[\ast \to S^0]$ (with $S^0 = \ast \coprod \ast$ the [[0-sphere]]) with its canonical map from $[\ast \to \ast]$.

In summary, the generic subterminal object and the generic pointed object fit into a sequence of the form

$$
  \array{
     \emptyset &\hookrightarrow& \ast &\longrightarrow& \ast
     \\
     \downarrow && \downarrow && \downarrow
     \\
     \ast &\longrightarrow& \ast &\longrightarrow& S^0
  }
$$

###Cohomology

According to the general idea of [[cohomology]], for $\mathbf{H}$ an [[(∞,1)-topos]], and $X, A \in \mathbf{H}$ two [[objects]], cohomology classes of $X$ with coefficients in $A$ are the connected components

$$
  H(X,A) := \pi_0 \mathbf{H}(X,A)
  \,.
$$

Cohomology in the Sierpinski $(\infty, 1)$-topos, $\mathbf{H}^{I}$, corresponds to [[relative cohomology]] in $\mathbf{H}$.

Indeed, let $i : Y \to X$ and $f : B \to A$ be two [[morphisms]] in $\mathbf{H}$. Then the **relative cohomology** of $X$ with coefficients in $A$ relative to these morphisms is the connected components of the $\infty$-groupoid of relative cocycles

$$
  H_{Y}^B(X,A) := \pi_0 \mathbf{H}^I(Y \stackrel{i}{\to} X\;,\; B \stackrel{f}{\to} A)
  \,.
$$

In terms of the [[codomain fibration]], $\mathbf{H}^I \to \mathbf{H}$, intrinsic cohomology can be considered as nonabelian [[twisted cohomology]] (see there).

## References

The Sierpinski topos is mentioned around remarks A2.1.12, B3.2.11 (p.83, p.387f) in

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. I_, Oxford UP 2002.

See also

* {#Johnstone77} [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp.

The [[homotopy type theory]] of the Sierpinski $(\infty,1)$-topos is discussed in

* {#Shulman} [[Mike Shulman]], _The univalence axiom for inverse diagrams_ ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253))

Cohesion of the Sierpinski $\infty$-topos is discussed in section 2.2.4 of

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

[[!redirects Sierpinski (∞,1)-topos]]
[[!redirects Sierpinski (infinity,1)-topos]]


sieve

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The notion of _sieve_ is a generalization of that of (right) _[[ideal in a monoid]]_ from [[monoids]] to [[categories]]:
a **sieve on** an [[object]] $X$ in a [[category]] $\mathcal{C}$ is a collection of [[morphisms]] with [[codomain]] $X$ that are closed under [[composition|precomposition]] with morphisms in $\mathcal{C}$.

Sometimes one says that a **sieve in** a category $\mathcal{C}$ is a [[full subcategory]] closed under precomposition with morphisms in $\mathcal{C}$ ([Lurie, def. 6.2.2.1](#Lurie)). If so, then a _sieve on_ an object $X$ is a _sieve in_ the [[slice category]] $\mathcal{C}_{/X}$. But a _sieve in_ a category is also naturally taken to be a collection of _sieves on_ various objects. On the other hand, most authors speak just about _sieves on_ an object anyway.

_Sieves on_  objects are an equivalent way to talk about [[subobjects]] of [[representable functors]] in a [[presheaf]] category in terms of the total sets of _elements_ of such a [[subfunctor]].

{#Attitude} In this sense "sieve" is a [[concept with an attitude]]:
One refers to a [[subobject]] of a [[representable functor]] as a *sieve* to indicate that one is interested in regarding it as a [[cover]] of a (sifted) [[coverage]]: the singled out subobjects then correspond to _covering sieves_.

Accordingly, the classical examples of _sieves on_ an object are [[Grothendieck topologies]], used to present the [[presheaves]] that behave like [[coverings]]. They are used to say which [[presheaves]] are actually [[sheaves]] with respect to a given [[coverage]] or [[Grothendieck topology]].

A choice of collections of morphism $d \to c$ into an object $c \in C$  for each $d \in C$ reminds one of the [[representable functor]] [[presheaf]]
$Y(c) := \hom_C(-, c): C^{op} \to Set$ which assigns to each $d \in C$ the set of _all_ morphisms from $d$ to $c$. Every choice of covers of $c$ is therefore for each $d \in C$ a subset of the value of this functor evaluated at $d$. This begins to look like an [[monomorphism|monic]] [[natural transformation]] into this functor. Indeed, it turns out that one can assume without restriction of generality that the assignment of covers of $c$ can always be extended to such a monic natural transformation, and hence one formalizes the notion of "collection of covers" of $c$ as a [[subobject]] $i: F \hookrightarrow \hom(-, c)$ of the functor represented by $c$: a _sieve_ at $c$.

A dual notion is a [[cosieve]].

## Definition

Let $C$ be a [[small category]].

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

A _sieve $S$ in $C$_ is a functor $S \hookrightarrow C$ that is both [[fully faithful]] and a [[discrete fibration]].

A _sieve on an object $c \in C$_ is a sieve in the [[slice category]] $C/c$.
=--

Spelling this out, we arrive at the traditional definition of a sieve on an object:

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

A _sieve_ $S$ on an object $c \in C$ is a subset $S \subset Ob(C/c)$ of the set of objects of the [[over category]] over $c$ which is _closed under precomposition_:  it has the property that whenever $(d \to c) \in S$ and $(e \to d) \in Mor(C)$ then the composition $(e \to d \to c)$ is in $S$.
=--

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

This is probably called a _sieve_ because it "sifts out" the 'special' maps into $c$ from the set of _all_ maps into $c$.  (Note that 'sieve' is the noun, while 'sift' is the verb.)

The French term for a sieve is _crible_.

=--

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

Sometimes the condition of a sieve being closed under the operation of precomposing with an arbitrary morphism $g: e \to d$ is called a "saturation condition". Given any collection of morphisms targeted at $c$, one can always close it up, or _saturate_ it, to obtain a sieve on $c$.

=--

## Properties

### Relation to fibrations

Just as a [[Grothendieck fibration]] is equivalent to a functor $C^{op} \to Cat$ and a [[discrete fibration]] is equivalent to a functor $C^{op} \to Set$, a sieve in $C$ is equivalent to a functor $C^{op} \to \Omega$, where $\Omega$ is the preordered category of [[truth values]]. This makes it obvious that a sieve picks out a subset of the objects in a way that is "downward closed". This is a form of [[negative thinking]], as $\Omega, Set, Cat$ are the categories of all [[(-1)-categories]], [[0-categories]] and [[1-categories]] respectively.

### Relation to subfunctors

There is a canonical way to create subfunctors from sieves and sieves from subfunctors.

A [[subfunctor]] is a [[subobject]] in a [[functor category]]. Here, specifically, one is interested in [[subobjects]] in a [[presheaf]] category of [[representable functors]]. It's these subfunctors of representable functors that are in bijection with sieves.

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

Given a sieve $S$ on $c$, the **subfunctor** $F_S \hookrightarrow Y(c)$ **defined by the sieve**  is the [[presheaf]]

* that assigns to each object $d \in C$ the set $F_S(d) = \{(d \to c) \in S\}$;

* that assigns to each morphism $(d \to d') \in C$ the function $F_S(d') \to F_S(d)$ induced on elements by precomposition with $d \to d'$.
=--

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

Given a subfunctor $F \hookrightarrow Y(c)$, the **sieve defined by the subfunctor** is given by

* $S_F \coloneqq \{g: d \to c \in Mor(C) | (Y(d) \stackrel{Y(g)}\to Y(c)) = (Y(d)  \to F \to Y(c)) \}$

or equivalently

* $S_F = \coprod_{d \in Obj(C)}F(d)$.
=--

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

These two definitions establish a bijection between sieves on $c$ and subobjects of $Y(c)$.

For every sieve $S$ we have

$$
  S_{F_S} = S
$$

and for every subfunctor we have

$$
  F_{S_F} = F
  \,.
$$
=--

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

* The construction of $S_F$ makes sense for every morphism of presheaves $F \to Y(c)$. The sieve is sensitive precisely to the [[image]] of this map,
$$
  S_{F} = S_{im (F \to Y(c))}
  \,.
$$

* In the presence of a [[Grothendieck topology]] a morphism $F \to Y(c)$ is sometimes called a [[local epimorphism]] if the sieve $S_F$ is a covering sieve. If $F \to Y(c)$ is actually a subfunctor, then it is called a [[dense monomorphism]].

* The [[pullback]] of a subfunctor $i: F_S \hookrightarrow Y(c) = \hom(-, c)$ along any morphism $\hom(-, g): \hom(-, d) \to \hom(-, c)$ is again a subfunctor $g^* F$ of $d$, hence sieves are closed under pulling back. Concretely,
$$g^* S = \bigcup_{e \in Ob(C)} \{f: e \to d: g f \in S\}
\,.$$

* A sieve $S_F$ on $c$, for $i: F \hookrightarrow \hom(-, c)$ a subfunctor, may be described as a function which assigns to each object $d$ a collection of morphisms $f: d \to c$ into $c$. Naturality of the inclusion $i$ means that whenever $f: d \to c$ belongs to the sieve and $g: e \to d$ is any morphism, then $f g: e \to c$ also belongs to the sieve.
=--

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

The subfunctor $F_S \hookrightarrow X$ corresponding to a sieve $S$ is the [[coimage]] of the morphism out of the disjoint union of all objects (regarded as representable presheaves) in the sieve:

$$
 (U = \coprod_{(U_\alpha \to X) \in S} U_\alpha)  \to X
$$

in that

$$
  (F_S \to X) = ((colim(U \times_X U \stackrel{\to}{\to} U)   \to X)
  \,.
$$
=--

If the sieve is generated by (is the saturation of) a collection of morphisms $\{U_\alpha \to U\}$ then the same statement remains true with $U$ being the coproduct over just these $U_\alpha$.

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

As described at [[limits and colimits by example]], the colimit of presheaves may be computed objectwise in $Set$. Doing so and using the [[Yoneda lemma]] tells us that for each object $V$ we have

$$
  \begin{aligned}
    colim(U \times_X U \stackrel{\to}{\to} U)(V)
    & \simeq
    colim(Hom(V,U \times_X U) \stackrel{\to}{\to} Hom(V,U))
    \\
    & \simeq
    colim( Hom(V,U) \times_{Hom(V,X)} Hom(V,U)  \stackrel{\to}{\to} Hom(V,U))
  \end{aligned}
$$

where all objects appearing (at least in the first lines) are implicitly regarded as presheaves under the [[Yoneda embedding]].

But this colimit now manifestly computes the set

$$
  \{\text{maps from }\,V\,\text{ to }\,X\,\text{ that factor through }\,U\}_\sim
$$

where the equivalence relation is

$$
  ((f \to U \to X) \sim (g \to U \to X))
  \Leftrightarrow
  (f\,\text{ and }\,g\,\text{ coincide as maps to }\,X)
  \,.
$$

So the set is just the set of maps from $V$ to $X$ that factor through one of the $U_\alpha$, which is precisely the set $F_S(V)$ assigned by the subfunctor corresponding to the sieve.
=--

### Sieves as covers

#### Introduction and overview

The following is a pedagogical step-by-step description of the crucial aspects of sieves as covers.

To start with the simplest example that already contains in it all the relevant aspects, consider a [[topological space]] $X$ with an open subset $V \subset X$ that is covered by two open subsets $U_1, U_2 \subset X$ in that the [[union]] $U_1 \cup U_2$ in $X$ coincides with $V$:

$$
  U_1 \cup U_2 = V
  \,.
$$

This is the [[coproduct]] in the [[category of open subsets]] of $X$, but that behaves very differently from the [[disjoint coproducts]] that we are used to from categories like [[Set]].  On the other hand, when one comes to a category of [[presheaves]] or [[sheaves]] (a [[Grothendieck topos]]), this [[topos]] *does* have disjoint coproducts.

Another way to think of this is obtained by first
forming the [[fiber product]] of $U_1$ with $U_2$ over $V$ in $Op(X)$,  which is the intersection $U_1 \cap U_2$ sitting in the [[pullback]] diagram

$$
  \array{
    U_1 \times_V U_2 &\to& U_1
    \\
    \downarrow && \downarrow
    \\
    U_2 &\to& V
  }
$$

in $Op(X)$.

The union $U_1 \cup U_2$ is again obtained from this by removing in the above diagram the bottom right corner and then forming the [[pushout]] over the resulting diagram: this is again $V$, i.e. the diagram

$$
  \array{
    U_1 \times_V U_2 &\to& U_1
    \\
    \downarrow && \downarrow
    \\
    U_2 &\to& U_1 \cup U_2 = V
  }
$$

is not only a [[pullback]] also a [[pushout]] diagram.

The important point about (covering) sieves is that they show up when the above situation is sent via the [[Yoneda embedding]] from $Op(X)$ to [[presheaf|presheaves]] on $Op(X)$. The crucial aspect here that gives rise to the peculiarities of sieves is that

* the [[Yoneda embedding]] preserves [[limits]], but not, in general, [[colimits]]

As a result, the above discussion goes through equivalently for the [[presheaf|presheaves]] [[representable functor|represented]] by our open subsets all the way up to the last pushout. In $Op(X)$ that last pushout reproduced the open subset $V$. In $PSh(X) = [Op(X)^{op}, Set]$ it instead reproduces the _sieve_ on $V$ generated by $U_1$ and $U_2$.

Let's go through this in detail. First of all notice that in $PSh(X)$ all limits and colimits do exist (see [[limits and colimits by example]] for more on that); in particular the coproduct

$$
  Y(U_1) \amalg Y(U_2)
$$

exists. Here $Y$ denotes the [[Yoneda embedding]] which we here indicate explicitly, even though often and elsewhere, notably elsewhere in this entry here, it is notationally suppressed.

For the following it is helpful to say explicitly what the presheaf $Y(U_1) \amalg Y(U_2)$ is like. Since, as described at [[limits and colimits by example]], colimits of presheaves are computed objectwise, we know that this presheaf evaluated on any open set $W \subset X$ yields the set

$$
  \begin{aligned}
   (Y(U_1) \amalg Y(U_2))(W)
   &=
    Y(U_1)(W) \amalg Y(U_2)(W)
   \\
   &=
   Hom(W,U_1) \amalg Hom(W, U_2)
  \end{aligned}
$$

where the coproducts on the right are just those in [[Set]] which are just ordinary disjoint unions of sets.

So this says that $Y(U_1) \amalg Y(U_2)$ is the presheaf that assigns to any open set $W$ the disjoint union of the collections of maps from $W$ to $U_1$ and those from $W$ to $U_2$ in $X$. (Since $Op(X)$ is a [[poset]] there is either none or one such map in each case, but it is helpful to speak generally of "sets of all maps", since that is the general intuition useful for presheaf categories. $Op(X)$ just happens to be a particularly simple example.)

Notice that in particular a given map $W \to V$ which factors both through $U_1 \to V$ as well as through $U_2 \to V$ will appear as two distinct elements in the set
$(Y(U_1) \amalg Y(U_2))(W)$. This we'll come back to in a minute.

But first consider the fiber product from before, now after having applied the [[Yoneda embedding]]. Since we know from general nonsense that this preserves fiber products, we know that the [[pullback]] presheaf $Y(U_1) \times_{Y(V)} Y(U_2)$ in

$$
  \array{
   Y(U_1) \times_{Y(V)} Y(U_2) &\to& Y(U_2)
   \\
   \downarrow && \downarrow
   \\
   Y(U_1)
   &\to&
   Y(V)
  }
$$

is the same as $Y(U_1 \times_V U_2) = Y(U_1 \cap U_2)$.

But this is also easily checked explicitly. We go through this because this kind of reasoning for computing limits and colimits of presheaves will be needed throughout here: since for any $W$ the covariant [[hom-functor]] $Psh(Y(W),-) : Psh \to PSh$ preserves limits (by the very definition of [[limit]]!) we have for every $W$ a pullback diagram of sets

$$
 \array{
  Hom(Y(W),Y(U_1) \times_{Y(V)} Y(U_2))
   &\to& Hom(Y(W),Y(U_2))
  \\
  \downarrow && \downarrow
  \\
  Hom(Y(W),Y(U_1))
  &\to&
  Hom(Y(W),Y(V))
 }
$$

Again by the [[Yoneda lemma]] this is simply

$$
 \array{
  (Y(U_1) \times_{Y(V)} Y(U_2) )(W)
   &\to&
   Hom(W,U_2)
  \\
  \downarrow && \downarrow
  \\
  Hom(W,U_1)
  &\to&
  Hom(W,V)
 }
  \,.
$$

This being a pullback diagram now says in words:

> The set $(Y(U_1) \times_{Y(V)} Y(U_2) )(W)$ is the set of those pairs of maps $W \to U_1$ and $W \to U_2$ that coincide as maps $W \to U_1 \to V$ and $W \to U_2 \to V$ to $V$.

Clearly, this set is the same as the set of maps into the intersection $U_1 \cap U_2$, so indeed

$$
  Y(U_1) \times_{Y(V)} Y(U_2) = Y(U_1 \times_V U_2)
  = Y(U_1 \cap U_2)
  \,.
$$

So far so long-winded. Now let's see what happens when we now form the pushout over

$$
  \array{
    Y(U_1 \times_V U_2) &\to& Y(U_2)
    \\
    \downarrow
    \\
    Y(U_1)
  }
$$

that will go, for a moment, by its canonical but lengthy name
$
  Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2)
$

$$
  \array{
    Y(U_1 \times_V U_2) &\to& Y(U_2)
    \\
    \downarrow && \downarrow
    \\
    Y(U_1)
    &\to&
    Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2)
  }
  \,.
$$

Again, we can figure out what this presheaf is by computing objectwise what it does to any open subset $W$: since colimits of presheaves are computed objectwise, the diagram

$$
  \array{
    Hom(W, U_1 \times_V U_2)
    &\to&
    Hom(W, U_2)
    \\
    \downarrow && \downarrow
    \\
    Hom(W,U_1)
    &\to&
    (Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W)
  }
$$

must be a colimit in [[Set]]. Again, this is easily read
out in words:

> The set $(Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W)$ is the quotient of the disjoint union of the collection of maps from $W$ into $U_1$ and those from $W$ into $U_2$, by the equivalence relation which identifies two such maps $W \to U_1$  and $W \to U_2$ if they both factor through a map $W \to U_1 \times_X U_2$, i.e. if they both land in the intersection $U_1 \cap U_2$ and coincide there.

But this just means that contrary to the plain coproduct $Y(U_1) \amalg Y(U_2)$, two maps $W \to U_1$ and $W \to U_2$ that coincide as maps $W \to X$ are no longer regarded as different elements of our set given by the pushout presheaf, but are regarded as being the same.

So this means we find that

$$
  (Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2))(W)
  =
  \{
    \text{ maps } \,W \to V \,\text{ that factor through either }\, U_1
    \,\text{ or }\, U_2
  \}
  \,.
$$

But this is by definition the assignment of the subfunctor corresponding to the sieve on $V$ generated by $U_1 \to V$ and $U_2 \to V$.

So we find that

$$
  F_{sieve(U_1,U_2)}
  =
  Y(U_1) \coprod_{Y(U_1 \times_V U_2)} Y(U_2)
  \,.
$$

Given that we made it to this point, we should go one small step further that will be very useful.

In the present simple example we worked with a cover given by just two objects $U_1$ and $U_2$. Of course in general the cover will consist of more than just two objects. Then the above kind of notation becomes a bit cumbersome.  But there is a simple reformulation that makes everything look nice again.

Namely, let's come back to the observation that the coproduct $Y(U_1) \amalg Y(U_2)$ does exist. Let's just call this presheaf $\mathbf{U}$ (not in general a [[representable functor|representable]]!).

Then it is easy to see by the same kind of objectwise reasoning that the colimiting presheaf that we are after is equivalently the colimit over the pair of [[parallel morphisms]]

$$
  \mathbf{U} \times_{Y(V)} \mathbf{U}
  \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}}
  \mathbf{U}
$$

in that

$$
  F_{sieve(U_1,U_2)} \simeq
  colim (
   \mathbf{U} \times_{Y(V)} \mathbf{U}
   \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}}
   \mathbf{U}
  )
  \,.
$$

This description now has an evident direct generalization to the case where instead of just $U_1 \to V$ and $U_2 \to V$ we have an arbitrary collection $\{U_i \to V\}$ of open sets $U$ covering $V$. One finds again with

$$
  \mathbf{U} := \coprod_i Y(U(i))
$$

that

$$
  F_{sieve(\{U_i\})} \simeq
  colim (
   \mathbf{U} \times_{Y(V)} \mathbf{U}
   \stackrel{\stackrel{p_2}{\to}}{\stackrel{p_1}{\to}}
   \mathbf{U}
  )
$$

is the presheaf that to every $W$ assigns the set of all maps $W \to V$ that factor through any one of the $U_i$.

It is in this way that sieves and their associated subfunctors encapsulate the notion of _cover_ of an object $V$: they tell us which of all the maps into $V$ do factor through the cover.

And, to end this pedagogical piece with an outlook to indicate the gain in understanding this achieves:

once we start forming $\mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\to} \mathbf{U}$ there is no stopping. We can keep forming higher and higher such fiber products

$$
 \cdots
\mathbf{U} \times_{Y(V)} \mathbf{U}
 \times_{Y(V)} \mathbf{U}
 \times_{Y(V)} \mathbf{U}
\stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}}
\mathbf{U} \times_{Y(V)} \mathbf{U}
 \times_{Y(V)} \mathbf{U}
\stackrel{\to}{\stackrel{\to}{\to}}
\mathbf{U} \times_{Y(V)} \mathbf{U} \stackrel{\to}{\to} \mathbf{U}
  \,.
$$

When one passes from just presheaves to [[(∞,1)-presheaves]], then [[cover]]ing presheaves will be given by the right kind of colimit over these [[simplicial object|simplicial]] diagrams (namely the [[homotopy limit|homotopy colimit]]). More on that is at [[descent]].

#### The sheaf condition from morphisms out of a sieve

We now show that the subfunctor $F_S$ associated with a sieve $S$ coming from a cover $\{U_i \to X\}$ is the _right_ kind of morphism to require a [[sheaf]] to be a [[local object]] for, by demonstrating that using it the usual [[sheaf|sheaf condition]] on a presheaf with respect to the cover $\coprod_i U_i$ is reproduced:

From the above detailed discussion, recall that $F_{sieve(\{U_i\})}$ is precisely the [[coequalizer]] of the obvious pair of morphisms

$$
  \coprod_{i, j} \hom(-, U_i \cap U_j)
    \stackrel{\to}{\to}
  \coprod_i hom(-, U_i)
$$

with $hom(-, U_i) := Y(U_i)$ denoting the presheaf [[representable functor|represented]] under the [[Yoneda embedding]] by $U_i$, as usual.

Here the domain of this parallel pair is the [[pullback]] of the evident map
$\coprod_i hom(-, U_i) \to hom(-,X)$
along itself, and the two parallel arrows are the projection maps out of this pullback:

$$
  \array{
     \coprod_{i,j} hom(-,U_i \cap U_j)
     &\to&
     \coprod_j hom(-, U_j)
     \\
     \downarrow && \downarrow
     \\
     \coprod_i hom(-, U_i)
     &\to&
     hom(-,X)
  }
$$

Thus for $G$ any [[presheaf]],
maps $F \to G$ are precisely the same as maps
$\coprod_i \hom(-, U_i) \to G$
which coequalize the parallel pair. Applying $Hom_{Set^{C^{op}}}(-,G)$ to the colimit diagram

$$
  \coprod_{i, j} \hom(-, U_i \cap U_j)
    \stackrel{\to}{\to}
  \coprod_i hom(-, U_i)
  \to
  F
$$

yields the limit diagram

$$
  Hom(F, G)
  \to
  Hom(\coprod_i hom(-, U_i), G)
  \stackrel{\to}{\to}
  Hom(\coprod_{i, j} \hom(-, U_i \cap U_j), G)
$$

which using [[Yoneda lemma|Yoneda]] is the equalizer diagram

$$
  Hom(F,G)
  \to
 \prod_i G(U_i) \stackrel{\to}{\to} \prod_{i, j} G(U_i \cap U_j)
$$

and hence identifies $Hom(F,G)$ indeed as the set of [[descent]] data for the [[sheaf]] condition on $G$.

## Examples

* For $X$ a [[topological space]] let $Op(X)$ be the [[category of open subsets]] of $X$ and consider presheaves $PSh(X) := [Op(X)^{op}, Set]$ on $X$. For any open subset $c = V \in Op(X)$ let $\{d_i\} = \{U_i\}$ be a cover of $V$ by open subsets $U_i$ in the ordinary sense (i.e. each $U_i$ is an open subset of $V$ and their joint union is $V$, $\bigcup_i U_i = V$), then  $\pi : (\coprod_i Y(U_i)) \stackrel{\coprod_i U_i \hookrightarrow_i X}{\to} Y(V)$ (with $Y$ the [[Yoneda embedding]]) is a [[local epimorphism]] of presheaves on $V$ and its [[image]] -- or equivalently its [[coimage]] -- is the subfunctor $(F := \bigcup_i Y(U_i)) \hookrightarrow Y(V)$ that sends each $W \in Op(X)$ to the set of maps $W \to V$ that factor through one of the $U_i$. The collection of all such maps for all choice of $W$ is the corresponding _covering sieve_ $\{ f : W \to V  \in Mor(S) \;|\; f = W \to U_i \to V  \}$.

* The situation for more general [[sites]] $S$ other than $Op(X)$ is literally the same as above, with $U_i, W, V$ etc. objects of $S$.

## References

A standard textbook account on sieves in [[category theory]] is in

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

In the context of [[(∞,1)-categories]] sieves are discussed around def. 6.2.2.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

[[!redirects sieves]]


sifted (infinity,1)-category

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

#Contents#
* table of contents
{:toc}

## Definition

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

An [[(∞,1)-category]] is **sifted** if a [[quasi-category]] $K \in sSet$ modelling it has the property that

1. it is not empty, $K \neq \emptyset$

1. The diagonal $K \to K \times K$ (in [[sSet]]) models a [[cofinal (∞,1)-functor]].

=--

## Properties

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

Let $C$ be an [[(∞,1)-category]] such that [[products]] preserve [[sifted (∞,1)-colimits]] (for instance an [[(∞,1)-topos]], see [[universal colimits]]).

Then [[sifted (∞,1)-colimits]] preserve [[finite products]].

=--

This is ([Lurie, lemma 5.5.8.11](#Lurie)).

## Examples

### General

\begin{proposition}
The [[opposite category]] $\Delta^{op}$ of the [[simplex category]] is a sifted $(\infty,1)$-category.
\end{proposition}
([Lurie HTT, prop. 5.5.8.4](#LurieHTT)).

\begin{proposition}

Every [[filtered (∞,1)-category]] is sifted.

\end{proposition}
([Lurie HTT, prop. 5.3.1.20](#LurieHTT)).

### In categories of commutative monoids

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

In a category of commutative monoids in a symmetric monoidal $(\infty,1)$-category, sifted colimits are computed as sifted colimits in the underlying $(\infty,1)$-category.

=--

See [[commutative monoid in a symmetric monoidal (∞,1)-category]] for details.

## Related concepts

* [[sifted category]], **sifted $(\infty,1)$-category**

* [[filtered category]], [[filtered (∞,1)-category]]

## References

* {#LurieHTT} [[Jacob Lurie]], Section 5.5.8 of: _[[Higher Topos Theory]]_

[[!redirects sifted (infinity,1)-categories]]
[[!redirects sifted (∞,1)-category]]
[[!redirects sifted (∞,1)-categories]]

[[!redirects cosifted (infinity,1)-category]]
[[!redirects cosifted (infinity,1)-categories]]
[[!redirects cosifted (∞,1)-category]]
[[!redirects cosifted (∞,1)-categories]]


sifted (infinity,1)-colimit

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

#Contents#
* table of contents
{:toc}

## Definition

A _sifted $(\infty,1)$-colimit_ is an [[(∞,1)-colimit]] over a [[diagram]] that is a [[sifted (∞,1)-category]].

## Properties

\begin{prop}
Let $C$ be an [[(∞,1)-category]] such that [[products]] preserve [[sifted (∞,1)-colimits]] (for instance an [[(∞,1)-topos]], see [[universal colimits]]).

Then [[sifted (∞,1)-colimits]] preserve [[finite products|finite]] [[homotopy products]].
\end{prop}

([Lurie HTT, lemma 5.5.8.11](#LurieHTT)).

## Examples

### Simplicial $\infty$-colimits

\begin{prop}\label{SimplicialInfinityColimitsAreSifted}
**(simplicial $\infty$-colimits are sifted)**
\linebreak
  The $(\infty,1)$-colimits of [[simplicial object in an (infinity,1)-category|simplicial objects in an $(\infty,1)$-category]] are sifted.
\end{prop}
This is because the [[opposite category|opposite]] of the [[simplex category]] is a [[sifted (∞,1)-category]] ([Lurie HTT, Prop. 5.5.8.4](#LurieHTT)).

\begin{remark}
Simplicial $\infty$-colimits preserve even [[homotopy fiber products]], under mild conditions: see at *[[geometric realization of simplicial topological spaces]]* the section *[Preservation of homotopy limits](https://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces#PreservationOfHomotopyLimits)*.
\end{remark}

## Related concepts

* [[sifted colimit]], **sifted $(\infty,1)$-colimit**

* [[filtered colimit]], [[filtered (∞,1)-colimit]]

## References

* {#LurieHTT} [[Jacob Lurie]], *[[Higher Topos Theory]]*

[[!redirects sifted (∞,1)-colimit]]
[[!redirects sifted (∞,1)-colimits]]

[[!redirects sifted (infinity,1)-colimits]]

[[!redirects sifted infinity-colimit]]
[[!redirects sifted infinity-colimits]]