concordance

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

## Idea

A concordance between [[cocycle]]s in [[cohomology]] is a relation similar to but different from a plain [[coboundary]], it is a "coboundary after [[geometric realization]]".

A concordance is a left [[homotopy]] in an [[(∞,1)-topos]] with respect to a _topological_ [[interval object]], not with respect to the _[[interval category|categorical interval]]_ .

For instance for $S = Diff$ the [[site]] of smooth manifolds, there is

* the "topological interval" $I \in \mathbf{H}_{diff}$ which is the [[smooth ∞-stack]] on $Diff$ [[representable functor|represented by]] the manifold $I = [0,1]$;

* the "categorical interval" $Ex^\infty \Delta^1 \in \mathbf{H}_{Diff}$ is the [[smooth ∞-stack]] that is constant on the free groupoid on a single morphism.

## Definition

For $\mathbf{H}$ and [[(∞,1)-topos]] with a fixed notion of topological [[interval object]] $I$, for $A \in \mathbf{A}$ any coefficient object and $X \in \mathbf{H}$ any other object, a **concordance** between two objects

$$
  c,d \in \mathbf{H}(X,A)
$$

(two cocycles in $A$-[[cohomology]] on $X$)

is an object $\eta \in A(X \times I)$ such that

$$
\begin{matrix}
X&&\\
\downarrow&\searrow^{c}&\\
X \times I&\stackrel{\eta}{\to}& A\\
\uparrow& \nearrow_{d}&\\
X&&
\end{matrix}
\,.
$$

## Examples

### For topological principal bundles
 {#ForTopologicalPrincipalBundles}

\begin{proposition}\label{ConcordantTopologicalPrincipalBundlesAreIsomorphic}
**(concordant topological principal bundles are isomorphic)**
\linebreak
  With $kTopSp$ denoting the category of [[compactly generated weakly Hausdorff spaces]],
  for $X \,\in\, kTopSp$ a [[k-topological space]]
  and $G \,\in\, Grp(kTopSp)$ a $k$-[[topological group]],
  consider a [[concordance]] between a [[pair]] of
  $G$-[[principal bundles]] over $X$,
\begin{tikzcd}
  P_0
  \ar[r]
  \ar[d, "{p_0}"]
  \ar[dr,phantom,"\mbox{\tiny(pb)}"]
  &
  P
  \ar[d, "{p}"]
  \ar[dr,phantom,"\mbox{\tiny(pb)}"]
  &
  P_1
  \ar[l]
  \ar[d, "{p_1}"]
  \\
  X \times \{0\}
  \ar[r, hook]
  &
  X \times [0,1]
  &
  X \times \{1\}
  \ar[l, hook']
\end{tikzcd}

If

* {#FirstConditionForConcordanceOfTopologicalPrincipalBundlesToInduceIso} $X$ is a [[CW-complex]], or, more generally, a [[retract]] of a [[cell complex]]

or

* {#SecondConditionForConcordanceOfTopologicalPrincipalBundlesToInduceIso} $P$ is [[numerable bundle|numerable]], e.g. in that $X$ is a [[paracompact topological space]]

(e.g. if $X$ admits the structure of a [[smooth manifold]])

then there exists already an [[isomorphism]] of [[principal bundles]]
\begin{tikzcd}
  P_0
  \ar[dr, "{p_0}"{below, xshift=-2pt}]
  \ar[rr, "{\sim}"]
  &&
  P_1
  \ar[dl, "{p_1}"]
  \\
  &
  X
\end{tikzcd}
\end{proposition}
(e.g. [Roberts & Stevenson 2016, Cor. 15](#RobertsStevenson16))
\begin{proof}

{#PrincipalBundleIsosAsSections}
Observe that isomorphisms $f \,\colon\, P \xrightarrow{\;} P'$ between principal bundles over $X$ are equivalently [[global sections]] of the fiber bundle $(P \times_X P')/G$:
\begin{tikzcd}[column sep={between origins, 50pt}]
    &[-10pt]
    &[-10pt]
    &&[+30pt]
    P \times G
    \ar[d, shift right=3pt, "\mathrm{pr}_1"{swap}]
    \ar[d, shift left=3pt, "\rho"]
    \ar[rr]
    &&
    \big(
      P \times_X P'
    \big) \times G
    \ar[d, shift right=3pt, "\mathrm{pr}_1"{swap}]
    \ar[d, shift left=3pt, "\rho"]
    \\
    &&&&
    P
    \ar[d]
    \ar[rr, "{ (\mathrm{id},f) }"{description}]
    &&
    P \times_X P'
    \ar[d]
    \\
    P
      \ar[out=180-66, in=66, looseness=3.5, "\scalebox{.77}{$\,\mathclap{
        G
      }\,$}"{description},shift right=1]
    \ar[rr, "f"]
    \ar[dr]
    &
    &
    P'
      \ar[out=180-66, in=66, looseness=3.5, "\scalebox{.77}{$\,\mathclap{
        G
      }\,$}"{description},shift right=1]
    \ar[dl]
    &\leftrightarrow&
    P/G
    \ar[rr, dashed]
    \ar[dr, -, shift left=1pt]
    \ar[dr, -, shift right=1pt]
    &&
    \big(
      P
        \times_{X}
      P'
    \big) / G
    \ar[dl]
    \\
    &
    X
    &
    &&
    &
    X
\end{tikzcd}

Here, from left to right, the dashed section follows by the [[universal property]] of the [[quotient space]] $X = P/G$. From right to left, the top morphism follows by [[pullback]] along the dashed section, using that

1. the bundle projections are [[effective epimorphisms]] by [[locally trivial bundle|local triviality]],

1. their [[kernel pairs]] are as shown, by [[principal bundle|principality]],

1. [[compactly generated topological spaces]] form a [[regular category]] (by [this Prop.](compactly+generated+topological+space#kHausIsRegular)),

1. in a regular category pullback preserves effective epimorphisms ([this Prop.](regular+epimorphism#InRegularCategoryRegularEpisAreStableUnderPullback)) and, of course, their kernel pairs.

In particular, for every $P$ the [[identity morphism]] on it corresponds to the canonical section of $(P \times_X P)/G$.

{#TheLiftingDiagram} In the given situation, this means that we have a canonical local section $\sigma_0$ making the following solid diagram [[commuting square|commute]], exhibiting that the restriction of the bundle $P_0 \times [0,1]$ to $\{0\} \subset [0,1]$ is isomorphic to $P_0$, by construction:

\begin{tikzcd}
  X \times \{0\}
  \ar[rr, "\sigma_0"]
  \ar[dd, "\in \mathrm{Cof} \cap \mathrm{W}"]
  &&
  \big(
    P \underset{X \times [0,1]}{\times} (P_0 \times [0,1])
  \big)/G
  \ar[dd, "\in \mathrm{Fib}"]
  \\
  \\
  X \times [0,1]
  \ar[rr,-, shift right=1pt]
  \ar[rr,-, shift left=1pt]
  \ar[uurr, dashed, "\exists"]
  &&
  X \times [0,1]
\end{tikzcd}

Now

* assuming the [first condition](#FirstConditionForConcordanceOfTopologicalPrincipalBundlesToInduceIso):

  1. the right vertical [[map]] is a [[Serre fibration]], as all [[locally trivial bundle|locally trivial]] [[fiber bundles]] are [[Serre fibrations]] (by [this Prop.](fiber+bundle#RelationToFibrations));

  1. the left vertical [[continuous function|map]] is a [[Serre-Quillen model structure on topological spaces|Serre-Quillen]]-[[acyclic cofibration]] -- since (see [this Prop.](classical+model+structure+on+topological+spaces#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen)) it is the [[product]]  $id_X \times (D^0 \hookrightarrow D^0 \times [0,1] )$ of the [[cofibrant object]] $X$ with a generating [[acyclic cofibration]] (see [this Def.](classical+model+structure+on+topological+spaces#TopologicalGeneratingAcyclicCofibrations)) --, hence is a [[Serre-Quillen model structure on topological spaces|Serre-Quillen]]-[[acyclic cofibration]] and as such has the [[left lifting property]] against the right map;

or

* assuming the [second condition](#SecondConditionForConcordanceOfTopologicalPrincipalBundlesToInduceIso):

  1. the right vertical map is a [[Hurewicz fibration]], by [this Prop.](Hurewicz+fibration#NumerableFiberBundlesAreSerreFibrations),

  1. hence it has the [[right lifting property]] against the left map, by definition of [[Hurewicz fibrations]].

In either case, this implies that a [[lift]] exists, as shown by the dashed arrow above.

The resulting [[commuting diagram|commutativity]] of the bottom right triangle says that this lift is a [[global section]] which hence exhibits an [[isomorphism]] of principal bundles (over $X \times [0,1]$) of this form:

$$
  P
  \;\;
  \simeq_X
  \;\;
  P_0 \times [0,1]
  \,.
$$

The [[restriction]] of this isomorphism to $\{1\} \subset [0,1]$ is hence an isomorphism of the form $P_1 \,\simeq_X\, P_0$, as required.
\end{proof}

### For topological vector bundles
 {#ForTopologicalVectorBundles}

For [[topological vector bundles]] over [[paracompact Hausdorff spaces]], concordance classes coincide with plain [[isomorphism classes]]:

+-- {: .num_prop #ConcondanceOfTopologicalVectorBundles}
###### Proposition
**([[concordance]] of [[topological vector bundles]])**

Let $X$ be a [[paracompact Hausdorff space]]. If $E \to X \times [0,1]$ is a [[topological vector bundle]] over the [[product space]] of $X$ with the [[closed interval]] (hence a _[[concordance]]_ of topological vector bundles on $X$), then the two endpoint-restrictions

$$
  E|_{X \times \{0\}}
  \phantom{AA}
  \text{and}
  \phantom{AA}
  E|_{X \times \{1\}}
$$

are [[isomorphism|isomorphic]] [[topological vector bundles]] over $X$.

=--

For **proof** see at _[[topological vector bundle]]_ [this Prop.](topological+vector+bundle#ConcondanceOfTopologicalVectorBundles).

### More examples

* For $A = VectrBund(-)$ the difference between concordance of [[vectorial bundles]] and isomorphism of vectorial bundles plays a crucial rule in the construction of [[K-theory]] from this model.

* The notions of coboundary and concordance exist in every [[cohesive (∞,1)-topos]].

## References

Discussion of concordance of topological [[principal bundles]] (in fact for [[simplicial principal bundles]] [[parameterized homotopy theory|parameterized]] over some base space):

* {#RobertsStevenson16} [[David Roberts]], [[Danny Stevenson]], Cor. 15 in: _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](https://arxiv.org/abs/1203.2460), [nyjm:22-19](http://nyjm.albany.edu/j/2016/22-19.html))

Discussion of concordance in terms of the [[shape modality]] in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]] (see at *[[shape via cohesive path ∞-groupoid]]* for more):

* {#Pavlov14} [[Dmitri Pavlov]], _Structured Brown representability via concordance_, 2014 ([pdf](https://dmitripavlov.org/concordance.pdf), [[Pavlov_Concordance.pdf:file]])

* {#BEBP} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

[[!redirects concordances]]

[[!redirects concordance class]]
[[!redirects concordance classes]]


concrete (2,1)-category

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###Context###
#### 2-Category theory
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[[!include 2-category theory - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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#### Discrete and concrete objects
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[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of *[[concrete category]]* from [[category theory]] to [[(2,1)-category theory]].

## Definition

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

A __concrete (2,1)-category__ is a [[(2,1)-category]] $C$ equipped with a [[k-surjective functor|3-surjective functor]]

  $$
    U \colon C \to Grpd
  $$

to the [[large]] [[(2,1)-category]] [[Grpd]] of [[groupoids]]. We say a (2,1)-category $C$ is _concretizable_ if and only if it admits a 3-surjective functor $U \colon C \to Grpd$.
=--

## Examples

* The (2,1)-category [[Grpd|$Grpd$]] of [[groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $MonGrpd$ of [[monoidal groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $BraidedMonGrpd$ of [[braided monoidal groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $SymmetricMonGrpd$ of [[symmetric monoidal groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $RingGrpd$ of [[ring groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $SymmetricRingGrpd$ of [[symmetric ring groupoids]] is a concrete (2,1)-category.

* The (2,1)-category $2Grp$ of [[2-groups]] is a concrete (2,1)-category.

* The (2,1)-category $Smooth2Grp$ of [[smooth 2-groups]] is a concrete (2,1)-category.

* The (2,1)-category $Braided2Grp$ of [[braided 2-groups]] is a concrete (2,1)-category.

* The (2,1)-category $Symmetric2Grp$ of [[symmetric 2-groups]] is a concrete (2,1)-category.

* The (2,1)-category [[Cat|$Cat$]] of [[weak categories]] (with [[functors]] between these and [[natural isomorphisms]] between those) is a concrete (2,1)-category.

## Related concepts

* [[(2,1)-category]]
* [[concrete object]]
* [[concrete category]]
* [[concrete (n,1)-category]]

[[!redirects concrete (2,1)-categories]]


concrete (infinity,1)-category

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###Context###
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of *[[concrete category]]* from [[category theory]] to [[(infinity,1)-category theory]].

## Definition

\begin{definition}
A __concrete (infinity,1)-category__ is a [[(infinity,1)-category]] $C$ equipped with an [[k-surjective functor|essentially $\infty$-surjective]] [[(infinity,1)-functor]]

  $$
    U \colon C \to \infty Grpd
  $$

to the [[(infinity,1)-category]] $\infty Grpd$ of [[infinity-groupoids]]. However every (infinity,1)-functor functor is essentially $\infty$-surjective, so the surjectivity requirement is redundant. We say an (infinity,1)-category $C$ is _concretizable_ if and only if it admits a [[(infinity,1)-functor]] $U \colon C \to \infty Grpd$.
\end{definition}

## Related concepts

* [[(infinity,1)-category]]
* [[concrete object]]
* [[concrete category]]
* [[concrete (n,1)-category]]

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


concrete (infinity,1)-sheaf

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###Context###
#### Discrete and concrete objects
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[[!include discrete and concrete objects - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Inside a [[local (∞,1)-topos]] $\mathbf{H}$ there are objects that may be thought of as [[∞-groupoid]]s equipped with [[stuff, structure, property|extra structure]] ("cohesive structure" if $\mathbf{H}$ is even a [[cohesive (∞,1)-topos]]). These are the _concrete objects_ in $\mathbf{H}$.

## Definition

Let $\mathbf{H}$ be a [[local (∞,1)-topos]].

$$
  \mathbf{H}
  \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}}
  \infty Grpd
  \,.
$$

Since $Codisc$ is by definition a [[full and faithful (∞,1)-functor]] this means that

$$
  \infty Grpd
  \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}}
  \mathbf{H}
$$

is a [[geometric embedding]]. By the discussion at [[reflective sub-(∞,1)-category]] this means that $\Gamma$ is the [[localization of an (∞,1)-category]] at a class $S \in Mor \mathbf{H}$ of [[morphism]]s.  It factors therefore canonically through the [[(∞,1)-quasitopos]] of $S$-separated $(\infty,1)$-sheaves

$$
  \infty Grpd
   \stackrel{\overset{\Gamma}{\leftarrow}}{\hookrightarrow}
  Conc(\mathbf{H})
   \stackrel{\overset{concretize}{\leftarrow}}{\hookrightarrow}
   \mathbf{H}
  \,.
$$

We call $Conc(\mathbf{H})$ the [[(∞,1)-quasitopos]] of **concrete objects** of the local $(\infty,1)$-topos $\mathbf{H}$.

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

We say an object $X$ is **$n$-concrete** if the canonical morphism
$X \to coDisc \Gamma X$ is [[n-truncated|(n-1)-truncated]].

If a [[0-truncated]] object $X$ is $0$-concrete, we call it just **concrete**.

=--

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

For $C$ an [[∞-cohesive site]], a 0-truncated object in the
[[(∞,1)-topos]] over $C$ is concrete precisely if it is
a [[concrete sheaf]] in the traditional sense.

=--

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

For $X \in \mathbf{H}$ and $n \in \mathbb{N}$, the
**$(n+1)$-concretification** of $X$ is the morphism

$$
  X \to conc_{n+1} X
$$

that is the left factor in the decomposition with respect to the
[[n-connected/n-truncated factorization system]] of the
$(\Gamma \dashv coDisc)$-[[unit of an adjunction|unit]]

$$
  \array{
    && conc_{n+1} X
    \\
    & \nearrow && \searrow
    \\
    X &&\to&& coDisc \Gamma X
  }
  \,.
$$

=--

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

By that very [[n-connected/n-truncated factorization system]]
we have that $conc_{n+1} X$ is an ${n+1}$-concrete object.

=--

## Examples

* [[concrete smooth ∞-groupoid]]

## Related concepts

* [[concrete sheaf]]

* **concrete $(\infty,1)$-sheaf**

## References

This entry goes back to some observations by [[David Carchedi]].

[[!redirects concrete (∞,1)-sheaves]]
[[!redirects concrete (∞,1)-sheaf]]


concrete (n,1)-category

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of *[[concrete category]]* from [[category theory]] to (n,1)-category theory.

## Definition

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

A __concrete (n,1)-category__ is a [[(n,1)-category]] $C$ equipped with a [[k-surjective functor|(n+1)-surjective functor]]

  $$
    U \colon C \to nGrpd
  $$

to the [[large]] [[(n,1)-category]] nGrpd of [[n-groupoids]]. We say a (n,1)-category $C$ is _concretizable_ if and only if it admits a $(n+1)$-surjective functor $U \colon C \to Grpd$.
=--

## Related concepts

* [[(n,1)-category]]
* [[concrete object]]
* [[concrete category]]
* [[concrete (2,1)-category]]
* [[concrete (infinity,1)-category]]

[[!redirects concrete (n,1)-categories]]


concrete category

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _concrete category_ is a [[category]] that looks like a category of "[[sets]] equipped with [[extra structure|extra]] [[structure]]", hence like a category of [[structured sets]].

## Definition

### With a family of collection of elements

+-- {: .num_defn}
###### Definition
Given a category $C$ with a type of objects $Ob(C)$ and for every object $a:Ob(C)$ and $b:Ob(C)$ a set of morphisms $Mor_C(a, b)$, $C$ is a __concrete category__ if for every object $a:Ob(C)$ there is a set of elements $U(a)$ and for every object $a:Ob(C)$ and $b:Ob(C)$, there is an [[injection]] $i_{a,b} \colon Mor_C(a,b) \to (U(a) \to U(b))$.
=--

### With one collection of elements

+-- {: .num_defn}
###### Definition
Given a category $C$ with a type of objects $Ob(C)$ and a set of morphisms $Mor(C)$ with source and target functions $s:Mor(C) \to Ob(C)$ and $t:Mor(C) \to Ob(C)$, $C$ is a __concrete category__ if there is a set of elements $U(C)$ with a function $o:U(C) \to Ob(C)$ with an [[injection]] $i:Mor(C) \to Func(Set)$ (and functions $s_{Set}:Func(Set) \to U(C)$ and $t_{Set}:Func(Set) \to U(C)$) such that for every term $f:Mor(C)$, $s(f) = o(s_{Set}(i(f)))$ and $t(f) = o(t_{Set}(i(f)))$.
=--

### With a functor into Set

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

A __concrete category__ is a [[category]] $C$ equipped with a [[faithful functor]]

  $$
    U : C \to Set
  $$

to the [[large category]] [[Set]]. We say a category $C$ is _concretizable_ if and only if it admits a faithful functor $U: C \to Set$.
=--

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

Very often it is useful to consider the case where $U$ is [[representable functor|representable]] by some [[object]] $c_0 \in C$, in that $U \simeq C(c_0,-)$. For example, this is important for the statement of various concrete [[duality|dualities]] induced by [[dual adjunctions]]. We say in this case that $(C, U: C \to Set)$ is _representably concrete_. By definition, the object $c_0$ is then a [[separator]] of the category.

We remark that the existence of a left [[adjoint functor|adjoint]] $F$ to $U: C \to Set$ implies that $U$ is representable by $F(1)$. Conversely, if $C$ has [[coproducts]] or even just [[copowers]], then representability of $U$ implies that $U$ has a left adjoint.
=--

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

One can also consider concrete categories over any base category $X$ instead of necessarily over $Set$.  This is the approach taken in [The Joy of Cats](#JoyOfCats).  Then the (small) categories concrete over $X$ form a [[2-category]] $Cat(X)$.

=--

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

A concrete category is **univalent** if its underlying category (by forgetting the functor into $Set$) is a [[univalent category]].

=--

## Examples

* Any small category $C$ can be made into a concrete category by setting $U(X)$ to be the set of all arrows with codomain $X$, and $U(X \to Y)$ given by composition with $X \to Y$. This can be generalized to any locally small category with a small [[separator]].

The following furnish examples of concrete categories, with the first three representably concrete:

* $C = Set$ itself with [[separator]] $c_0 = \{\bullet\}$ the singleton set.

* $C = Top$ with the separator $c_0$ taken to be the one-point space.

* Any [[monadic functor]] $U: C \to Set$ is faithful (because it preserves equalizers and reflects isomorphisms) and has a left adjoint. As special cases, we have the usual collection of examples of concrete categories: [[monoid]]s, [[group]]s, [[ring]]s, [[algebra]]s, etc.

A category may be concretizable in more than one way:

* Take $ C $ to be the category of [[Banach space]]s with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by $ 1 $ (so $ \| T v \| \leq \| v \| $ for all $ v $ in the source).  Then there are two versions of $ U $ that one may use: one where $ U ( V ) $ (for $ V $ a Banach space) consists of *every* vector in $ V $, and one where $ U ( V ) $ consists of those vectors bounded by $ 1 $ (so the closed unit ball in $ V $).  The first may seem more obvious at first, but only the second is representable (by a $ 1 $-dimensional Banach space).

* Insofar as categories such as $Set$, $Top$, $Vect_k$, etc. admit many [[separators]], these categories may be rendered representably concrete in a variety of ways. Indeed, the category $Vect_k$ may be _monadic_ over $Set$ in many different ways. For example, if $V$ is $n$-dimensional, the functor $\hom(V, -): Vect_k \to Set$ is monadic and realizes $Vect_k$ as equivalent to the category of [[modules]] over the matrix algebra $\hom(V, V)$.

* Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If $E = Sh(C, J)$ is the category of sheaves on a small site $(C, J)$, we have a familiar string of faithful functors
$$Sh(C, J) \hookrightarrow Set^{C^{op}} \stackrel{monadic}{\to} Set/C_0 \stackrel{\Sigma_{C_0}}{\to} Set.$$
But if for example $E$ is the category of sheaves over $\mathbb{R}$, then no object $X$ can serve as a single separator of $E$, since it cannot detect differences between arrows $Y \stackrel{\to}{\to} Z$ whenever the support of $Y$ is strictly contained in the support of $X$.

* A concrete category that is equipped with the structure of a [[site]] in a compatible way is a [[concrete site]]. The category of [[concrete sheaves]] on a concrete site is concrete.

Many familiar examples of "sets with additional structure" provide examples of concrete categories where $U$ is the usual 'underlying set':

* The category $Mon$ of [[monoids]] and monoid [[homomorphisms]] is a concrete category.

* The category $Ab$ of [[abelian groups]] and abelian group homomorphisms is a concrete category.

* Given a commutative ring $R$, the category $R Mod$ of $R$-[[modules]] and $R$-[[linear maps]] is a concrete category.

* Given a commutative ring $R$, the category $R Alg$ of $R$-[[algebras]] and $R$-algebra homomorphisms is a concrete category.

* The category $CRing$ of [[commutative rings]] and commutative ring homomorphisms is a concrete category.

* The category $Field$ of [[fields]] and field homomorphisms is a concrete category.

* The category $HeytAlg$ of [[Heyting algebras]] and Heyting algebra homomorphisms is a concrete category.

* The category $Frm$ of [[frames]] and frame homomorphisms is a concrete category.

* The category $Conv$ of [[convergence spaces]] and [[continuous functions]] is a concrete category.

* The category $Top$ of [[topological spaces]] and continuous functions is a concrete category.

* The category $Met$ of [[metric spaces]] and [[isometries]] is a concrete category.

* The category $StrictCat$ of [[strict categories]] and [[strict functors]] is a concrete category.

There are other examples of concretizable categories where the objects are described as sets, but one cannot choose $U$ satisfying $U(X) = X$

* The category $Set_\bot$ of sets and [[partial functions]] is a concrete category when equipped with the functor $U(X) = X \amalg \{ * \}$ that adds a disjoint point, and sends a partial function to the total function whose undefined values are set to the point.

* The category $Rel$ of sets and [[relations]] has a separator given by the singleton set. Thus, it is a concrete category when equipped with the functor $U(X) = PowerSet(X)$, and $U(X \to Y)$ given by composition of relations (viewing a subset of $X$ as a relation on $X$). This is faithful since for any relation $\Phi \in Rel(X, Y)$ we have $(x,y) \in \Phi$ iff $y \in \Phi \circ \{x\}$.

## Non-examples ##

* The [[classical homotopy category]] [[Ho(Top)]] of [[topological spaces]] is _not_ concretizable

* The opposite category of [[commutative rings]] $CRing^{op}$ equipped with the [[prime spectrum]] functor $CRing^{op} \to Set$ is not concrete, since the prime spectrum is not faithful. This is one of the reasons for the use of [[scheme]]s in algebraic geometry.

* The category $Prefunc$ of sets and [[prefunctions]] is not a concrete category.

## Properties

+-- {: .num_prop}
###### Proposition
Every small category $C$ is concretizable (since it fully and faithfully embeds in the concrete category $Set^{C^{op}}$).
=--

+-- {: .num_prop}
###### Proposition
If $C$ is concretizable, so is $C^{op}$.
=--

+-- {: .proof}
###### Proof
By assumption, there is a faithful functor $U^{op}: C^{op} \to Set^{op}$, and $\hom(-, \mathbf{2}): Set^{op} \to Set$ is monadic.
=--

+-- {: .num_remark}
###### Remark
Of course, since a category $C$ may possess a separator but no [[coseparator]], it does not follow that $C^{op}$ is representably concrete if $C$ is.
=--

### Morphism evaluation and extensionality

+-- {: .num_prop}
###### Proposition
In any concrete category $(C, U:C \to Set)$, there is an [[evaluation map]]
$$(-)((-))\colon Hom(a,b) \times U(a) \to U(b)$$
such that for every morphism $f \colon Hom(a,b)$ and $g \colon Mor_C(b,c)$ and every element $x:U(a)$, $(g \circ f)(x) = g(f(x))$ and $id_A(x) = x$.
=--

+-- {: .proof}
###### Proof
Because [[Set]] is a [[cartesian closed category]], [[currying]] the injective function $f_{a,b}$ of the functor $F$ in [[Set]] means that there is an [[evaluation map]] $(-)((-))\colon Hom(a,b) \times U(a) \to U(b)$ which satisfies the above axioms.
=--

+-- {: .num_prop}
###### Proposition
In any concrete category $(C, U:C \to Set)$, the morphisms satisfy [[function extensionality]] with respect to the [[evaluation map]]: for all morphisms $f \colon Hom(A,B)$ and $g \colon Hom(A,B)$, if $f(x) = g(x)$ for all [[elements]] $x \colon El(A)$, then $f = g$.
=--

+-- {: .proof}
###### Proof
Since [[Set]] is a [[well-pointed category]], and there is a bijection between $\mathbb{1} \to U(a)$ and $U(a)$, function extensionality follows.
=--

### Relationship with well-pointedness

The category [[Set]] of sets and functions is both concrete and well-pointed. However, not every [[well-pointed category]] is an concrete category, as well-pointed categories are not required to be [[concrete categories]]: most models of [[ETCS]] aren't defined to be concrete. Moreover, not every concrete category is a [[well-pointed category]]: the category $Field$ of [[fields]] and field [[homomorphisms]] is concrete, but is not well-pointed because it doesn't have a [[terminal object]].

The distinction between concreteness and well-pointedness is the distinction between [[elements]] and [[global elements]] in a concrete category with a [[terminal object]], as it is not true that elements and global elements (if they exist) coincide in general.

### Characterization

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

A [[finitely complete category]] is concretizable, i.e., admits a [[faithful functor]] to $Set$, if and only if it is [[well-powered category|well-powered]] with respect to [[regular monomorphism|regular]] [[subobject]]s.

=--

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

"Only if" was proven in ([Isbell](#Isbell)).  To prove it, note that if $F: C\to D$ is a faithful functor, then it is injective on equivalence classes of regular subobjects.  For suppose that $m\colon a \to x$ is the [[equalizer]] of $f,g\colon x\rightrightarrows y$, and $n\colon b\to x$ is the equalizer of $h,k\colon x\rightrightarrows z$.  If $F(a) \cong F(b)$ as subobjects of $F(x)$, then since $f m = g m$ and so $F(f)\circ F(m) = F(g)\circ F(m)$, we must also have  $F(f)\circ F(n) = F(g)\circ F(n)$; hence (since $F$ is faithful) $f n = g n$, so that $n$ factors through $m$.  Conversely, $n$ factors through $m$, so we have $a\cong b$ as subobjects of $x$.  Since $Set$ is regularly well-powered, it follows that any category admitting a faithful functor to $Set$ must also be so.

(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)

"If" was proven in ([Freyd](#Freyd)).  The argument is rather more involved, passing through [[additive categories]], and is not reproduced here.

=--

+-- {: .num_remark}
###### Remark
A relatively deep application of Isbell's result is that the [[classical homotopy category]] [[Ho(Top)]] of [[topological spaces]] is _not_ concretizable, even though it is a quotient of $Top$ which _is_ concretizable. ([Freyd 70](#Freyd70))

A similar way to use Isbell's result applies to show that a really vast number of model categories can not have a concrete localization at weak equivalences: see [Di Liberti and Loregian, 2017](#DiliLore)

=--

## Related concepts

* [[concrete object]]
* [[amnestic functor]]  (for many usual concrete categories, the forgetful functor happens to be amnestic)
* [[function extensionality]]
* [[well-pointed category]]
* [[family of objects in a concrete category]]
* [[concrete (2,1)-category]]
* [[concrete (n,1)-category]]
* [[concrete (infinity,1)-category]]

## References

* [[Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], *Abstract and Concrete Categories*, Wiley 1990, reprinted as: Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html))

* _[[The Joy of Cats]]_
{#JoyOfCats}

* [[John Isbell]], _Two set-theoretical theorems in categories_, Fund. Math 53 (1963)
{#Isbell}

* {#Freyd} [[Peter Freyd]], _Concreteness_, JPAA 3 (1973)

* {#Freyd70} [[Peter Freyd]], _Homotopy is not concrete_, in _The Steenrod Algebra and its Applications_, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 ([web](http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html))

* {#DiliLore} [[Ivan di Liberti]], [[Fosco Loregian]] "Homotopical algebra is not concrete." Journal of Homotopy and Related Structures (2017): 1-15.

[[!redirects concrete category]]
[[!redirects concrete categories]]
[[!redirects CAT(X)]]
[[!redirects Cat(X)]]


concrete object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $\Gamma \;\colon\; \mathcal{E} \to \mathcal{B}$ a [[functor]] we say that it _has [[codiscrete object]]s_ if it has a  [[full and faithful functor|full and faithful]] [[right adjoint]] $coDisc \,\colon\, \mathcal{B} \hookrightarrow \mathcal{E}$.

This is for instance the case for the [[global section]] [[geometric morphism]] of a [[local topos]] $ (Disc \dashv \Gamma \dashv coDisc) \;\colon\; \mathcal{E} \to \mathcal{B}$.

In this situation, we say that a  **concrete object** $X \in \mathcal{E}$ is one for which the $(\Gamma \dashv coDisc)$-[[unit of an adjunction]] is a [[monomorphism]].

If $\mathcal{E}$ is a [[sheaf topos]], this is called a [[concrete sheaf]].

If $\mathcal{E}$ is a [[cohesive (∞,1)-topos]] then this is called a [[concrete (∞,1)-sheaf]] or the like.

The dual notion is that of a [[co-concrete object]].

## Properties

$\Gamma$ is a [[faithful functor]] on [[morphisms]] whose [[codomain]] is concrete.

### Concretification factorization

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

For $\mathbf{H}$ a [[local topos]], write

$$
  \mathbf{H}_{conc}
  \overset{ \phantom{AAAA} }{\hookrightarrow}
  \mathbf{H}
$$

for its [[full subcategory]] of [[concrete objects]].

Then there is a sequence of [[reflective subcategory]]-inclusions that factor the $(\Gamma \dashv coDisc)$-adjunction as

$$
  \Gamma \;\dashv\; coDisc
  \;\;\colon\;\;
  \mathbf{H}
  \array{
    \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow}
    \\
    \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow}
  }
  \mathbf{H}_{conc}
  \array{
    \overset{\phantom{AAA}}{\longrightarrow}
    \\
    \overset{\phantom{AAA}}{\hookleftarrow}
  }
  Set
$$

Here morphisms on top are left adjoint to morphisms below, hence

$$
  conc \dashv \iota_{conc}
$$

exhibits the concrete objects as a [[reflective subcategory]], the reflector $conc$ being "[[concretification]]".

=--

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

For the adjunction on the right, we just need to observe that for every [[set]] $S \in Set$, the [[codiscrete object]] $coDisc(S)$ is [[concrete object|concrete]], which is immediate by [[idempotent monad|idempotency]] of $\sharp$ and the fact that every [[isomorphism]] is also a [[monomorphism]].

For the adjunction on the left we claim that the [[left adjoint]] $conc$ ([[concretification]]), is given by sending each [[object]] to the [[image]] of its $(\Gamma \dashv coDisc)$ [[adjunction unit]] $\eta^\sharp$:

$$
  conc \;\colon\; X \mapsto im(\eta^\sharp_X)
$$

hence to the object which exhibits the [[(epi, mono) factorization system|epi/mono-factorization]] of $\eta^\sharp_X$

\[
  \label{ConcretificationUnitFromEpiMonoFactorization}
  \eta^\sharp_X
  \;\colon\;
  X
    \underoverset{epi}{ \eta^{conc}_X }{\longrightarrow}
  conc X
    \underoverset{mono}{}{\longrightarrow}
  \sharp X
  \,.
\]

First we need to show that $conc X$, thus defined, is indeed concrete, hence that $\eta^\sharp_{im(\eta^\sharp_X)}$ is a monomorphism. For this, consider the following [[naturality square]] of the $\Gamma \dashv coDisc$-adjunction hom-isomorphism

\[
  \label{ConcNatur}
  \array{
    {\phantom{A}}
    \\
    Hom_{Set}( \Gamma im(\eta^\sharp_X), \Gamma im(\eta^\sharp_X) )
    &\simeq&
    Hom_{\mathbf{H}}( im(\eta^\sharp_X), \sharp im(\eta^\sharp_X) )
    \\
    {}^{ \mathllap{ (-) \circ \Gamma(\eta^{conc}_X) } }\big\downarrow
      &&
    \big\downarrow^{
      \mathrlap{ (-) \circ \eta^{conc}_X }
    }
    \\
    Hom_{Set}( \Gamma X, \Gamma im(\eta^\sharp_X) )
    &\simeq&
    Hom_{\mathbf{H}}( X, \sharp im(\eta^\sharp_X) )
  }
  \phantom{AAAA}
  \array{
    {\phantom{A}}
    \\
    \left\{ id_{\Gamma im(\eta^\sharp_X)} \right\}
      &\longrightarrow&
    \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }}
    \left\{ \eta^{\sharp}_{im(\eta^\sharp_X)} \right\}
    \\
    \big\downarrow
      &&
    \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }}
    \big\downarrow
    \\
    \left\{ \Gamma(\eta^{conc}_X) \right\}
      &\longrightarrow&
    \left\{
      \underset{
        iso
      }{
      \underbrace{
         \sharp(\eta^{conc}_X)
      }}
      \circ \eta^\sharp_{ X }
      \;=\;
    \eta^{\sharp}_{ im(\eta^\sharp_X) } \circ \eta^{conc}_X
    \right\}
  }
\]

By chasing the [[identity morphism]] on $\Gamma im(\eta^\sharp_X)$ through this diagram, as shown by the diagram on the right, we obtain the equality displayed in the bottom right entry, where we used the general formula for [[adjuncts]] and the definition $\sharp \coloneqq coDisc \circ \Gamma$.

But observe that $\Gamma (\eta^{conc}_X)$, and hence also $\sharp(\eta^{conc}_X)$, is an [[isomorphism]], as indicated above: Since $\Gamma$ is both a [[left adjoint]] as well as a [[right adjoint]], it preserves both [[epimorphisms]] as well as [[monomorphisms]], hence it preserves [[image]] factorizations. This implies that $\Gamma \eta^{conc}_X$ is the epimorphism onto the image of $\Gamma( \eta^\sharp_X )$. But by [[idempotent monad|idempotency]] of $\sharp$, the latter is an [[isomorphism]], and hence so is the epimorphism in its image factorization.

Therefore the equality in (eq:ConcNatur) says that

$$
  \begin{aligned}
    \eta^\sharp_{ X }
      & =
    \left(
      iso \circ \eta^{\sharp}_{ im(\eta^\sharp_X)}
    \right)
      \circ
    \eta^{conc}_X
    \\
    & =
    mono \circ \eta^{conc}_X
    \,,
  \end{aligned}
$$

where in the second line we remembered that $\eta^{conc}_X$ is, by definition, the epimorphism in the epi/mono-factorization of $\eta^\sharp_X$.

Now the defining property of epimorphisms allows to cancel this commmon factor on both sides, which yields

$$
  \eta^{\sharp}_{ im(\eta^\sharp_X) }
  \;=\;
  iso \circ mono
  \;=\;
  mono.
$$

This shows that $conc X \coloneqq im(\eta^\sharp_X)$ is indeed concrete.

It remains to show that this construction is [[left adjoint]] to the inclusion.
We claim that the [[adjunction unit]] of $(conc \dashv \iota_{conc})$ is provided by $\eta^{conc}$ (eq:ConcretificationUnitFromEpiMonoFactorization).

To see this, first notice that, since the [[(epi, mono) factorization system|epi/mono-factorization]] is [[orthogonal factorization system|orthogonal]] and hence [[functorial factorization|functorial]], we have [[commuting diagrams]] of the form

\[
  \label{NaturalitySquareForConcretificationUnit}
  \array{
    X_1
      &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}&
    im(\eta^\sharp_{X_1})
      &\underset{mono}{\longrightarrow}&
    \sharp X_1
    \\
    \big\downarrow && \big\downarrow && \big\downarrow
    \\
    X_2
      &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}&
    im(\eta^\sharp_{X_2})
      &\underset{mono}{\longrightarrow}&
    \sharp X_2
  }
\]

Now to demonstrate the adjunction, it is sufficient, to show that $\eta^{conc}$ is a [[universal morphism]] in the sense discussed at _[[adjoint functors]]_. Hence consider any morphism $f \;\colon\; X_1 \to X_2$ with $X_2 \in \mathbf{H}_{conc} \hookrightarrow \mathbf{H}$. Then we need to show that there is a unique diagonal morphism as below, that makes the following _top left triangle_ [[commuting diagram|commute]]:

$$
  \array{
    X_1
      &\overset{\phantom{AA} f \phantom{AA}}{\longrightarrow}&
    X_2
    \\
    {}^{\mathllap{epi}}\big\downarrow^{\mathrlap{\eta^{conc}_{X_1}}}
      &{}^{\mathllap{\exists !}}\nearrow&
    \big\downarrow^{\mathrlap{mono}}
    \\
    im(\eta^\sharp_{X_1})
      &\underset{}{\longrightarrow}&
    \sharp X_2
  }
$$

Now, from (eq:NaturalitySquareForConcretificationUnit), we have a [[commuting square]] as shown. Here the left morphism is an [[epimorphism]] by construction, while the right morphism is a [[monomorphism]] by assumption on $X_2$. With this, the [[(epi, mono) factorization system|epi/mono-factorization]] says that there is a diagonal [[lift]] which makes _both_ triangles [[commuting diagram|commute]].

It remains to see that the lift is unique with just the property of making the top left triangle commute. But this is equivalently the statement that the left morphism is an [[epimorphism]].

=--

### Further properties
 {#FurtherProperties}

\begin{proposition}\label{CohesiveMapsToConcreteObjectsGlue}
**(cohesive maps to concrete objects glue)**
\linebreak
  Let $\mathbf{H}$ be a [[cohesive (infinity,1)-topos|cohesive $\infty$-topos]], and consider $Y \,\in\, \mathbf{H}_{\sharp_1} \xhookrightarrow{\;} \mathbf{H}$ a [[concrete object]], in that its [[sharp modality]] [[unit of a monad|unit]] is [[n-truncated object of an (infinity,1)-category|(-1)-truncated]]/[[monomorphism in an (infinity,1)-category|monomorphic]]: $Y \xhookrightarrow{ \;\;\eta^\sharp_Y\;\; } \sharp Y$.  Then *cohesive maps to $X$ glue* (satisfy the respective sheaf property):

For any $Y \,\in\, \mathbf{H}$ and any open cover, namely any [[n-connected object of an (infinity,1)-topos|(-1)-connected]]/[[effective epimorphism in an (infinity,1)-category|effective epi-morphism]] $U \twoheadrightarrow X$, cohesive maps $U \xrightarrow{\;} X$ whose maps of [[underlying]] $\infty$-groupoids descend/extend to $Y$, then they also descend/extend to $Y$ as cohesive maps, in an essentially unique way, in that all solid [[homotopy]]-[[commutative square]] as follows have essentially unique dashed lifts:

\begin{tikzcd}[row sep=22pt, column sep=22pt]
  U
  \ar[
    rr,
    "{ \forall }"{above}
  ]
  \ar[
    dd,
    ->>
  ]
  &&
  X
  \ar[
    dd,
    hook,
    "{ \eta^\sharp_X }"
  ]
  \\
  \\
  Y
  \ar[
    rr,
    "{ \forall }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists ! }"{description}
  ]
  &&
  \sharp X
\end{tikzcd}

\end{proposition}
The point here is the given interpretation of this [[lifting problem]], but the proof of the latter is immediate:
\begin{proof}
Under the given assumptions, the essentially unique existence of the lift is an instance of the [[(n-connected, n-truncated) factorization system]] for $n = (-1)$.
\end{proof}

## Related concepts

* [[concretification]]

[[!include cohesion - table]]

## References

* {#Shulman} [[Mike Shulman]], _Discreteness, Concreteness, Fibrations, and Scones_ ([blog post](http://golem.ph.utexas.edu/category/2011/11/discreteness_concreteness_fibr.html))

* [[Urs Schreiber]], Section 3.7.2 in *[[schreiber:differential cohomology in a cohesive topos|Differential cohomology in a cohesive $\infty$-topos]]* ([arXiv:1310.7930](https://arxiv.org/abs/1310.7930))

[[!redirects concrete objects]]


concrete sheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

When regarding a [[sheaf]] as a [[space]] defined by how it is probed by test spaces, a _concrete sheaf_ is a generalized space that has (at least) an _underlying [[set]] of points_ out of which it is built.

So a concrete sheaf models a [[space]] that is given by a set of points and a choice of which morphisms of [[set]]s from concrete test spaces into it count as "structure preserving" (e.g. count as smooth, when the sheaf models a [[smooth space]]).

More in the intrinsic language of [[sheaves]], a _concrete sheaf_ is a [[sheaf]] on a [[concrete site]] $C$ that, while perhaps not [[representable functor|representable]], is "quasi-representable" in that it is a [[subobject]] of a sheaf of the form

$$
  U \mapsto Hom_{Set}(|U|, S)
$$

where $S$ is a [[set]] and $|U|$ is the [[set]] $|U| \coloneqq Hom_C({*}, U)$ of points underlying the object $U$ in the [[concrete site]] $C$.

For the category of concrete sheaves $Conc(Sh(C))$, the [[global sections]] functor

$$Hom_{Conc(Sh(C))}(1,-):Conc(Sh(C))\to Set$$

is [[faithful functor|faithful]], i.e. $Conc(Sh(C))$ is a [[concrete category]].

## Definition

We discuss two definitions: the first one is more elementary and describes concrete sheaves explicitly in terms of properties of the underlying [[site]].

The second one is more abstract and more general, and describes them entirely [[topos theory|topos theoretically]].

### On a concrete site

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

A **[[concrete site]]** is a [[site]] $C$ with a [[terminal object]] $*$ such that

1. the functor $Hom_C(*,-) : C \to Set$ is a [[faithful functor]];

1. for every [[coverage|covering family]] $\{f_i : U_i \to U\}$ in $C$ the morphism

   $$
      \coprod_i Hom_C(*,f_i) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)
   $$

   is [[surjective]].

=--

For $X \in PSh(C)$ any [[presheaf]], write

$$
  \tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))
$$

for the [[adjunct]] of the restriction map

$$
  X(U) \times Hom_C(*,U) \to X(*)
  \,,
$$

which in turn is the adjunct of the component map of the functor

$$
  X_{*,U} . Hom_C(*,U) \to Hom_{Set}(X(U), X(*))
  \,.
$$

{#ConcSheafOnConcSite}
+-- {: .un_def}
###### Definition

A [[presheaf]] $X : C^{op} \to Set$ on a [[concrete site]] is a **concrete presheaf** if for each $U \in C$ the map $\tilde X_U : X(U) \to Hom_{Set}(Hom_C(*,U), X(*))$ is [[injective]].

A **concrete sheaf** is a presheaf that is both concrete and a [[sheaf]].

=--

So a concrete presheaf $X$ is a [[subobject]] of the presheaf $U \mapsto Hom_{Set}(Hom_C(*,U), X(*))$.

Write $Conc(Sh(C)) \hookrightarrow Sh(C)$ for the [[full subcategory]] of the [[category of sheaves]] on concrete sheaves.

### In a local topos

A more abstract perspective on the [previous definition](#ConcSheafOnConcSite) is obtained by noticing the following.

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

The [[category of sheaves]] on a [[concrete site]] is a [[local topos]].

=--

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

Taking $U=*$ in the second condition defining a concrete site implies that any [[covering]] family of $*$ contains a [[split epimorphism]], or equivalently that the only covering [[sieve]] of $*$ is the maximal sieve consisting of all morphisms with [[target]] $*$.  This means that a concrete site is in particular a [[local site]], which implies that its topos of sheaves is a [[local topos]].

=--

In fact, we can formulate the definition of concrete sheaf inside any [[local topos]] $E$ over any base topos $S$:

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

Let

$$(Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}} S$$

be a [[local geometric morphism]]. Since then by definition $S \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}} E$ is a [[subtopos]] the morphisms $V = \Gamma^{-1}(isos(S)) \subset Mor E$ that are inverted by $\Gamma$ are the [[local isomorphism]]s with respect to which the objects of $S$ are [[sheaves]]/$V$-[[local object]]s in $E$.

The **concrete sheaves** are the objects of $E$ that are the $V$-[[separated objects]].

=--

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

For $E = Sh(C) \stackrel{\Gamma = Hom(*,-)}{\to} Set$ the [[category of sheaves]] on a [[concrete site]], this is equivalent to the
[previous definition](#ConcSheafOnConcSite).

=--

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

Since $C$ is concrete, in the [[global sections]] [[geometric morphism]] $(Disc,\Gamma)\colon Sh(C) \to Set$, the [[direct image]] $\Gamma$ is evaluation on the point: $X\mapsto X(*)$. The further [[right adjoint]] $Codisc \colon Set\to Sh(C)$, sends a set $A$ to the functor $U\mapsto Hom_{Set}(Hom_C(*,U),A)$.  Moreover, this right adjoint $Codisc$ is [[full and faithful functor|fully faithful]] and thus embeds $Set$ as a [[subtopos]] of $Sh(C)$.

We observe that $ (\Gamma \dashv Codisc) : Set \to  Sh(C)$ is the [[localization]] of $Sh(C)$ at the set $\{Disc \Gamma U \to U | U \in C\}$ of counits of the adjunction $(Disc \dashv \Gamma)$ on representables: because if for $X \in Sh(C)$ we have that

$$
  \begin{aligned}
     Hom_{Sh(C)}(Disc \Gamma U \to U, X)
     & = (X(U) \to Hom_{Set}(\Gamma(U), \Gamma(X)))
     \\
     & = (X(U) \to Hom_{Set}(Hom_C(*,U)), X(*))
  \end{aligned}
$$

is an [[isomorphism]], then clearly $X = Codisc(X(*))$.

On the other hand, comparison with the [previous definition](#ConcSheafOnConcSite) shows that this is a [[monomorphism]] precisely if $X$ is a concrete sheaf.  But this is also the definition of a separated object.

=--

So the concrete sheaves on $C$ are precisely the [[separated presheaf|separated objects]] for this [[Lawvere-Tierney topology]] on $Sh(C)$ that corresponds to the subtopos $Codisc : Set \hookrightarrow Sh(C)$.

## Properties

### General

Let $\Gamma : E \to S$ be a [[local topos]]. From the definition of concrete sheaves as [[separated presheaves]] it follows immediately that

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

The category of concrete sheaves $Conc_\Gamma(E)$ forms a [[reflective subcategory]] of $E$

$$
  S
   \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}}
  Conc_\Gamma(E)
    \stackrel{\overset{Conc}{\leftarrow}}{\hookrightarrow}
  E
$$

which is a [[quasitopos]].

The left adjoint $Conc$ is **concretization** which sends a sheaf $X$ to the [[image]] sheaf

$$
  Conc X : U \mapsto Im(X(U) \to Hom_{Set}(Hom_C(*,U), X(*)))
  \,.
$$

=--

### Slice topos over a concrete object {#SliceOverConcreteObject}

Let $(Disc \dashv \Gamma \dashv coDisc) :  \mathcal{E} \to \mathcal{S}$ be a [[Grothendieck topos]] that is a  [[local topos]] over $\mathcal{S}$ and let $X \in \mathcal{E}$ be a concrete object, equivalently an object such that the $(\Gamma \dashv coDisc)$-[[unit of an adjunction|counit]] $X \to coDisc \Gamma U$ is a [[monomorphism]].

We discuss properties of the [[over-topos]] $\mathcal{E}/X$.

Notice that

$$
  e_0 \coloneqq (\Gamma \dashv coDisc) : \mathcal{S} \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\to}} \mathcal{E}
$$

is the canonical [[point of a topos|topos point]] of $\mathcal{E}$.

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

For every [[global element]] $(x \in \Gamma(X)) : * \to X$ (for every $X \in \mathcal{E}$) there is a [[point of a topos|topos point]] of the form

$$
  (e_0,x)
  :
  \mathcal{S}
    \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  \mathcal{S}/\Gamma(X)
    \stackrel{\overset{\Gamma/X}{\leftarrow}}{\underset{coDisc/X}{\to}}
  \mathcal{E}/X
  \,.
$$

=--

This is discussed in detail at over-topos -- points.

+-- {: .un_prop}
###### Observation
**(relative concretization)**

Let $X \in \mathcal{E}$ be concrete. Then the image under the
$coDisc/X \circ \Gamma/X$-[[monad]] of any object $(A \to X) \in \mathcal{E}/X$ is an object $(\tilde A \to X)$ with $\tilde A$ being concrete.

This $\tilde A$ is the finest concrete sheaf structure on $\Gamma A$ that extends $\Gamma A \to \Gamma X$ to a morphism of concrete sheaves.

=--

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

By definition of the slice geometric morphism we have that $coDisc/X \circ \Gamma/X (A \stackrel{f}{\to} X)$ is the [[pullback]] $\tilde A \to X$ in

$$
  \array{
    \tilde A &\to & coDisc \Gamma A
    \\
    \downarrow && \downarrow^{\mathrlap{coDisc \Gamma A}}
    \\
    X &\to& coDisc \Gamma X
  }
  \,,
$$

where the bottom morphism is the $(\Gamma \dashv coDisc)$-[[unit of an adjunction|unit]]. Since this is a [[monomorphism]] by assumption on $X$ it follows that $\tilde A\to coDisc \Gamma A$ is a monomorphism. Since $coDisc$ is a [[full and faithful functor]] by assumption on $\mathcal{E}$ and $\Gamma$ is a [[right adjoint]] it follows that the [[adjunct]] $\Gamma \tilde A \to \Gamma coDisc\Gamma A \stackrel{\simeq}{\to} \Gamma A$ is a monomorphism, as is its image $coDisc \Gamma \tilde A\to coDisc \Gamma A$ under the [[right adjoint]] $coDisc$.

Then by the universal property of the unit we have a commuting diagram

$$
  \array{
    && coDisc \Gamma \tilde A
    \\
    & \nearrow & \downarrow
    \\
    \tilde A &\to& coDisc \Gamma A
  }
  \,,
$$

where the bottom and the right morphisms are [[monomorphism]]s. Therefore also the diagonal morphism, the $(\Gamma \dashv coDisc)$-unit on $\tilde A$, is a monomorphism, and hence $\tilde A$ is concrete.

=--

## Examples

* The concrete sheaves on the [[concrete site]] [[CartSp]] are the [[diffeological spaces]].

* The category of [[quasi-Borel space|quasi-Borel spaces]] is the category of concrete sheaves on the category of standard Borel spaces considered with the [[extensive category#Superextensive sites|extensive coverage]].

* [[quasi-topological spaces]]

## Related concepts

* [[plot]]

* [[concrete (∞,1)-sheaf]]

## References

The notion of quasitoposes of concrete sheaves goes back to

* [[Eduardo Dubuc]], _Concrete quasitopoi_, Lecture Notes in Math. 753 (1979), 239–254 ([doi:10.1007/BFb0061821](https://doi.org/10.1007/BFb0061821))

and is further developed in

* [[Eduardo Dubuc]], [[Luis Espanol]], _Quasitopoi over a base category_  ([arXiv:math.CT/0612727](http://arxiv.org/abs/math.CT/0612727)) .

A review of categories of concrete sheaves, with special attention to sheaves on [[CartSp]], i.e. to [[diffeological space]]s is in

* [[John Baez]], [[Alex Hoffnung]], _Convenient Categories of Smooth Spaces_, Trans. Amer. Math. Soc. 363 (11), 2011 ([arXiv](http://arxiv.org/abs/0807.1704))

The characterization of concrete sheaves in terms of the extra right adjoint of a [[local topos]] originated in [discussion](http://nforum.ncatlab.org/comments.php?DiscussionID=1955) with [[David Carchedi]].

Concrete sheaves over trivial sites are called *extensional presheaves* in the following references.

* [[Giuseppe Rosolini]], [[Thomas Streicher]], [Comparing models of higher type computation](https://www.researchgate.net/publication/220366978_Comparing_models_of_higher_type_computation), 1999

* [[Thomas Ehrhard]], [On finiteness spaces and extensional presheaves over the Lawvere theory of polynomials](https://www.researchgate.net/publication/29650819_On_finiteness_spaces_and_extensional_presheaves_over_the_Lawvere_theory_of_polynomials), 2007

[[!redirects concrete sheaves]]
[[!redirects concrete presheaf]]
[[!redirects extensional presheaf]]
[[!redirects concrete presheaves]]
[[!redirects concretization]]
[[!redirects concretizations]]
[[!redirects concretification]]
[[!redirects concretifications]]


concrete site

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A **concrete site** is a [[site]] whose objects can be thought of as [[set]]s with extra [[stuff, structure, property|structure]]: it is a [[category]] that is a [[concrete category]] and a [[site]] in a compatible way.

In a [[category of presheaves]] on a concrete site one can consider [[concrete presheaves]].

## Definition

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

A **[[concrete site]]** is a [[site]] $C$ with a [[terminal object]] $*$ such that

1. the functor $Hom_C(*,-) : C \to Set$ is a [[faithful functor]];

1. for every [[coverage|covering family]] $\{f_i : U_i \to U\}$ in $C$ the morphism

   $$
     \coprod_i Hom_C(*,f_i) : \coprod_i Hom_C(*, U_i) \to Hom_C(*, U)
   $$

   is [[surjective]].

=--

## Examples

* every ([[small category|small]]) [[concrete category]] becomes a concrete site when equipped with the trivial [[coverage]] (every covering family consists of just an [[identity]] morphism);

  * for instance the [[simplex category]] $\Delta$, which is the site for [[simplicial set]]s.

* Any small subcategory of [[concrete categories]] such as [[Top]], [[Diff]], etc, with their standard [[coverage]]s (by [[open cover]])s;

  * for instance [[CartSp]] (covering families are open covers);

    which is the site for [[smooth space]]s, including [[diffeological space]]s.

[[!redirects concrete sites]]


concrete smooth infinity-groupoid

> This is a sub-entry of [[smooth ∞-groupoid -- structures]]. See there for more context.

--

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In every [[cohesive (∞,1)-topos]] there is a notion of concrete cohesive ∞-groupoids. Here we discuss concrete [[nLab:smooth ∞-groupoid]]s. These are the [[(∞,1)-category theory|higher generalization]] of [[nLab:diffeological space]]s.

## Definition

The general abstract definition is:

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

A **concrete smooth $\infty$-groupoid** is a concrete cohesive ∞-groupoids in [[Smooth∞Grpd]].

=--

We will compare this intrinsic definition with more concrete models:

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

A [[diffeological space]] is a [[concrete sheaf]] on the [[site]] [[CartSp]]${}_{smooth}$.

Write

$$
  DiffeolSpace \hookrightarrow Sh(CartSp)
$$

for the [[full subcategory]] on diffeological spaces. Notice that is a [[quasitopos]].

=--

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

A **diffeological groupoid** is an [[internal groupoid]] in the [[category]] of [[diffeological space]]s.

=--

(...)

## Special cases

We discuss the special cases of [[n-truncated]] concrete smooth $\infty$-groupoids:

* 0-truncated: [diffeological space](0TruncAndDiffeolSpace)

* 1-truncated: [diffeological groupoids](#1TruncDiffolGroupoid).

### Diffeological spaces
 {#0TruncAndDiffeolSpace}

+-- {: .num_prop #Concrete0TruncatedIsDiffeological}
###### Proposition

Concrete smooth 0-groupoids are equivalently [[diffeological space]]s.

More in detail, write $Conc(\tau_{\leq 0} Smooth \infty Grpd)$ for the full [[subcategory]] on the [[concrete sheaf|concrete]] [[0-truncated]] objects. This is [[equivalence of categories|equivalent]] to the category of [[diffeological space]]s

$$
  DiffeolSp
   \simeq
  Conc(\tau_{\leq 0} Smooth \infty Grpd)
  \,.
$$

=--

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

Let $X \in Sh(CartSp) \hookrightarrow Smooth \infty Grpd$ be a [[sheaf]] on $CartSp$. The condition for it to be concrete is that the [[unit of an adjunction|unit]]

$$
  X \to coDisc \Gamma X
$$

is a [[monomorphism]]. Since monomorphisms of sheaves are detected objectwise (see [[category of sheaves]]) this is equivalent to the statement that for all $U \in CartSp$ the morphism

$$
  X(U) \simeq Smooth\infty Grpd(U, X) \to Smooth \infty Grpd(U, coDisc \Gamma X)
  \simeq
  \infty Grpd(\Gamma U, \Gamma X)
$$

is a monomorphism of sets, where in the first step we used the [[(∞,1)-Yoneda lemma]] and in the last one the $(\Gamma \dashv coDisc)$-[[adjunction]].

That this morphism is indeed $\Gamma : Sh(U,X) \to Set(\Gamma(U), \Gamma(X)) \hookrightarrow \infty Grpd(\Gamma(U), \Gamma(X))$ follows by chasing the identity on $\Gamma X$ through the adjunction naturality square for any morphism $f : U \to X$

$$
  \array{
    Set(\Gamma X, \Gamma X) &\stackrel{\simeq}{\to}&
    Sh(X, coDisc \Gamma X)
    \\
    \downarrow^{\mathrlap{\Gamma(f)^*}} && \downarrow^{\mathrlap{f^*}}
    \\
    Set(\Gamma U, \Gamma X) &\stackrel{\simeq}{\leftarrow}& Sh(U, coDisc \Gamma X)
  }
  \,.
$$

So this is indeed the defining condition for [[concrete sheaves]] that defines [[diffeological space]]s.

=--

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

The canonical embedding $SmoothMfd \hookrightarrow Smooth \infty Grpd$
from [above](#EmbeddingOfSmoothManifolds) factors through [[diffeological spaces]]: we have a sequence of [[full and faithful (∞,1)-functor]]s

$$
  SmoothMfd \hookrightarrow DiffeolSp \hookrightarrow Smooth \infty Grpd
  \,.
$$

=--

### Diffeological groupoids
 {#1TruncDiffolGroupoid}

We want to say the following

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

Concrete [[1-truncated]] smooth $\infty$-groupoids are equivalent to diffeological groupoids, def. \ref{DiffeologicalGroupoid}.

(...)

=--

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

Let $A$ be 1-truncated and concrete. Then by definition there is
a concrete 0-truncated object $A_0$ and an [[effective epimorphism]] $A_0 \to A$ -- an [[atlas]] -- , such that the [[(∞,1)-pullback]]

$$
  A_1 :=  A_0 \times_A A_0
$$

is itself concrete.

Since $A$ is assumed to be [[1-truncated]], it follows that $A_1$ is [[0-truncated]]. By [[Giraud's axioms]] in the [[(∞,1)-topos]] [[Smooth∞Grpd]] we have that $A$ is [[equivalence in an (∞,1)-category|equivalent]] to the [[groupoid object in an (∞,1)-category]] $A_1 \stackrel{\to}{\to} A_0$:

\[
  A \simeq \lim_\to (A_1 \stackrel{\to}{\to} A_0)
  \,.
\]

Now by prop. \ref{Concrete0TruncatedIsDiffeological} both $A_0$ and $A_1$ are [[diffeological space]]s. Hence the above exhibits $A$ as equivalent to a diffeological groupoid.

(...)

=--

## Transgression of differential cocycles to mapping spaces
 {#TransgressionOfDifferentialCocycles}

For $n \in \mathbb{N}$, write $\mathbf{B}^n U(1)_{conn} \in Smooth\infty Grpd$ for the smooth $\infty$-groupoid given under the [[Dold-Kan correspondence]] by the [[Deligne complex]]. Over [[smooth manifold]]s this is the coefficient object for [[circle n-bundles with connection]].

At [[schreiber:∞-Chern-Simons theory]] the following fact is proven:

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

Let $\Sigma$ be a closed [[smooth manifold]] of [[dimension]] $dim \Sigma \leq n$. Then there is an equivalence

$$
  hol
    :
  \tau_{n - dim \Sigma} \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})
   \to
  B^{n-dim \Sigma} U(1)
$$

of [[discrete ∞-groupoid]]s such that for $dim \sigma  = n$ this computes the $n$-volume [[holonomy]] of circle $n$-bundles with connection.

=--

Using concretization we want to refine this from discrete to smooth $\infty$-groupoids.

Write $[\Sigma, \mathbf{B}^n U(1)_{conn}]$ for the [[internal hom]] in the [[(∞,1)-topos]] (see there).

We first look at this for $n = dim \Sigma$

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

For $dim \Sigma = n$ there is an equivalence

$$
  hol
   :
  Conc \tau_0 [\Sigma, \mathbf{B}^nU(1)_{conn}]
   \to
   U(1)
$$

in [[Smooth∞Grpd]].

=--

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

Since $\tau_0 [\Sigma, \mathbf{B}^n U(1)_{conn}]$ is [[0-truncated]], hence a [[sheaf]], concretification is that discussed at [[concrete sheaves]]:

$$
  Conc(F)(U) = image( F(U) = \mathbf{H}(U,F) \to
  \mathbf{H}(U, coDisc \Gamma F) =  Set(\Gamma(U), \Gamma(F)) )
  \,.
$$

So for $U = *$ first of all we have by prop. \ref{IntrinsicHolonomyTakingValuesInDiscrete} that

$$
  Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(*) = U(1)
$$

in $Set$.

Generally $Conc \tau_0 \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn})(U)$
is therefore a subset of the set of functions of sets $U \to U(1)$. We need to show that it is precisely the set of smooth such functions.

But this is clear: holonomy of a smoth family of smoth circle $n$-bundles is a smooth function. Moreover, every smooth function arises this way: for $f : U \to U(1)$ any smooth function, pick a trivial family of trivial circle $n$-bundles with connection and then rescale the connection form using $f$.

=--

## Related concepts

* [[concrete sheaf]], [[concrete (∞,1)-sheaf]]

* [[diffeological space]], [[Lie groupoid]]

## References

Many of the ideas involved here are due to [[Dave Carchedi]].

A writeup of some aspects is in section 3.3.1 of

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

[[!redirects concrete smooth ∞-groupoid]]

[[!redirects concrete smooth infinity-groupoid]]
[[!redirects concrete smooth infinity-groupoids]]

[[!redirects diffeological groupoid]]
[[!redirects diffeological ∞-groupoid]]
[[!redirects diffeological infinity-groupoid]]

[[!redirects diffeological groupoids]]
[[!redirects diffeological ∞-groupoids]]
[[!redirects diffeological infinity-groupoids]]


concrete structure

# Concrete and abstract structures
* table of contents
{: toc}

## Idea

Many concepts from ordinary mathematics were originally developed in the course of a particular application, such that the original definition of the concept is tied to that application.  The modern definition (typically some kind of [[structured set]]) is then obtained by a process of abstraction.  Sometimes this history is remembered in the terminology, often with the original notion being called _concrete_ and the later notion called _abstract_.

It then becomes possible to define the concrete notion in terms of the abstract one (so giving an *abstract* definition of the original *concrete* concept); usually, a concrete structure is then an abstract structure equipped with some [[extra stuff]].  Sometimes there is also a [[representation theorem]] showing that every abstract structure arises from some concrete structure (but sometimes this is false).

## Examples

We list here only examples where the concrete meaning has historically been the default meaning for at least some authors or where at least one of the adjectives 'concrete' or 'abstract' have been used.

### Groups

Given a [[set]] $X$, the [[permutation group]] on $X$ is the set of [[permutations]] on $X$; consider a [[subset]] of this that includes the [[identity function]] and is closed under [[composition]] and taking [[inverse functions]].  This is the original notion of __concrete group__ due to [[Évariste Galois]].  Abstracting from this, we get the modern notion of __abstract [[group]]__ as a set equipped with an appropriate operation (taking the place of composition of permutations).

Much of the early work on group theory dealt with symmetry groups in [[geometry]], giving a geometric notion of concrete group in which $X$ is a [[space]] instead of a [[set]], although (except perhaps in [[constructive mathematics]] without the [[fan theorem]]) the relevant spaces form a [[concrete category]] and so the elements of the group can still be viewed as permutations of an [[underlying set]].

We can now give an *abstract* definition of the notion of *concrete* group:  An abstract group $G$ becomes a __concrete group__ on the set $X$ once it is given a [[free action]] of $G$ on $X$.  The [[Cayley theorem]] shows that every group may be made into a concrete group.  Of course, if we generalise from [[Set]] to an arbitrary [[category]] (whether thought of as a category of spaces or not), then every group is *already* a concrete group, identifying a group with the [[automorphism group]] of a [[pointed category|pointed]] [[connected category|connected]] [[groupoid]].

### Categories

Unlike groups, [[categories]] were first defined in full modern abstraction.  (At least, modern for the 20th century; by the end of the 21st century, it may seem old-fashioned not to start with weak $(\infty,1)$-[[(infinity,1)-category|categories]].)  But in light of the motivating examples and [[Bourbaki]]\'s prior theory of structures, we can anachronistically define a __concrete category__ as any category of [[structured sets]] (as defined there).

Of course, an __abstract category__ is the usual notion of [[category]].  Then the abstract definition of a __[[concrete category]]__ is a category equipped with a [[faithful functor]] to [[Set]]; see [[structured set]] again for the equivalence.

### Vector spaces

The original [[vector spaces]] were [[Cartesian spaces]]: $\mathbb{R}^n$, where $\mathbb{R}$ is the [[real line]] and $n$ is a [[natural number]].  This generalises easily to $K^c$, where $K$ is any [[field]] and $c$ is any [[cardinal number]]; we may call such a __concrete vector space__.  Unlike with groups, there is no need (in [[classical mathematics]]) to consider [[subspaces]] of $K^c$ closed under [[linear combinations]], since these are all isomorphic to $K^d$ for $d \leq c$.

Then an __abstract vector space__, which came later, is a [[module]] over $K$ thought of as a [[commutative ring]].  The abstract definition of a __concrete vector space__ is an abstract vector space equipped with a [[basis]].  Using the [[axiom of choice]], we may prove that every abstract vector space has such a concrete structure.

### Hilbert spaces

The original notion of [[Hilbert space]] (the one used by [[David Hilbert]]) was $L^2(\mathbb{R})$, the [[Lebesgue space]] on the [[real line]] (with [[Lebesgue measure]]) of exponent $2$.  This immediately generalises to $L^2(X)$, where $X$ is any [[measure space]].  As a special case, if $X$ is a [[discrete space|discrete]] measure space (a [[set]] equipped with [[counting measure]]), then we have a topological version of the concrete vector space $K^c$.

Either of these (any measure space or only a discrete measure space) may be taken as a __concrete Hilbert space__, while the modern notion is an __abstract Hilbert space__.  An [[orthonormal basis]] on an abstract Hilbert space gives it the structure of a __concrete Hilbert space__ (in the stricter sense).

### Operator algebras

With [[operator algebras]], we have the curious situation that special names may still be found in the literature to distinguish the concrete and abstract structures.

Let $H$ be a [[Hilbert space]] over the [[complex numbers]] and consider the $*$-[[star-algebra|algebra]] $B(H)$ of [[bounded linear operators]] from $H$ to itself.  A __$C^*$-[[C-star-algebra|algebra]]__ is a sub-$*$-algebra of $B(H)$ that is [[closed subspace|closed]] in the [[norm topology]]; a __[[von Neumann algebra]]__ is a sub-$*$-algebra of $B(H)$ that is closed in the [[weak operator topology]] (a stronger condition).

On the abstract side, a __$B^*$-[[B-star-algebra|algebra]]__ is a [[Banach algebra|Banach]] $*$-algebra such that ${\|x^* x\|} = {\|x\|}^2$ always holds, while a __$W^*$-[[W-star-algebra|algebra]]__ is a $B^*$-algebra with a [[predual]] as a [[Banach space]].  It is then a theorem that (as defined above) every $C^*$-algebra is a $B^*$-algebra, and every von Neumann algebra is a $W^*$-algebra, so we may abstractly define a __$C^*$-algebra__ to be a $B^*$-algebra with a [[faithful representation|faithful]] [[representation]] on a Hilbert space, and similarly define a __von Neumann algebra__ to be a $W^*$-algebra with a free action on a Hilbert space.

The representation theorem here is that every $B^*$-algebra may be given the structure of a $C^*$-algebra, and in fact the term '$B^*$-algebra' is nearly obsolete.  Similarly, every $W^*$-algebra may be given the structure of a von Neumann algebra, but here both terms may yet be found (and even distinguished such that a von Neumann algebra comes with a representation on a Hilbert space but a $W^*$-algebra does not).

### Manifolds and varieties

The original [[algebraic varieties]] were [[subspaces]] of [[affine spaces]] or [[projective spaces]] given as the [[zero set]]s of [[algebraic function]]s; these are __concrete varieties__.  The modern notion of varieties as certain [[schemes]], the __abstract varieties__, is much more general.

Similarly, [[manifolds]] can be viewed as [[subspaces]] of [[Cartesian spaces]] with locally invertible local parametrisations ---**concrete manifolds**--- or as abstract sets of points equipped with an [[atlas]] of locally invertible local charts ---**abstract manifolds**.  Both concepts were used from the earliest days; the [[Whitney embedding theorem]] shows their equivalence (at least if the abstract manifolds are assumed to be [[second-countable space|second-countable]] and [[Hausdorff space|Hausdorff]], as is common).

### Sets

One might view the [[sets]] of [[material set theory]] as __concrete sets__ and the sets of [[structural set theory]] as __abstract sets__ (a term used at least by [[Lawvere]]).  The abstract definition of a concrete set is then that given at [[pure set]].  There is also a more naïve version of a __concrete set__ as a [[subset]] of a given [[ambient set]]; these were the first sets studied, predating [[set theory]] as such.

### Points

A __concrete [[point]]__ in a [[set]] $X$ is an [[element]] of $X$, an __abstract point__ is a [[singleton]], and the abstract definition of __concrete point__ in $X$ is a [[function]] to $X$ from an abstract point.  The concrete definition of concrete point doesn\'t generalise from [[Set]] to arbitrary [[categories]], but the others do: an __abstract point__ is a [[terminal object]], and a __concrete point__ is a [[global element]].

## Concrete objects

Since all of these concrete and abstract objects literally are [[objects]] in various [[categories]], it would be nice to use the terms 'concrete object' and 'abstract object' to refer to them collectively.  However, there is another meaning of '[[concrete object]]', so I have gone for the vaguer term 'structure' instead (justified since they are all objects in [[concrete categories]] and so are sets with [[extra structure]] ... with the exception of the concrete and abstract categories themselves!).

There is a relationship: concrete objects generalise [[concrete sheaves]], which are concrete in the sense of having an [[underlying set]] of points, similar to how a concrete variety has a set of points from an affine or projective space.  However, it\'s not really an example of the concept on this page, as far as I can tell.

[[!redirects concrete structure]]
[[!redirects concrete structures]]
[[!redirects abstract structure]]
[[!redirects abstract structures]]
[[!redirects concrete or abstract structure]]
[[!redirects concrete or abstract structures]]
[[!redirects concrete and abstract structure]]
[[!redirects concrete and abstract structures]]

[[!redirects concrete group]]
[[!redirects concrete groups]]


concrete subobject

#Contents#
* toc
{:toc}

This entry is meant as a complement to [[subobject]].  In particular, the initial author knows that it is there, and that it really doesn't mean what he intends by "concrete subobject".

## Idea

In a [[concrete category]] $|-| : \mathcal{C} \to Set $, for an object $X$, when an injection $i : A \to | X | $ naturally induces (this is deliberately vague)

 * an object $ A^\vee $ with $ A = | A ^ \vee | $
 * a monomorphism $ i^\vee : A^\vee \to X $ with $ | i^\vee | = i $

then we say $A^\vee$ is a (concrete) subobject of $X$.

Compare with [[Grothendieck fibration]]

## Examples

### Algebra

 * In the category $Gp$ of groups, a concrete subgroup is exactly a monomorphism in $Gp$, which is exactly a map in $Gp$ whose underlying function is an injection; a map of sets $f: A \to |G|$ underlies a subgroup inclusion $ ? \to G $ iff there is *exactly one* group structure on $A$ such that $f$ underlies a homomorphism.

### Topology

 * For every topological space $X$ and every set $A$ and every map $f : A \to |X|$, there is a *terminal* (coarsest) topology on $A$ with a continuous map to $|X|$ over the function $f$; a *subspace* ("embedded" subspace) is exactly an injection of underlying sets where the domain is given this terminal compatible topology.
 * A sub-$\beta$-module is a *closed* subset; that is, an embedding of a compact space.

### Geometry

 * An injection of a topological manifold to a smooth manifold may be a topological embedding, but there may be many (or no) compatible smooth structures on the domain; a submanifold is a *smooth* embedding *with injective differential*, which is exactly an embedding of a topological manifold such that smooth functions on the codomain restrict to give a smooth atlas on the domain.


concretification

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a [[local topos]] there is a notion of _[[concrete objects]]_. These form a [[reflective subcategory]]. The corresponding reflector is the _concretification map_ which universally approximates any object by a concrete object.

## Definition

There is a unique evident notion of concretification in any [[local topos]], this we discuss first in

* _[In a local topos](#InALocalTopos)_

This involves an [[image]] factorization. Since in [[higher category theory]]/[[homotopy theory]] image factorization refines to a tower of notions of [[n-image]] factorization in a [[local (∞,1)-topos]] there are different constructions that one may all think of as concretification. This we discuss in

* _[In a higher cohesive topos](#InAHigherCohesiveTopos)_

### In a local topos
 {#InALocalTopos}

A [[local topos]] is a [[topos]] equipped with a [[sharp modality]] $\sharp$.

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

For $X$ any object of the topos, the [[image]] projection of the [[unit of a monad|unit]] $\iota_X \colon X \to \sharp X$ is the _concretification_ of $X$

$$
  (X \to Conc X)
  \coloneqq
  (X \to im(\iota_X))
  \,.
$$

=--

### In a cohesive $(\infty,1)$-topos
 {#InAHigherCohesiveTopos}

In a [[local (∞,1)-topos]] there is still the [[sharp modality]], but here the [[1-image]]-factorization of its unit is rarely of interest, for this concretifies an object in degree 0 but makes it [[codiscrete object|codiscrete]] in all higher degrees. Typically one is interested in concretifying in all degrees. One needs to specify extra data to say what this means.

One case where this is arises is the _differential concretification_ of [[moduli ∞-stacks]] of [[principal ∞-connections]].

## Examples

### Concretification of differential moduli
 {#ConcretificationOfDifferentialModuli}

#### Models for $n$-image factorization

The following gives a sufficient condition for modeling [[n-image]] factorizations in some [[(∞,1)-toposes]] with particularly convenient presentation.

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

Let $C$ be a site with [[point of a topos|enough points]], so that the weak equivalences in $sPSh(C)_{\mathrm{loc}}$ are detected on [[stalks]] ([this prop.](model+structure+on+simplicial+presheaves#OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks)). Then given a morphism of [[Kan complex]]-valued [[simplicial presheaves]]

$$
  f \colon X \longrightarrow Y$ in $sPSh(C)
$$

such that both  $X$ and $Y$ are [[homotopy n-types|homotopy k-types]] for some finite $k \in \mathbb{N}$, then its [[n-image]] factorization in the [[(∞,1)-topos]] $L_{lwhe} sPSh(C)_{loc}$ for any $n \in \mathbb{N}$ is presented by any factorization $X \longrightarrow im_{n}(f) \longrightarrow Y$ in $sPSh(C)$ through some Kan-complex valued simplicial presheaf $im_n(f)$ such that for each object $U \in C$ the [[simplicial homotopy groups]] satisfy the following conditions:

1. $\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are [[isomorphisms]];

1. $\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the [[(epi,mono) factorization]] of $\pi_n(f(U))$;

1. $\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are [[isomorphisms]].

=--

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

Evalutation on [[stalks]] is a [[filtered colimit]] which preserves the [[finite limits]] and [[finite colimits]] that go into the definition of [[simplicial homotopy groups]]. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the [[categorical homotopy groups in an infinity-topos|categorical homotopy groups]] in $L_{lwhe} sPSh(C)_{loc}$. By [this prop.](n-truncated+object+of+an+infinity-category#RecognizngnTuncationOnSimplicialHomotopyGroups) and [this def.](n-connected+object+of+an+infinity-topos#Connectedness) we may recognize $n$-truncation of morphisms on categorical homotopy groups (using the assumption that $X$ and $Y$ are $k$-truncated for some $k$). Therefore the claim now follows from the stalkwise [[long exact sequence of homotopy groups]].

=--

In order to appeal to prop. \ref{nImageFactroizationModeledOnSimplicialPrsheaves} we are interested in explicit models for $n$-image factorization of morphisms of [[Kan complexes]]. The following gives such for the special case that the the morphism of Kan complexes is the image under the [[Dold-Kan correspondence]] of a [[chain map]] between [[chain complexes]].

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

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$
be a [[chain map]] between [[chain complexes]]

For $n \in \mathbb{N}$, consider the abelian group

$$
  (im_{n+1}(f))_n
    \;\coloneqq\;
  coker(\,
     ker(\partial_V) \cap ker(f_n)
     \to
     V_n
  \,)
$$

For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an [[equivalence relation]] on $V_n$ by
$$
  \left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    (\partial_V v_n = \partial_V v'_n)
    \;\text{and}\;
    (f_n(v_n) = f_n(v'_n))
  \right)
  \,.
$$

Then
$$
  (im_{n+1}(f))_n \simeq V_n/_\sim
$$
is equivalently the set of [[equivalence classes]] of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that
$$
  \left(
     v_n \sim v'_n
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n)
  \right)
$$
and hence is generated under linearity by
$$
  \left(
     v_n \sim 0
  \right)
  \;\Leftrightarrow\;
  \left(
    v_n  \in ker(\partial_V) \cap ker(f_n)
  \right)
  \,.
$$

Moreover, notice that the [[Dold-Kan correspondence]]
$$
  DK
  \;\colon\;
  Ch_{\bullet \geq 0}
    \longrightarrow
  KanCplx
$$
factors through [[globe|globular]] [[strict omega-groupoids]] ([here](Dold-Kan+correspondence#GlobularAndCubical)). An [[n-morphism]] in the [[strict omega-groupoid]] $DK(V_\bullet)$ is of the form
$$
  (v_{n-1})
    \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow}
  (v_{n-1} + \partial v_n)
  \,.
$$

In terms of these morphisms the [[equivalence relation]] above says that two of them are equivalent precisely if

1. they are "[[parallel morphisms]]" in that they have the same [[source]] and [[target]];

1. they have the same image under $f$ in the [[n-morphisms]] of  $DK(W_\bullet)$.

This suggests yet another equivalent way to think of $(im_{n+1}(f))_n$: it is the [[disjoint union]] over the [[target]] $(n-1)$-cells in $V_{n-1}$ of the images under $f$ of the sets of $n$-cells from zero to that target:
$$
  (im_{n+1}(f))_n
   \simeq
   \underset{v_{n-1} \in V_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1}
    \right\}
  \,.
$$

=--

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

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$
be a [[chain map]] between [[chain complexes]] and let
$n \in \mathbb{N}$. Recall the abelian group
 $\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\}$ from remark \ref{ChainComplexnPlusOneImageInDegreen}.

The following [[diagram]] of [[abelian groups]]
[[commuting diagram|commutes]]:
$$
  \array{
    \vdots && \vdots && \vdots
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+3}
      &\overset{f_{n+3}}{\longrightarrow}&
    W_{n+3}
      &\overset{=}{\longrightarrow}&
    W_{n+3}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+2}
      &\overset{f_{n+2}}{\longrightarrow}&
    W_{n+2}
      &\overset{=}{\longrightarrow}&
    W_{n+2}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\mathrlap{ \partial_W  } }
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_{n+1}
      &\overset{f_{n+1}}{\longrightarrow}&
    \left\{
      w_{n+1} | \exists v_n :  \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0,
    \right\}
      &\overset{}{\longrightarrow}&
    W_{n+1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
     &&
    \downarrow^{\partial_W}
     &&
    \downarrow^{\mathrlap{\partial_{W}}}
    \\
    V_n
      &\overset{ (f_n, \partial_V) }{\longrightarrow}&
    \underset{v_{n-1}}{\sqcup}
    \left\{
      f_n(v_n) \vert \partial_V v_n = v_{n-1}
    \right\}
     &\overset{  }{\longrightarrow}&
    W_n
    \\
    \downarrow^{\mathrlap{\partial_V}}
    &&
    \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-1}
      &\overset{=}{\longrightarrow}&
    V_{n-1}
      &\overset{f_{n-1}}{\longrightarrow}&
    W_{n-1}
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    V_{n-2}
      &\overset{=}{\longrightarrow}&
    V_{n-2}
      &\overset{f_{n-2}}{\longrightarrow}&
    W_{n-2}
    \\
    \\
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_{V}}}
    &&
    \downarrow^{\mathrlap{\partial_W}}
    \\
    \vdots && \vdots && \vdots
  }
$$

Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps
$$
  f_\bullet
   \;\colon\;
  V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet
  \,.
$$

Finally, this is a model for the [[n-image|(n+1)-image factorization]] of $f$ in that on [[homology groups]] the following holds:

1. $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are [[isomorphisms]];

1. $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the [[image|image factorization]] of $H_n(f)$;

1. $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$   are [[isomorphisms]].

=--

+-- {: .proof}
###### Proof (but check)

This follows by elementary and straightforward direct inspection.

=--

#### Moduli of circle $n$-connections

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

For $p \in \mathbb{N}$ and $k \leq p+1$ write
$$
  \mathbf{B}^{p+1}U(1)_{conn^k}
  \coloneqq
  DK
  \left(
    U(1) \to \Omega^1 \to \Omega^2 \to \cdots \to \Omega^k \to 0 \to \cdots \to 0
  \right)
  \;\in\;
  sPSh(CartSp)
$$

for the [[simplicial presheaf]] which is the image under the [[Dold-Kan correspondence]] of the presheaf of chain complexes which is the [[Deligne complex]] starting with the presheaf represented by $U(1)$ in degree $p+1$ and truncated to the differential $k$-forms, as shown.

Since the $DK$ map sends surjections of chain complexes to [[Kan fibrations]],
the canonical projection maps yield a tower of objectwise [[Kan fibrations]] of the following form:
$$
  \mathbf{B}^{p+1}U(1)_{conn}
  =
  \mathbf{B}^{p+1}U(1)_{conn^{p+1}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^{p}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^{p-1}}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^1}
    \longrightarrow
  \mathbf{B}^{p+1}U(1)_{conn^0}
    =
  \mathbf{B}^{p+1}U(1)
  \,.
$$

=--

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

For $\Sigma$ a [[smooth manifold]], write
$$
  (\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma)
  \in
  sPSh(CartSp)
$$

for the image under the [[Dold-Kan correspondence]] of the presheaf of chain complexes which to $U \in CartSp$  assigns the _vertical_ [[Cech cohomology|Cech]]-[[Deligne complex]] on $\Sigma \times U \to U$ in the given degree, i.e. the Cech-Deligne complex involving differential forms on $\Sigma \times U$ that have no leg along $U$, i.e. those in $\Omega^{\bullet,0}(\Sigma \times U)$.

=--

#### Differential concretification on contractibles

We first consider differential concretification on geometrically contractible base spaces. Once this is established, then the general differential concretification follows simply by stackifying along the base space.

+-- {: .num_defn #DifferentialConcretificationForCirclenConnectionsOnContractible}
###### Definition
**(differential concretification for higher circle connections on contractibles)**

Let $\Sigma$ be a [[contractible topological space|contractible]] [[smooth manifold]].
For $p \in \mathbb{N}$ write
$$
  (\mathbf{B}^p U(1))\mathbf{Conn}_0(\Sigma)
    \coloneqq
  [\Sigma, \mathbf{B}^{p+1}U(1)]
$$
and then for $0 \leq k \leq p$ define [[induction|inductively]]
$$
  (\mathbf{B}^p U(1))\mathbf{Conn}_{k+1}(\Sigma)
   \coloneqq
  im_{p+1-k}
  \left(
    [\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}}]
      \longrightarrow
    \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^{k+1}} ]
      \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn^k}]}{\times^h}
    (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
  \right)
 \,.
$$

=--

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

Let $\Sigma$ be a [[contractible topological space|contractible]] [[smooth manifold]].  Then there is a weak equivalence
$$
  (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma)
    \simeq
  (\mathbf{B}^p U(1)) \mathbf{Conn}(\Sigma)
  \,,
$$
from the inductively defined object from def. \ref{DifferentialConcretificationForCirclenConnectionsOnContractible} to the moduli object from def. \ref{ModuliOfnFormConnection}.

=--

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

By the assumption that $\Sigma$ is contractible, the Cech-direction of the Cech-Deligne double complex is trivial and so we have for all $U \in CartSp$ and $0 \leq k \leq p$ weak equivalences of the form
$$
  [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}](U)
  \;\simeq\;
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^1(\Sigma \times U)
      \to
    \Omega^2(\Sigma \times U)
     \to
    \cdots
      \to
    \Omega^{p+1}(\Sigma \times U)
  \right)
$$
and
$$
  (\mathbf{B}^p U(1))\mathbf{Conn}(\Sigma)
    \simeq
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^{1,0}(\Sigma \times U)
     \to
    \Omega^{2,0}(\Sigma \times U)
      \to
    \cdots
      \to
    \Omega^{p+1,0}(\Sigma \times U)
  \right)
  \,.
$$

We claim now for all $k \leq p$ that
$$
  (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
    \simeq
  DK\left(
    C^\infty(\Sigma \times U, U(1))
     \to
    \Omega^{1,0}(\Sigma \times U)
     \to
    \cdots
     \to
    \Omega^{k,0}(\Sigma \times U)
     \to
    0
      \to
    \cdots
      \to
    0
  \right)
  \,.
$$

For $k = p$ this is the statement to be shown.
Hence we may now prove this by [[induction]].

It is manifestly true for $k = 0$. Hence suppose it is true for some $k \lt p$.  Observe then that
$$
  \sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^{k+1}}]
    \longrightarrow
  \sharp [\Sigma, \mathbf{B}^{p+1}U(1)_{conn^k}]
$$
is an objectwise Kan fibration, because so is $\mathbf{B}^{p+1}U(1)_{conn^{k+1}} \to \mathbf{B}^{p+1}U(1)_{conn^k}$ by def. \ref{TruncatedDelgneComplexes}, and both $[\Sigma,-]$ as well as $\sharp$ are right Quillen functors from $sPSh(C)$ with its global projective model structre to itself.

It follows ([this prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)) that the homotopy fiber product in question
is presented by the plain 1-categorical fiber product. Since $DK$ is right adjoint, this in turn is given by the degreewise fiber product of the corresponding chain complexes. By direct inspection this means that
$$
  \begin{aligned}
    &
    \sharp [ \Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_{k+1}} ]
      \underset{\sharp[\Sigma, \mathbf{B}(\mathbf{B}^p U(1))_{conn_k}]}{\times^h}
    (\mathbf{B}^p U(1))\mathbf{Conn}_k(\Sigma)
   \\
   & \simeq
   DK
   \left(
      C^\infty(\Sigma \times U, U(1))
        \to
      \Omega^{1,0}(\Sigma \times U)
        \to
      \cdots
        \to
      \Omega^{k,0}(\Sigma \times U)
        \to
      (\sharp \Omega^{k+1}(\Sigma \times -))(U)
        \to
      0
       \to
      \cdots
        \to
      0
   \right)
  \end{aligned}
$$

Hence we are now reduced to computing the $(p+1-k)$ image of
$$
  \array{
  DK
  (
    C^\infty(\Sigma \times U)
     &\to&
    \Omega^1(\Sigma \times U)
      &\to&
    \cdots
       &\to&
    \Omega^{k}(\Sigma \times U)
      &\to&
    \Omega^{k+1}(\Sigma \times U)
      &\to&
    0
      &\to&
    \cdots
      &\to&
    0
  )
  \\
    \downarrow
    &&
    \downarrow
    &&
    &&
    \downarrow
    &&
    \downarrow
    &&
    \downarrow
    &&
    &&
    \downarrow
  \\
   DK
   (
      C^\infty(\Sigma \times U, U(1))
        &\to&
      \Omega^{1,0}(\Sigma \times U)
        &\to&
      \cdots
        &\to&
      \Omega^{k,0}(\Sigma \times U)
        &\to&
      (\sharp \Omega^{k+1}(\Sigma \times -))(U)
        &\to&
      0
       &\to&
      \cdots
        &\to&
      0
   )
  }
$$

Observe that in degree $(p+1)-(k+1)$ the ordinary image is
$$
  im\left(
    \Omega^{k+1}(\Sigma \times U)
      \to
     (\sharp \Omega^{k+1}(\Sigma \times -))(U)
  \right)
   \simeq
  \Omega^{k+1,0}(\Sigma \times U)
$$

With this the induction follows by prop. \ref{nImageFactroizationModeledOnSimplicialPrsheaves} and prop. \ref{ImageFactorizationForChainComplexes}.

=--

#### General differential concretification
 {#GeneralDifferentialConcretification}

+-- {: .num_defn #DifferentialConcretificationOfCirclenConnections}
###### Definition
**(differential concretification of moduli for higher connection)**

For $\Sigma$ a [[smooth manifold]], define for $p \in \mathbb{N}$
$$
  (\mathbf{B}^{p}U(1))
  \mathbf{Conn}_{p+1}(\Sigma)
    \;\simeq\;
  \underset{\longleftarrow}{\lim}^h_i
  \;
  (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(U_i)
$$
to be the [[homotopy limit]] over the differential concretifications from def. \ref{DifferentialConcretificationForCirclenConnectionsOnContractible} of contractibles $U_i$, for
$$
  \Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i
$$
a presentation of $\Sigma$ as a homotopy colimit of contractible manifolds (e.g. the realization of the [[Cech nerve]] of a [[good open cover]]).

=--

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

For $\Sigma$ a [[smooth manifold]], then the differential concretifiction of def. \ref{DifferentialConcretificationOfCirclenConnections} is equivalent to the moduli object from def. \ref{ModuliOfnFormConnection}:
$$
  (\mathbf{B}^p U(1)) \mathbf{Conn}_{p+1}(\Sigma)
    \simeq
  (\mathbf{B}^{p}U(1)) \mathbf{Conn}(\Sigma)
  \,.
$$

=--

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

Let $\Sigma \simeq \underset{\longrightarrow}{\lim}_i^h U_i$ be the realization of the Cech nerve of a good open cover. Then
$$
  \underset{\longleftarrow}{\lim}_i (\mathbf{B}^p  U(1))\mathbf{Conn}_{p+1}(U_i)
$$
is equivalently the image under DK of the corresponding Cech hypercomplex with coefficients in the presheaf of chain complexes $(\mathbf{B}^p  U(1))\mathbf{Conn}_{p+1}(-)$. By lemma \ref{DifferentialConcretificationFornConnectionsOnContractibles} this is the vertical Deligne complex, and hence the claim follows.

=--

## References

Introductory lecture notes with an eye towards applications in fundamental [[physics]] are at

* _[[geometry of physics]]_

The differential concretification of differential moduli is discussed in

* _[[schreiber:differential cohomology in a cohesive topos]]_.

The full proof of example \ref{DifferentialConcretificationOfCirclenconnections} is due to Joost Nuiten....

[[!redirects concretifications]]
[[!redirects differential concretification]]
[[!redirects differential concretifications]]


concurrency theory

# Concurrency theory
* table of contents
{: toc}

## Idea

*Concurrency* (or maybe rather _parallelism_ ([Harper](#Harper))) is a joint property of several (maybe interacting) [[event structure|processes]] proceeding simultaneously. Here the word "simultaneously" indicates that the evolution of the participating processes is indexed along an irreversible [[directed object]].

## Calculi

### [[π-calculus]]

The processes can either be the nil process $0$, parallel execution $P \mid Q$, sending on channel $c$ and then continuing $\overline{c}\langle x \rangle.P$, receiving on channel $c$ and then continuing $c(x).P$, replicating a process $!P$ or introduction of a new name $(\nu x)P$.

### Rho Calculus

## Related entries

* [[parallel computing]]

* [[models for concurrency]]

* [[Directed Algebraic Topology and Concurrency]]

* [[Some geometric perspectives in concurrency theory]]

* in a wider sense also articles related to [[directed homotopy theory]], cf.

  * [[Directed Algebraic Topology]]

  * [[directed homotopy type theory]]

## References

* wikipedia [process calculus](https://en.wikipedia.org/wiki/Process_calculus)
* [[Bob Harper]], _[Parallelism is not concurrency](http://existentialtype.wordpress.com/2011/03/17/parallelism-is-not-concurrency/)_
 {#Harper}

* [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], [[Samuel Mimram]], [[Martin Raussen]], [[Directed Algebraic Topology and Concurrency]]

* _The Pi calculus: toward global computing_ Rchain [blog](https://blog.rchain.coop/blog/2019/04/03/the-pi-calculus-toward-global-computing)
* zoranskoda:[[zoranskoda:distributed consensus]]

category:computer science

[[!redirects concurrency]]
[[!redirects concurrency theory]]
[[!redirects copncurrency theory]]


condensate

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

## Related concepts

* [[vacuum expectation value]]

* [[Bose-Einstein condensate]]

* [[inflaton]], [[Higgs field]]

* [[tachyon condensation]]

* [[gravitino condensation]]

* [[gaugino condensation]]

## References

* Wikipedia, _[Vacuum expectation value](http://en.wikipedia.org/wiki/Vacuum_expectation_value)_

* Shigeki Matsumoto, Takeo Moroi, _Decay of Scalar Condensation in Quantum Field Theory_, Phys.Rev.D77:045014, 2008 ([arXiv:0709.4338](http://arxiv.org/abs/0709.4338))

[[!redirects condensates]]

[[!redirects condensation]]
[[!redirects condensations]]


condensed abelian group

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

#Contents#
* table of contents
{:toc}

## Definition

A [[condensed object in an (infinity,1)-category|condensed object]] [[internalization|in]] the [[category]] [[Ab]] of [[abelian groups]].

## Properties

The category of condensed abelian groups $CondAb$ enjoys excellent categorical properties for [[homological algebra]]:

* It is an [[abelian category]] that admits all [[small limit|small]] [[limits]] and [[colimits]];

* In this category, [[filtered colimits]] and infinite [[products]] are [[exact functor|exact]].
The latter property is rather rare.

* It has enough [[compact projective objects]]: [[free construction|free]] condensed abelian groups on [[extremally disconnected]] [[compact Hausdorff topological spaces]] generate all condensed abelian groups under small [[colimits]] and their [[corepresentable functors]] [[conservative functor|reflect isomorphisms]];

* The previous property implies that condensed abelian groups have the same exactness properties as the category of [[abelian groups]].

## Related concepts

* [[condensed set]]

* [[condensed spectrum]]

* [[solid abelian group]]

* [[liquid vector space]]

* [[solid module]]

## References

See [[condensed mathematics]].

[[!redirects condensed abelian groups]]
[[!redirects CondAb]]
[[!redirects Cond(Ab)]]