local addition

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## Idea

At its simplest, a [[smooth manifold]] is a place where one can do calculus without worries.  Let us consider the formula for a derivative:

$$
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
$$

From the fractional part of this, it appears that to be able to discuss calculus, we need to be working in a space which allows addition and scalar multiplication.  However, the fact that we are considering a limit means that we only need to be able to do such things in a neighbourhood of a given point.  That is, we need the concept of a _local_ addition.

Thus at its most general, a local addition at a point is simply the structure of a (piece of a) vector space near that point.  This is, of course, the notion of a [[chart]].  At this point, one starts to think about notions of compatibility between different local additions.  By concentrating on what it means for two charts to be compatible at a point, one soon arrives at the notion of an [[atlas]].

Another direction that one could take is to ask for a system of local additions, one for each point, and ask that these fit together in some nice manner as one moves over the manifold.  This leads to the notion of a **local addition**.

## Definition

The broadest definition is the following.

+-- {: .num_defn #locadd}
###### Definition
A **local addition** on $M$ consists of a vector bundle $\pi \colon E \to M$, an open neighbourhood of the zero section, $U \subseteq E$, and a smooth map $\eta \colon U \to M$ such that

1. The composition of $\eta$ with the zero section of $E$ is the identity on $M$, and
2. there exists an open neighbourhood, say $V$, of the diagonal in $M \times M$ such that the map $\pi \times \eta \colon U \to M \times M$ is a diffeomorphism onto $V$.
=--

### Remarks

1. By considering the derivative of $\eta$, one can see that the bundle $E$ must be isomorphic to $T M$.  However, it is not necessary to specify an isomorphism in advance since $\eta$ naturally defines one.  Nonetheless, it is common simply to take $E = T M$.

2. Another way to simplify the definition is to take $U = E$.  In [KM §42.4](#km), this is called a _globally defined_ local addition (though such a conjunction of "global" and "local" may grate).  Given a local addition with arbitrary $U$, it is possible to define a "globally defined" one simply by using a smooth fibrewise embedding (preserving the zero section) of $E$ onto an open subset of $U$.

3. A useful source of local additions is from Riemannian geometry: the exponential map coming from a Riemannian structure defines a local addition.

4. Another source of local additions is to apply the tubular neighbourhood theorem to the embedding of the diagonal $M \to M \times M$.

## Charts from Local Additions

### Charts for $M$

Local additions are used in constructing the manifold structure on certain mapping spaces.  In brief, if $M$ is a manifold (possibly infinite dimensional) admitting a local addition, and if $N$ is a compact manifold (possibly with boundary) then $C^\infty(N,M)$ can be given the structure of a smooth manifold and the construction of the charts uses the local addition on $M$.  For details, see [KM §42](#km).  Applying this to the case $N = pt$, we obtain charts for $M$ itself.  These charts are useful since then we have charts on both $M$ and the various mapping spaces which are all defined using the same method and so the relationships between them are that much clearer.

+-- {: .num_prop #Mcharts}
###### Proposition
Let $M$ be a smooth manifold, $\eta \colon E \supseteq U \to M$ a local addition on $M$.  Let $V \subseteq M \times M$ be the image of $\pi \times \eta$.  Let $p \in M$.  Let $U_p \coloneqq E_p \cap U$ be the fibre of $U$ over $p$.  Let $V_p \subseteq M$ be such that $\{p\} \times V_p = V \cap \left(\{p\} \times M\right)$.  Then the restriction of $\eta$ to $U_p$ defines a diffeomorphism $\eta_p \colon U_p \to
V_p$.
=--

+-- {: .proof}
###### Proof
Let us start by showing that $\eta_p$ is well-defined.  Clearly, we can restrict $\eta$ to $U_p$ to get _something_, the only question is its image.  To find that, we look at the image of $U_p$ under $\pi \times \eta$.  Since $U_p \subseteq E_p = \pi^{-1}(p)$, the image of $U_p$ is contained in $V \cap \left(\{p\} \times M\right)$, which is what we have called $V_p$.  Conversely, if $q \in V_p$ then $(p,q) \in V$ and so $(\pi \times \eta)^{-1}(p,q) \in U$.  Since $\pi(\pi \times \eta)^{-1}(p,q) = p$, we have $(\pi \times \eta)^{-1}(p,q) \in U \cap E_p$ and thus there is some $r \in U_p$ such that $\eta(r) = q$.  Hence the image of $\eta_p$ is $V_p$ as claimed.

The rest follows from the simple fact that we can also define $\eta_p$ as the restriction of the diffeomorphism $(\pi \times \eta) \colon U \to V$ to the subset $U_p$ on the source and $\{p\} \times V_p$ on the target.
=--

### Charts for Mapping Spaces

Local additions are used to great effect in constructing charts for mapping spaces.

Let $M$ be a smooth manifold (possibly infinite dimensional).  Let $N$ be a functionally compact [[Frölicher space]].  Let $P \subseteq M$ be a submanifold.  Let $Q \subseteq N$ be a subset.  We consider the space $C^\infty(N,M;Q,P)$ of smooth maps $N \to M$ which map $Q$ into $P$.  As a smooth manifold, $M$ naturally has the structure of a Frölicher space so this mapping space is well-defined.

We assume that the _pair_ $(M,P)$ admits local addition.  By that, we mean that $M$ admits a local addition, say $\eta$, with the property that it restricts to a local addition on $P$.  We shall also assume, for simplicity, that the domain of $\eta$ is $T M$.

Let $g \colon N \to M$ be a smooth map with $g(Q) \subseteq P$.  Let $E_g$ be the space of sections of $g^* T M$ with the property that the sections over $Q$ are constrained to lie in $g^* T P$.  In more detail, we define $g^* T M$ in the usual manner:

$$
g^* T M \coloneqq \{(x,v) \in N \times T M : g(x) = \pi(v)\}
$$

and then take the space of smooth maps $f \colon N \to g^* T M$ with the property that the composition $N \to g^* T M \to N$ is the identity.  Within that space, we further restrict to those $f$ such that the image of the map $Q \to g^* T M \to T M$ lies in $T P$.

Although $N$ could be quite complicated, because $T M \to M$ is a vector bundle, $E_g$ is a vector space.  Furthermore, by trivialising $g^* T M$ using a finite number of trivialisations (possible as $N$ is functionally compact), we can embed $E_p$ as a closed subspace of $C^\infty(N,\mathbb{R}^n)$ for some $n$.  This embedding shows that $E_p$ is a convenient vector space, in the sense of [Kriegl and Michor](#km).

We define a map for $\Phi \colon E_g \to C^\infty(N,M;Q,P)$ as follows.  Let $f \in E_p$.  Then $f$ is a section of $g^* T M$ and so is a map $N \to g^* T M$.  By the definition of $g^* T M$, we can think of $f$ as a map $N \to N \times T M$ which projects to the identity on the first factor.  By applying the projection to the second factor, we obtain a map $\hat{f} \colon N \to T M$.  Composing with $\eta$ produces a map $\eta \circ \hat{f} \colon N \to M$.  As $f \in E_g$, the restriction of $\hat{f}$ to $Q$ lands in $T P$, whence $\eta \circ \hat{f}$ takes $Q$ into $P$.  The map $f \mapsto \eta \circ \hat{f}$ is what we call $\Phi$.

Let us identify its image.  Let $V \subseteq M \times M$ be the image of the local addition.  Define $U_g \subseteq C^\infty(N,M;Q,P)$ to be the set of those functions $h$ such that $(g,h) \colon N \to M \times M$ takes values in $V$.  We claim that the image of $\Phi$ is $U_g$ and that $\Phi$ is a bijection $E_g \to U_g$.

Let us start with the image.  Let $h \in U_g$.  Then $(g,h) \colon N \to M \times M$ takes values in $V$, so we can compose with $(\pi \times \eta)^{-1}$ to get a map $\check{h} \colon N \to T M$.  Together with the identity on $N$, we get a map $N \to N \times T M$.  By construction, $\pi \check{h} = g$ and so this map ends up in $g^* T M$ (which has the subspace structure).  Again by construction, the projection of this map to $N$ is the identity and so it is a section of $g^* T M$.  That it takes $Q$ to $T P$ follows from the fact that $\eta$ restricts to a local addition on $P$, whence as $h(Q) \subseteq P$, $\check{h}(Q) \subseteq T P$.  Hence $\Phi$ is onto.  Moreover, this construction yields the inverse of $\Phi$ and so it is a bijection.

Thus we have charts for $C^\infty(N,M;Q,P)$.  The next step is the transition functions.  To prove this in full generality, we assume not just two different functions at which to base our charts, but also two different local additions to define them.  This will show that our resulting manifold structure is independent of this choice.

_... to be continued ..._

## Diffeomorphisms from Local Additions

Another useful construction from a local addition relates to diffeomorphisms.  In differential topology, manifolds have a certain amount of "flexibility": a common picture is of a rubber sheet that can be deformed.  The amount of flexibility is less than that allowed in algebraic topology, but certainly more than that in differential geometry.  Being able to deform a manifold a little is a very useful trick in establishing the relationships between the various mapping spaces (for example, it is used in showing that the loop space fibration $\Omega M \to L M \to M$ is a [[fibre bundle]] and not just a fibration) and so it is useful to have a consistent way to perform these deformations that is compatible with a given local addition.

The aim of this construction is to produce, for each $v \in U$, a 1-parameter family of diffeomorphisms, $\phi_{(t,v)}$, such that $\phi_{(t,v)}(\pi(v)) = \eta(t v)$.  That is, the basepoint $p = \pi(v)$ is deformed along the path $\eta(t v)$ to $\eta(v)$, and the rest of the manifold is dragged along behind it.  (In actual fact, only a very small amount of the manifold is dragged along - although our manifolds are flexible, they have a high amount of tension.)

So let $\eta \colon U \to M$ be a local addition.  Let $\nu \colon U \to \mathbb{R}$ be a smooth function with the following fibrewise property: for each $r \in \mathbb{R}$, the set $\{q \in U_p : \nu(q) \le r\}$ is a compact, [[radial]] subset.  The square of the norm coming from a smooth orthogonal structure would suffice.  Let $\rho \colon \mathbb{R} \to [0,1]$ be a smooth function such that $\rho(t) = 1$ for $t \le 0$ and $\rho(t) = 0$ for $t \ge 1$.

Now for $v \in U$, we define a vector field $\hat{X}_v$ on $U_p$, where $p = \pi(v)$, by

$$
\hat{X}_v(w) \coloneqq \rho(\nu(w) - \nu(v))v
$$

Notice that if $\nu(w) \ge \nu(v) + 1$ then $\hat{X}_v(w) = 0$ whilst if $\nu(w) \le \nu(v)$ (so in particular if $w = t v$ for some $t \in [0,1]$) then $\hat{X}_v(w) = v$.  By the first of these, we see that $\hat{X}_v$ is compactly supported on $U_p$.  We can transfer it via $\eta_p$ to $V_p$ and, as it is compactly supported, extend it to all of $M$ by defining it to be $0$ outside $V_p$.  Let us write $X_v$ for this vector field.

By construction, the map $v \to X_v$ is smooth.  As $X_v$ is compactly supported, we can apply the exponentiation map $\mathcal{X}_c(M) \to Diff(M)$ and so obtain, for each $v \in U$, a 1-parameter family of diffeomorphisms of $M$, say $\phi_{(t,v)}$.  From the construction of $X_v$, it is clear that all the "action" of this family of diffeomorphisms takes place in $V_p$.  Moreover, by construction, the point $p$ moves along the line $t \mapsto \eta(t v)$ for $t \in [0,1]$.

## References

* Kriegl and Michor, _A Convenient Setting of Global Analysis_
{#km}
* [[Andrew Stacey]], _The Differential Topology of Loop Spaces_ [main page](http://www.math.ntnu.no/~stacey/Seminars/diffloop.html) (Note: this was designed as an "easy reader" version of [KM](#km))
* [[Andrew Stacey]], _Constructing Smooth Manifolds of Loop Spaces_ [main page](http://www.math.ntnu.no/~stacey/Research/Papers/smooth.html)


local adjunction

> This page is about a [[structure]] on [[pairs]] of [[bicategories]]. Beware that [[parametric right adjoints]] are, confusingly, also sometimes called "local right adjoints". See at *[[locally]]*.

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## Idea

Let $\mathscr{A}$ and $\mathscr{B}$ be [[bicategories]].

A **local adjunction** between $\mathscr{A}$ and $\mathscr{B}$ comprises [[lax functors]] $L \colon \mathscr{A} \to \mathscr{B}$ and $R \colon \mathscr{B} \to \mathscr{A}$ and a [[family]] of [[adjunctions]]
$$
  \mathscr{B}(L A, B)
  \rightleftarrows
  \mathscr{A}(A, R B)
$$

between [[hom-categories]], [[natural transformation|natural]] in $A$ and $B$.

## Related concepts

- [[adjunction]]

- [[2-adjunction]]

- [[biadjunction]]

- [[pseudoadjunction]]

## References

Local adjunctions are defined in:

* Colin Barry Jay. _Local adjunctions_. Journal of Pure and Applied Algebra **53** 3 (1988) 227-238 [doi:10.1016/0022-4049(88)90124-7]

[[!redirects local adjunctions]]


local algebra

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## Idea

A _local algebra over an algebraic theory_ is to an [[algebra over an algebraic theory]] as a [[local ring]] is to a [[ring]]:

a local algebra in a [[sheaf topos]] is an algebra object / [[sheaf]] of [[algebra over an algebraic theory|algebras]], which is determined by its local restrictions, for a sense of _local_ determined both by the [[Grothendieck topology]] of any [[site]] of definition of the topos, as well as by a [[coverage]] on the category of finitely presented algebras.

## Definition

Let $\mathbb{T}$ be an [[essentially algebraic theory]] and write $\mathcal{C}_{\mathbb{T}}$ for its [[syntactic category]]: the category of [[finitely presented object|finitely presented]] $\mathbb{T}$-algebras

$$
  \mathcal{C}_{\mathbb{T}}
    \simeq
  \mathbb{T}Alg^{fp}
  \,.
$$

Let $J$ be a [[coverage]] on $\mathcal{C}_{\mathbb{T}}$.

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

For $\mathcal{E}$ a [[topos]], a **$J$-local $\mathbb{T}$-algebra** in $\mathcal{E}$ is a [[functor]]

$$
  A : \mathcal{C}_{\mathbb{T}} \to \mathcal{E}
$$

that

1. preserves [[finite limit]];

1. sends $J$-[[covering]]s in $\mathcal{C}_{\mathbb{T}}$ to [[epimorphism]]s in $\mathcal{E}$.

A topos equipped with a local algebra object is a [[locally algebra-ed topos]].

=--

## Properties

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

A [[theory]] of local algebras is a [[geometric theory]] and every geometric theory is the theory of some local algebras.

=--

For the moment see [[classifying topos]] for details.

## Examples

* A [[local ring]] is a local algebra for the theory of rings.

  A topos equipped with a local ring is a [[locally ringed topos]].

## Related concepts

* [[algebraic theory]], [[essentially algebraic theory]]

* [[algebra over an algebraic theory]]

The [[(∞,1)-category theory]]-analog of a theory of local algebras is (except for the additional choice of "admissible morphisms") a

* [[geometry (for structured (∞,1)-toposes)]]

[[!redirects local algebras]]


local analytic geometry

__Local analytic geometry__ studies local properties in [[analytic geometry]].  In other words, it studies locally around a given point a space equipped with a sheaf of local topological rings of analytic functions (power series convergent in some neighborhood) over a fixed topological ground ring.

* related entries include [[Weierstrass preparation theorem]],

* Hans Grauert, Reinhold Remmert, _Analytische Stellenalgebren_

category: analysis, geometry


local and global geometry - table

| [[geometry|geometric]] context | [[gauge group]] | [[stabilizer subgroup]] | local model [[space]] | local [[geometry]] |   global [[geometry]]  |  [[differential cohomology]]  |  [[first order formulation of gravity]] |
|--|---|---|--|-----------------------|--------------------------|-------------------------------|--------------|
| [[differential geometry]] | [[Lie group]]/[[algebraic group]] $G$ | [[subgroup]] ([[monomorphism]]) $H \hookrightarrow G$ | [[quotient]] ("[[coset space]]") $G/H$ | [[Klein geometry]]    |    [[Cartan geometry]]  | [[Cartan connection]] |      |
| examples | [[Euclidean group]] $Iso(d)$ | [[rotation group]] $O(d)$ | [[Cartesian space]] $\mathbb{R}^d$ | [[Euclidean geometry]] | [[Riemannian geometry]]  |  [[affine connection]]  | [[Euclidean gravity]]     |
|  | [[Poincaré group]] $Iso(d-1,1)$ | [[Lorentz group]] $O(d-1,1)$ | [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$  | [[Lorentzian geometry]]| [[pseudo-Riemannian geometry]] | [[spin connection]] | [[Einstein gravity]] |
|  | [[anti de Sitter group]] $O(d-1,2)$ | $O(d-1,1)$ | [[anti de Sitter spacetime]] $AdS^d$ |  |  |  | [[AdS gravity]] |
|  | [[de Sitter group]] $O(d,1)$ | $O(d-1,1)$ | [[de Sitter spacetime]] $dS^d$ |  |  |  | [[deSitter gravity]] |
|       | [[linear algebraic group]] | [[parabolic subgroup]]/[[Borel subgroup]] | [[flag variety]] | [[parabolic geometry]] |  |  |  |
|       |  [[conformal group]] $O(d,t+1)$ | conformal parabolic subgroup  | [[Möbius space]] $S^{d,t}$  |  | [[conformal geometry]] | [[conformal connection]] | [[conformal gravity]] |
| [[supergeometry]] | [[super Lie group]] $G$ | [[subgroup]] ([[monomorphism]]) $H \hookrightarrow G$ | [[quotient]] ("[[coset space]]") $G/H$ | [[super Klein geometry]]    |    [[super Cartan geometry]]  | Cartan [[superconnection]] |      |
| examples      | [[super Poincaré group]]  | [[spin group]] | [[super Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$ | Lorentzian [[supergeometry]] |  [[supergeometry]]  | [[superconnection]]  | [[supergravity]] |
|     | [[super anti de Sitter group]]  | | [[super anti de Sitter spacetime]]  |  |   |   |  |
| [[higher differential geometry]] | [[smooth 2-group]] $G$ | [[2-monomorphism]] $H \to G$ | [[homotopy quotient]] $G//H$    | [[Klein 2-geometry]]   |    [[Cartan 2-geometry]] |   |      |
|  | [[cohesive]] [[∞-group]] | [[n-monomorphism|∞-monomorphism]] (i.e. any [[homomorphism]]) $H \to G$  | [[homotopy quotient]] $G//H$ of [[∞-action]]  | [[higher Klein geometry]]  |    [[higher Cartan geometry]] | [[higher Cartan connection]]  |   |
| examples |  | | [[extended superspacetime|extended super Minkowski spacetime]] |  | extended supergeometry  |    | [[higher gauge theory|higher]] [[supergravity]]: [[type II supergravity|type II]], [[heterotic supergravity|heterotic]], [[11-dimensional supergravity|11d]] |


local antibracket

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## Idea

In [[BV-BRST formalism]], the _local antibracket_ of a [[Lagrangian field theory]] is the incarnation [[antibracket]] of [[horizontal differential forms]] on the [[jet bundle]] of the [[field bundle]]. Under [[transgression of variational differential forms]] this yields the ordinary ("global") [[antibracket]] on [[polynomial observables]].

For details see _[[A first idea of quantum field theory]]_ the chapter _[Reduced phase space](A+first+idea+of+quantum+field+theory#ReducedPhaseSpace)_

## Related concepts

* [[local field theory]]

* [[BV-operator]]

## References

* {#BarnichHenneaux96} [[Glenn Barnich]], [[Marc Henneaux]], section 2 and appendix B of _Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket_, J. Math. Phys. 37 (1996) 5273-5296 ([arXiv:hep-th/9601124](https://arxiv.org/abs/hep-th/9601124))

[[!redirects local antibrackets]]

[[!redirects local anti-bracket]]
[[!redirects local anti-brackets]]


local BRST cohomology

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## Idea

Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ with [[field bundle]] $E \overset{fb}{\to} \Sigma$ over some [[spacetime]] $\Sigma$ and [[local Lagrangian density]] $\mathbf{L}$, then its _local BV-BRST complex_ (or _local BRST complex_, for short) is the realization of the [[BV-BRST complex]] not on [[local observables]] $A = \tau_{\Sigma} \alpha$ given by [[functionals]] on the [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL} = 0}$ which are [[transgression of variational differential forms|transgressions]] $\tau_{\Sigma}$ of [[variational differential forms]] $\alpha \in \Omega^{\bullet, \bullet}_\Sigma(E)$ on the jet bundle, but on these variational differential forms themselves (whence "local", i.e. before [[transgression of variational differential forms|transgression]]).

If $s$ denotes the [[BV-BRST differential]] in a BV-resolution $\Omega^{\bullet,\bullet}_\Sigma(E)\vert_{\mathcal{E}_{BV}}$ of the restriction to the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ of the [[variational bicomplex]] $\Omega^{\bullet,\bullet}_\Sigma(E)$ with its [[total spacetime derivative]] $d$ (horizontal derivative), then the _local BV-BRST cohomology_ is the [[cochain cohomology]] of $s + d$, hence of the [[total complex]] of the [[double complex]] given by $s$ and $d$.

Generally, considering [[variational differential forms]] up to $d$-exact terms is the "local" incarnation  of what under the [[integration of differential forms|integration]] involved in the [[transgression of variational differential forms|transgression]] is [[integration by parts]] and it is in this way that "local BV-BRST cohomology" knows about the actual BV-BRST cohomology on [[local observables]].

## Example

Consider local coordinates $(\phi^a)$ on the [[fibers]] of the [[field bundle]]. The corresponding [[antifield]] coordinates are to be denoted $\overline{\phi}_a$ and the [[BV-BRST differential]] takes them to the corresponding component

$$
  s(\overline{\phi}_a) = \frac{\delta_{El} L}{\delta \phi^a}
$$

of the [[Euler-Lagrange form]].

In degree $(p+1,0)$ the $s+d$-closed elements in vanishing [[ghost]] degree are [[pairs]] $(v,J_v)$ consisting of an [[infinitesimal symmetry of the Lagrangian]] $v$, regarded as an [[antifield]] [[density]] $v^a \overline{\phi}_a dvol_\Sigma$, together with a corresponding [[conserved current|conserved]] [[Noether's theorem|Noether current]] $J_v$:

$$
  \array{
    \{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \}
    \\
    && \uparrow\mathrlap{-s}
    \\
    && \{ v^a \overline{\phi}_a dvol_\Sigma\}
  }
$$

Such pairs are $(s+d)$-exact if [[on-shell]] the infintiesimal symmetry coincides with an [[infinitesimal gauge symmetry]]. To see this, recall:

An  [[infinitesimal gauge symmetry]] $v_\epsilon$  of [[gauge parameter]] $(\epsilon^\alpha)$ is a vector field on the jet bundle with components of the form

$$
  \mathcal{L}_{v_\epsilon} \phi^a
   \;\coloneqq\;
  R^a_\alpha \epsilon^\alpha
    +
  R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
$$

such that this is an [[infinitesimal symmetry of the Lagrangian]] in that

$$
  \begin{aligned}
    \iota_{v_\epsilon} \delta_{EL} \mathbf{L}
    & =
    v^a \frac{\delta_{EL} L}{\delta \phi^a} dvol_\Sigma
    \\
    & =
    \epsilon^\alpha
    \left(
       R^a_\alpha \frac{\delta_{EL} L}{ \delta \phi^a}
       -
       \frac{d}{d x^\mu}
       \left(
          R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a}
       \right)
    \right)
    dvol_\Sigma
    +
    d\left(
       \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a}
    \right)
    \iota_{\partial_\mu} dvol_\Sigma
    \\
    & =
    0 + d(...)
  \end{aligned}
$$

for all $(\epsilon^\alpha)$.

The corresponding [[antighosts]] $\overline{c}_\alpha$ are taken by the BV-BRST differential to the antifield-preimage of the term on the left:

$$
  s\left(\overline{c}_\alpha\right)
  \;=\;
  R^a_\alpha \overline{\phi}_a
  -
  \frac{d}{d x^\mu}
  \left(
    R^{a \mu}_\alpha \overline{\phi}_a
  \right)
  \,.
$$

Moreover, an [[on-shell]] vanishing [[infinitesimal symmetry of the Lagrangian]] is a vector field with components of the form

$$
  \kappa^{a b} \frac{\delta_{EL} L}{\delta \phi^a}
$$

for $\kappa^{a b} = - \kappa^{b a}$ a skew-symmetric system of smooth functions on the jet bundle.

The linear combination of such an infinitesimal gauge symmetry and an on-shell vanishing infinitesimal symmetry is $(s+d)$-exact:

$$
  \begin{aligned}
    v^a dvol_\Sigma
    & =
    \left(
      R^a_\alpha \epsilon^\alpha
      +
      R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
      +
      \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a }
    \right)
    dvol_\Sigma
    \\
    & =
    s
    \left(
      \epsilon^\alpha \overline{c}_\alpha
      -
      \tfrac{1}{2}\kappa^{a b} \overline{\phi}_a \overline{\phi}_b
    \right) dvol_\sigma
    +
    d\left(
      \epsilon^\alpha R^{a \mu}_\alpha
    \right)
    \iota_{\partial_\mu} dvol_\Sigma
  \end{aligned}
$$

([Barnich-Brandt-Henneaux 94, p. 20](#BarnichBrandtHenneaux94))

It may be useful to organize this expression into the $s+d$-[[bicomplex]] like so:

$$
  \array{
    \{K\}
     &\overset{d}{\longrightarrow}&
     \{ d K
       +
     \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{ \delta \phi^a}
     \}
     &\overset{d}{\longrightarrow}&
    \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \}
    \\
    && \mathllap{s}\uparrow && \uparrow\mathrlap{-s}
    \\
    &&
    \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a
    \iota_{\partial_\mu} dvol_\Sigma
      &\underset{d}{\longrightarrow}&
    \left\{
      d\left(
        \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a
      \right)
      \iota_{\partial_\mu} dvol_\Sigma
      +
      \left(
        R^a_\alpha \epsilon^\alpha
        +
        R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
        +
        \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a }
      \right)
      \overline{\phi}_a
      \,
      dvol_\Sigma
    \right\}
    \\
    && && \uparrow\mathrlap{-s}
    \\
    &&
    &&
    \left(
       - \epsilon^\alpha \overline{c}_\alpha
       +
      \tfrac{1}{2}\kappa^{a b } \overline{\phi}_a \overline{\phi}_b
    \right)
    dvol_\Sigma
  }
$$

## Related concepts

* [[variational BV-BRST bicomplex]]

## References
 {#References}

The general theory:

* {#BarnichBrandtHenneaux94} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in the antifield formalism: I. General theorems_, Commun. Math. Phys. **174** (1995) 57-92 [[arXiv:hep-th/9405109](https://arxiv.org/abs/hep-th/9405109), [doi:10.1007/BF02099464](https://doi.org/10.1007/BF02099464)]

Review:

* {#BarnichBrandtHenneaux00} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in gauge theories_, Phys. Rept. **338** (2000) 439-569 [doi:10.1016/S0370-1573(00)00049-1[arXiv:hep-th/0002245](https://arxiv.org/abs/hep-th/0002245)]

Details on the [[local antibracket]]:

* {#BarnichHenneaux96} [[Glenn Barnich]], [[Marc Henneaux]], section 2 and appendix B of: _Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket_, J. Math. Phys. **37** (1996) 5273-5296 [[arXiv:hep-th/9601124](https://arxiv.org/abs/hep-th/9601124), [doi:10.1063/1.531726](https://doi.org/10.1063/1.531726)]

Application to [[gravity]] and/or [[Yang-Mills theory]] ([[Einstein-Yang-Mills theory]]) is discussed in

* [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in Einstein-Yang-Mills theory_, Nucl. Phys. B **455** (1995) 357-408 [doi:10.1016/0550-3213(95)00471-4[arXiv:hep-th/9505173](https://arxiv.org/abs/hep-th/9505173)]

[[!redirects local BV-BRST cohomology]]

[[!redirects local BRST complex]]
[[!redirects local BRST complexes]]

[[!redirects local BV-BRST complex]]
[[!redirects local BV-BRST complexes]]

[[!redirects local BV complex]]
[[!redirects local BV complexes]]

[[!redirects local BV-complex]]
[[!redirects local BV-complexes]]

[[!redirects local BRST differential]]
[[!redirects local BRST differentials]]

[[!redirects local BV-BRST differential]]
[[!redirects local BV-BRST differentials]]

[[!redirects local BV-cohomology]]
[[!redirects local BV cohomology]]


local category

# Local categories
* table of contents
{: toc}

## Idea and warning

Here we study the notion of local category and a different notion of local Grothendieck category. Local Grothendieck category is not simply Grothendieck
category which is local in above sense, but has a different definition. However, both notions have the category of modules over a local commutative unital ring as the main example.

## Definitions

(RosenbergSpectraNSp 1.1) A category $C$ is __local__ if the full subcategory generated by all objects which are not initial, has itself an initial object. In particular, every local category has initial objects.

The center of a local category $C$
which is Abelian (that is the ring of
endomorphisms of the identity functor of $C$)
is a commutative [[local ring]].

(RosenbergSpectraNSp 1.2) The __local spectrum__ of an arbitrary small category $C$ is the full subcategory $\mathfrak{Spec}^1(C)\hookrightarrow C$ whose objects are all $x$ in $Ob C$ such that the [[undercategory]] $x\backslash C$ is local.

(Popescu 4.20) A __local Grothendieck category__ is a Grothendieck category (= Ab5-category with a generator) $A$
having a simple object $S$ whose [[injective envelope]] $E(S)$
is a cogenerator of $A$.

Let $R$ be a unital ring, and ${}_R Mod$ the category of left $R$-modules. Following (Goldman 1969), define a __left prime__ (left prime system) a [[Gabriel filter]] such that the corresponding Gabriel localization is a local Grothendieck category. All left primes form another spectrum, $Spel(R)$. Regarding that Gabriel filters of left ideals correspond to localizing subcategories of the category of left modules, it is obvious how to generalize this.
Following Popescu, a [[localizing subcategory]] $T$ of any
Grothendieck category $A$ is a __prime localizing subcategory__ if the Serre quotient category $A/T$ is local Grothendieck category. Prime localizing subcategories form
certain spectrum.

## Literature

* (RosenbergSpectraNSp) A. L. Rosenberg, _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945) (2003)
* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375
* N. Popescu, _Le spectre a gauche d'un anneau_, J. Algebra __18__ (1971), 213-228.
* (Goldman 1969) O. Goldman, _Rings and modules of quotients_, J. Algebra __13__, 1969 10--47, [MR245608](http://www.ams.org/mathscinet-getitem?mr=245608), [doi](http://dx.doi.org/10.1016/0021-8693%2869%2990004-0)

category: algebra

[[!redirects local category]]
[[!redirects local categories]]

[[!redirects local Grothendieck category]]
[[!redirects local Grothendieck categories]]


local coefficient bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[twisted cohomology]] a [[cocycle]] on a [[space]] $X$ with [[coefficients]] in a coefficient object $V$ is not quite a direct morphism $X \to V$ as in ordinary $G$-[[cohomology]], but is instead a [[section]] of a $V$-[[fiber ∞-bundle]] $E \to X$ over $X$. This is called the _local coefficient bundle_ for the given twisted cohomology. Its class $[E] \in H^1(X, \mathbf{Aut}(V))$ is the _twist_.

## Related concepts

* [[twisted cohomology]], [[local system of coefficients]]

* [[twisted infinity-bundle]]

## References

Local coefficient bundles in the context of [[twisted ordinary cohomology]]:

* [[Norman Steenrod]], _Homology With Local Coefficients_, Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627 ([jstor:1969099](https://www.jstor.org/stable/1969099))

* M. Bullejos, E. Faro, M. A. García-Muñoz, _Homotopy colimits and cohomology with local coefficients_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 ([numdam:CTGDC_2003__44_1_63_0](http://www.numdam.org/item?id=CTGDC_2003__44_1_63_0))

* [[Paul Goerss]], [[Rick Jardine]], p. 331 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999)  Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4))

[[!redirects local coefficients]]

[[!redirects local coefficient bundles]]

[[!redirects local coefficient ∞-bundle]]
[[!redirects local coefficient ∞-bundles]]

[[!redirects local coefficient infinity-bundle]]
[[!redirects local coefficient infinity-bundles]]


local colimit

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

# Local colimits

* table of contents
{: toc}

## Definition

A **local colimit** in a [[bicategory]] is a [[colimit]] in a [[hom-category]] that is [[preserved limit|preserved]] by the [[composition]] [[functor]].  A bicategory **has local colimits** of some shape if its hom-categories have colimits of those shapes that are preserved in each variable by the composition functors.  A bicategory with all (small) local colimits is called **locally cocomplete**.

## Examples

* In the [[delooping]] $\mathbf{B} V$ of a [[monoidal category]] $V$, any colimit preserved in each variable by the tensor product gives a local colimit.  In particular, in the delooping of a [[closed monoidal category]] all colimits are local colimits.
* More generally, in a [[closed bicategory]] all colimits in hom-categories are local colimits.
* If $B$ has local coproducts, then we can construct its bicategory $Mat(B)$, whose objects are families of objects of $B$ and whose morphisms are  [[matrices]], which also has local coproducts.
* If $B$ has local coequalizers, then we can construct its bicategory $Mod(B)$, whose objects are [[monads]] in $B$ and whose morphisms are [[bimodules]], which also has local coequalizers.
* Thus, if $B$ is locally cocomplete, we can construct its bicategory $Prof(B) = Mod(Mat(B))$, whose objects are [[category enriched in a bicategory|categories enriched in]] $B$ and whose morphisms are profunctors, which is also locally cocomplete.  In particular, [[Prof]] = $Prof(\mathbf{B} Set)$ is locally cocomplete.

[[!redirects local colimits]]
[[!redirects bicategory with local colimits]]
[[!redirects bicategories with local colimits]]
[[!redirects locally cocomplete bicategory]]
[[!redirects locally cocomplete bicategories]]


local diffeomorphism

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[smooth function]] $f : X \to Y$ between two [[smooth manifold]]s is a **local diffeomorphism** if the following equivalent conditions hold

* $f$ is both a [[submersion]] and an [[immersion]];

* for each point $x \in X$ the [[derivative]] $d f : T_x X \to T_{f(x)} Y$ is an [[isomorphism]] of [[tangent space|tangent]] [[vector space]]s;

* the canonical diagram

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

  (with the [[differential]] between the [[tangent bundles]]) on top is a [[pullback]];

* for each point $x \in X$ there exists an [[open subset]] $x \in U \subset X$ such that

  1. the [[image]] $f(U)$ is an [[open subset]] in $Y$;

  1. $f$ restricted to $U$ is a [[diffeomorphism]] onto its image

     $$
       f|_U : U \stackrel{\simeq}{\to}  f(U)
     $$

=--

The equivalence of the conditions on [[tangent space]] with the conditions on [[open subset]]s follows by the [[inverse function theorem]].

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

An analogous characterization of [[étale morphisms]] between [[affine algebraic varieties]] is given by [[tangent cones]]. See there.

=--

## Properties

### General

* A local diffeomorphism is a [[local homeomorphism]] and so also an [[open map]]. By the [[inverse function theorem]].

### Abstract characterization

The [[category]] [[SmoothMfd]] of [[smooth manifold]]s may naturally be thought of as sitting inside the more general context of the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. This is canonically equipped with a notion of [[differential cohesion]] exhibited by its inclusion into [[SynthDiff∞Grpd]]. This implies that there is an intrinsic notion of [[formally étale morphism]]s of smooth $\infty$-groupoids in general and of smooth manifolds in particular

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

A [[smooth function]] is a [[formally étale morphism]] in this sense precisely if it is a local diffeomorphism.

=--

See this section for more details.

## Related concepts

* [[immersion of smooth manifolds|immersion]], [[submersion]]

## References

Discussion in the [[synthetic differential geometry]] of the [[Cahiers topos]] is in

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], around def. 3.1 of_[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

[[!redirects local diffeomorphisms]]


local epimorphism

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[site]] $\mathcal{S}$, then a _local epimorphism_ is a [[morphism]] in the [[category of presheaves]] over the site which becomes an [[epimorphism]] under [[sheafification]].

More abstractly, for $\mathcal{S}$ a [[small category]], one says axiomatically that a system of _local epimorphisms_ is a system of [[morphisms]] in the [[presheaf]] category $[S^{op}, Set]$ that has the closure properties expected of [[epimorphisms]] under [[composition]] and under [[pullback]].

There is then a unique [[Grothendieck topology]] on $\mathcal{S}$ that induces this system of local epimorphism, see _[Relation to sieves](#RelationToSieves)_ below.

Moreover the [[local isomorphisms]] among the local epimorphisms admit a [[calculus of fractions]] which equips the [[category of presheaves]] with the structure of a [[category with weak equivalences]]. The corresponding [[reflective localization]] is the [[category of sheaves]] on the site $\mathcal{S}$.

## Definition

+-- {: .num_defn #SystemOfLocalEpimorphisms}
###### Definition
**(system of local epimorphisms)**

Let $\mathcal{S}$ be a [[small category]]. A _system of local epimorphisms_ on the [[presheaf category]] $[\mathcal{S}^{op}, Set]$ is a [[class]] of [[morphisms]] satisfying the following axioms:

**LE1** every [[epimorphism]] in $[\mathcal{S}^{op}, Set]$ is a local epimorphism;

**LE2** the [[composition|composite]] of two local epimorphisms is a local epimorphism;

**LE3** if the [[composition|composite]] $A_1 \stackrel{u}{\to} A_2 \stackrel{v}{\to} A_3$ is a local epimorphism, then so is $v$;

**LE4** a morphism $u \colon A \to B$ is a local epimorphism precisely if for all $U \in \mathcal{S}$ (regarded as a [[representable presheaf]]) and morphisms $y: U \to B$, the [[pullback]] morphism $A \times_B U \to U$ is a local epimorphism.

=--

## Properties

### Relation to sieves
 {#RelationToSieves}

The specification of a system of local epimorphisms is equivalent to a system of [[Grothendieck topology|Grothendieck covering]] [[sieves]].

To see this, translate between local epimorphisms to sieves as follows.

Throughout, let $\mathcal{S}$ be a [[small category]]. Write $[\mathcal{S}^{op}, Set]$ for its [[category of presheaves]] and write

$$
  y \;\colon\; \mathcal{S} \longrightarrow [\mathcal{S}^{op}, Set]
$$

for the [[Yoneda embedding]].

+-- {: .num_defn #LocalEpimorphismFromGrothendieckTopology}
###### Definition
**(local epimorphisms from [[Grothendieck topology]])**

Let the [[small category]] $\mathcal{S}$ be equipped with a [[Grothendieck topology]].

For $U \in \mathcal{S}$ an object in the site, a [[morphism]] of [[presheaves]] into the corresponding [[representable presheaf|represented presheaf]]

$$
  A \overset{f}{\longrightarrow} y(U)
  \;\;\;
  \in [\mathcal{S}^{op}, Set]
$$

is a _local epimorphism_ if the [[sieve]]

$$
  sieve_A \subset y(U) \in [S^{op}, Set]
$$

at $U$ which assigns to $V$ all morphisms from $V$ to $U$ that factor through $f$

$$
  sieve_f
    \;\colon\;
  V
   \;\mapsto\;
  \left\{  V \overset{g}{\to} U \,\in \mathcal{S}
      \;\Big\vert\;
      \array{
        && A
        \\
        & {}^{\mathllap{\exists}}\nearrow & \Big\downarrow{}^{f}
        \\
        y(V) & \underset{y(g)}{\longrightarrow} & y(U)
      }
  \right\}
$$

is a [[covering sieve]].

A general morphism of presheaves

$$
  A \overset{}{\longrightarrow} B \;\;\; \in [\mathcal{S}^{op}, Set]
$$

is a _local epimorphism_ if for every $U \in \mathcal{S}$ and
every $y(U) \to B$ the [[projection]] morphism  $y(U) \times_{B} A \overset{p_1}{\to} y(V)$ out of the [[pullback]]/[[fiber product]]

$$
  \array{
    y(U)\times_{B} A
      &\overset{}{\longrightarrow}&
    A
    \\
    {}^{\mathllap{p_1}}\Big\downarrow
    &{}^{(pb)}& \Big\downarrow{}^{\mathrlap{f}}
    \\
    y(U) &\underset{}{\longrightarrow}& B
  }
$$

is a local epimorphism as above. By the [[universal property]] of the [[fiber product]], this means equivalently that

$$
  sieve_f
    \;\colon\;
  V
   \;\mapsto\;
  \left\{  V \overset{g}{\to} U \,\in \mathcal{S}
      \;\Big\vert\;
      \array{
        y(V) &\overset{\exists}{\longrightarrow}& A
        \\
        {}^{\mathllap{g}}\Big\downarrow & & \Big\downarrow{}^{f}
        \\
        y(U) & \underset{}{\longrightarrow} & B
      }
  \right\}
$$

is a [[covering sieve]].

=--

+-- {: .num_remark #InTermsOfCoverages}
###### Remark
**(in terms of [[coverages]])**

If instead of a [[Grothendieck topology]] we are just given a [[coverage]], then Def. \ref{LocalEpimorphismFromGrothendieckTopology} becomes:

$$
  A \overset{f}{\longrightarrow} B
  \;\;\; \in
  [\mathcal{S}^{op}, Set]
$$

is a local epimorphism, if for all $y(U) \longrightarrow B$ there is a [[covering]] $\{ V_i \overset{\iota_i}{\longrightarrow} U \}$ in the coverage, such that for each $i$ there exists a lift

$$
  \array{
    y(V_i) &\overset{\exists}{\longrightarrow}& A
    \\
    {}^{\mathllap{\iota_i}}\Big\downarrow & & \Big\downarrow{}^{f}
    \\
    y(U) & \underset{}{\longrightarrow} & B
  }
$$

=--

+-- {: .num_defn #GrothendieckTopologyFromLocalEpimorphisms}
###### Definition
**(Grothendieck topology from local epimorphisms)**

Conversely, assume a system of local epimorphisms as in Def. \ref{SystemOfLocalEpimorphisms} is given.

Declare a [[sieve]] $F$ at $U$ to be a [[covering sieve]] precisely if the inclusion morphism $F \hookrightarrow U$ is a local epimorphism. Then this defines a [[Grothendieck topology]] encoded by the collection of local epimorphisms.

=--

### Relation to simplicial presheaves

+-- {: .num_prop #CechNerveProjectionOfLocalEpimorphismIsLocalWeakEquivalence}
###### Proposition
**([[Cech nerve]] [[projection]] of [[local epimorphism]] is [[local weak equivalence]])**

For $\mathcal{S}$ a [[site]], let

$$
  A \overset{f}{\longrightarrow} B
  \;\colon\;
  [\mathcal{S}^{op}, Set]
$$

be a local epimorphism (Def. \ref{LocalEpimorphismFromGrothendieckTopology}). Then the projection

$$
  C(f) \longrightarrow B
  \;\;\;\;
  \in [\mathcal{S}^{op}, sSet]
$$

out of the [[Cech nerve]] [[simplicial presheaf]]

$$
  C(f)_k \;\coloneqq\;
  \underset{
     k \; \text{factors}
  }{
  \underbrace{
    A \times_B \cdots \times_B A
  }}
$$

is a [[weak equivalence]] in the projective local [[model structure on simplicial presheaves]] $[\mathcal{S}^{op}, sSet_{Qu}]_{proj,loc}$.

=--

([Dugger-Hollander-Isaksen 02, corollary A.3](#DuggerHollanderIsaksen02))

$\,$

## References

* Kashiwara-Schapira,  section 16 of _[[Categories and Sheaves]]_

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], chapter III, section 8 of _[[Sheaves in Geometry and Logic]], Springer 1992

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1. Cambridge University Press, 2004 ([arXiv:math/0205027](https://arxiv.org/abs/math/0205027))

[[!redirects local epimorphisms]]
[[!redirects local epi]]
[[!redirects local epis]]

[[!redirects generalized cover]]
[[!redirects generalized covers]]


local equivalence

In the context of [[localization of an (∞,1)-category]] or the corresponding $1$-category-theoretic [[Bousfield localization]], _local equivalences_ are those morphisms that are seen as equivalences by [[local object]]s.

More concretely, let $S$ be a subset of morphisms. Recall that an $S$-[[local object]] $X$ is one such that for all $s : A \to B$ in $S$ the induced morphism

$$
  Hom(s,X) : Hom(B,X) \to Hom(A,X)
$$

is an equivalence.

Conversely, a morphism $f : V \to W$ is is an $S$-**local equivalence** if for every $S$-[[local object]] $X$ the induced morphism

$$
  Hom(f,X) : Hom(W,X) \to Hom(V,X)
$$

is an equivalence.

In the context of [[simplicial model category|simplicial model categories]]
"equivalence" means: [[model structure on simplicial sets|weak equivalence]] of simplicial sets.

#References#

The model category theoretic notion is discussed in section A.3.7 of

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

[[!redirects local equivalences]]

[[!redirects local weak equivalence]]
[[!redirects local weak equivalences]]


local fibration

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

#Contents#
* table of contents
{:toc}

## Idea

In a category of [[simplicial presheaves]] it makes sense to ask if a morphism is _locally_ a [[Kan fibration]], in that the required lifting property holds after refinement along some [[cover]].

These _local fibrations_ are typically not the fibrations in any of the [[model structures on simplicial presheaves]]. (They may instead form the fibrations in the structure of a [[category of fibrant objects]] on simplicial sheaves.) Nevertheless, it is useful to consider them, and they satisfy various properties otherwise known from genuine fibrations.

For instance a pullback diagram of simplicial presheaves is a [[homotopy pullback]]  already if one of the two morphisms in the cospan is a local fibration (e.g. [Jardine, lemma 5.16](#Jardine))

## References

* [[Rick Jardine]], _Local homotopy theory_ (2011) ([pdf](http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf))

[[!redirects local fibrations]]


local field

> This article concerns the notion of "local field" as it is commonly used in [[algebraic number theory]]. For another notion of "local field" in [[commutative algebra]], see [[local field (commutative algebra)]].

---

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **local field** is a [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] (non-[[discrete space|discrete]]) [[topological field]].

Basic examples are the [[p-adic numbers]] $\mathbb{Q}_p$ and the field of [[Laurent series]] $\mathbb{F}_q((t))$ over a [[finite field]] $\mathbb{F}_q$. Local fields are opposite to _[[global fields]]_ in that where (under the [[function field analogy]]) the latter may be thought of as fields of [[rational functions]] on [[arithmetic curves]], local fields are like fields of functions on [[formal disks]] inside such curves. Accordingly the [[Langlands correspondence]] for [[global fields]] has a "localization" to the [[local Langlands correspondence]] for local fields.

Note that for a [[topological field]], the [[closed subspace|topological closure]] of $\{0\}$ is an [[ideal]], which must therefore be either $\{0\}$ or the whole field. It follows that either a topological field is [[separation axioms|T]]$_1$ (and therefore Hausdorff or T$_2$; see [[uniform space]]), or has the [[discrete and codiscrete topology|codiscrete topology]].

## Properties

A local field $K$ carries a [[valuation]] ${\|-\|}: K \to \mathbb{R}_{\geq 0}$ defined by

$${\|a\|} = \frac{\mu(a X)}{\mu(X)}$$

where $\mu$ is any [[Haar measure]] defined on the underlying locally compact Hausdorff additive group of $K$, and $X$ is any set such that $0 \lt \mu(X) \lt \infty$.

By analyzing the possibilities for the valuation, any local field is one of the following types:

* [[characteristic|Characteristic zero]]. In this case local fields $F$ are [[complete space|completions]] of [[number fields]] with respect to [[metrics]] induced by [[valuations]]. The valuations may be

  * [[archimedean field|Archimedean]]. Here for every $x \in F$, there exists $n \in \mathbb{N}$ such that ${\|n x\|} \gt 1$, where ${\| \cdot \|}$ is the valuation. The local fields in this case are isomorphic as topological fields to $\mathbb{R}$ or $\mathbb{C}$.

  * [[nonarchimedean field|Nonarchimedean]]. Such valuations are [[discrete valuations]], and are the completions of discrete valuations induced by prime ideals $v$ of the ring of [[algebraic integers]] $\mathcal{O}_k$ in a number field $k$. The valuation on the [[number field]] is defined by ${\|x\|_v} = q^{-n}$ where $q$ is the cardinality of the [[finite field]] $\mathcal{O}_k/v$, and $n$ is the least integer such that $x \in v^n$. The completion is called the **$v$-adic completion** and is denoted $k_v$.

* [[characteristic|Characteristic]] $p \gt 1$. In this case local fields are fields of [[Laurent series]] $\mathbb{F}_q((t))$ over a finite field $\mathbb{F}_q$ of cardinality $q = p^n$; here ${\|f(t)\|} = q^{-n}$ where $f(t) = a_n t^n + a_{n+1}t^{n+1} + \ldots$. The valuation is nonarchimedean.

Local fields are technically useful in modern [[number theory]]; for example in formulating local-to-global principles, and in formulations of [[class field theory]] following Tate's thesis. Part of the technical convenience resides in the fact that one can effectively do [[Fourier analysis]] on them; as additive [[topological abelian group|topological groups]], they are self-dual locally compact abelian groups (in the sense of [[Pontrjagin dual|Pontryagin duality]]).

## Relation to local rings (warning)

It is possible to construe "local field" in at least two other ways, to wit:

* As meaning "[[field of fractions]] of an [[integral domain]] that is a [[local ring]]".

* As meaning "field of fractions of an integral domain that arises as the completion of a local ring with respect to its canonical valuation".

The first meaning is not too serious (and is seldom if ever considered seriously), since usually a field $F$ will not uniquely determine a local subring giving rise to it, nor does this meaning imply any tight connection to local topological conditions such as local compactness. Under this interpretation, $\mathbb{Q}$ would be a "local field", which is virtually unheard of.

The second meaning has more content, because the Cauchy completeness (with respect to an $\mathfrak{m}$-topology, where $\mathfrak{m}$ is the maximal ideal of some local ring) determines the local ring via the topology: the complement of $x$ such that $x^{-n}$ converges to $0$. There is nontrivial intersection with the notion of local field as defined above, since the _nonarchimedean_ local fields as defined above are conspicuous examples of this second meaning. Observe however that

* The archimedean local fields $\mathbb{R}$, $\mathbb{C}$ do _not_ arise this way;

* Under the $m$-[[adic topology]], the completion of a local ring $R$ with maximal ideal $m$, i.e., the [[inverse limit]] of the [[diagram]]
$$\ldots R/m^{n+1} \stackrel{proj}{\to} R/m^n \to \ldots \to R/m$$
is typically not compact (and its field of fractions is not locally compact). It is of course compact if each $R/m^n$ is finite with the discrete topology.

In any case, the second meaning certainly occurs in the literature, as in the famous text _Corps Loceaux_ by Serre. For more on this, see [[local field (commutative algebra)]].

## Related concepts

* [[local Langlands correspondence]]

* [[higher local field]]

* [[global field]]

## References

* {#Cassels86} [[J. W. S. Cassels]], *Local Fields*, Cambridge University Press, 1986 (ISBN:9781139171885, [doi:10.1017/CBO9781139171885](https://doi.org/10.1017/CBO9781139171885))

See also:

* Wikipedia, _[Local field](http://en.wikipedia.org/wiki/Local_field)_

[[!redirects local field]]
[[!redirects local fields]]


local field (commutative algebra)

> This article concerns the notion of "local field" in [[commutative algebra]]. For the notion of "local field" in [[algebraic number theory]], see [[local field]].

---

\tableofcontents

## Idea

In [[commutative algebra]], there are two notions of "local field", to wit:

* As meaning "[[field of fractions]] of an [[integral domain]] that is a [[local ring]]".

* As meaning "field of fractions of an integral domain that arises as the completion of a local ring with respect to its canonical valuation".

The first meaning is not too serious (and is seldom if ever considered seriously), since usually a field $F$ will not uniquely determine a local subring giving rise to it, nor does this meaning imply any tight connection to local topological conditions such as local compactness. Under this interpretation, $\mathbb{Q}$ would be a "local field", which is virtually unheard of.

The second meaning of a local field has more content, because the Cauchy completeness (with respect to an $\mathfrak{m}$-topology, where $\mathfrak{m}$ is the maximal ideal of some local ring) determines the local ring via the topology: the complement of $x$ such that $x^{-n}$ converges to $0$. The second meaning occurs in the literature, as in the famous text _Corps Loceaux_ by Serre.

## Relation to local fields in algebraic number theory

There is nontrivial intersection with the notion of [[local field]] as defined in algebraic number theory, since the _nonarchimedean_ local fields as defined in algebraic number theory are conspicuous examples of this second meaning. Observe however that

* The archimedean local fields in algebraic number theory $\mathbb{R}$, $\mathbb{C}$ do _not_ arise this way;

* Under the $m$-[[adic topology]], the completion of a local ring $R$ with maximal ideal $m$, i.e., the [[inverse limit]] of the [[diagram]]
$$\ldots R/m^{n+1} \stackrel{proj}{\to} R/m^n \to \ldots \to R/m$$
is typically not compact (and its field of fractions is not locally compact). It is of course compact if each $R/m^n$ is finite with the discrete topology.

## See also

* [[local ring]]

* [[valuation]]

* [[integral domain]]

* [[field of fractions]]

## References

* [[Jean-Pierre Serre]], *Corps Locaux*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237


Local field theory via Higher correspondences

> this is a sub-chapter of _[[geometry of physics]]_, see there for background, context, and for all the other chapters

## References

* _[[schreiber:Classical field theory via Cohesive homotopy types]]_

* section 1.2 of _[[schreiber:differential cohomology in a cohesive topos]]_


local formal group scheme

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

A $k$-formal group $G=Spf A$ is called  *local* (or *connected*) if the following two equivalent conditions hold:

1. $A$ is local

1. $G(k)=\{0\}$ for any field $K$.

=--

A morphism from a connected group to an étale group is zero.


local geometric morphism

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

#Contents#
* table of contents
{:toc}

## Definition

### Local geometric morphism and relative local topos

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

A **local geometric morphism** $f : E \to S$ between [[topos]]es $E,S$ is

*  a [[geometric morphism]]

   $$
    (f^* \dashv f_*) \colon E \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}
    {\to}}  S
   $$

*  such that a further [[right adjoint]] $f^! : S \to E$ exists

   $$
    (f^* \dashv f_* \dashv f^!) :
    E
    \stackrel{\overset{f^*}{\longleftarrow}}{\stackrel{\underset{f_*}{\longrightarrow}}{\underset{f^!}{\longleftarrow}}}
    S
   $$

*  and such that one, hence all, of the following equivalent conditions hold:

   1. The right adjoint $f^!$ is an $S$-[[indexed functor]].
   1. $f$ is [[connected geometric morphism|connected]], i.e. $f^*$ is [[full and faithful functor|fully faithful]].
   1. The right adjoint $f^!$ is fully faithful.
   1. The right adjoint $f^!$ is [[cartesian closed functor|cartesian closed]].

=--

When we regard $E$ as a topos over $S$, so that $f$ is regarded as its [[global section]] geometric morphism in the category of toposes over $S$, then we say that $E$ is a **local $S$-topos**.  In this case we may label the functors involved as

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

to indicate that if we think of $\Gamma$ as sending a [[space]] to its underlying $S$-object of points by forgetting [[cohesive topos|cohesion]], then $Disc$ creates the [[discrete space]]/[[discrete object]] and $Codisc$ the [[codiscrete space]]/[[codiscrete object]] on an object in $S$.

This is especially common when $S=$ [[Set]], in which case the final condition is automatic since all functors are $Set$-indexed. Hence in that case we have the following simpler [definition](local).

### Local topos
 {#LocalTopos}

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

A [[sheaf topos]] $\mathcal{T}$ is a **local topos** if the [[global section]] [[geometric morphism]] $\mathcal{T} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Set$ has a further [[right adjoint]] $CoDisc$, making an [[adjoint triple]]  $(LConst \dashv \Gamma \dashv CoDisc)$

$$
  CoDisc \colon Set \hookrightarrow \mathcal{T}.
$$

(As just stated, it is automatic in the case over $Set$ that this is furthermore a [[full and faithful functor]].)

=--

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

Another way of stating this is that a Grothendieck topos is local if and only if the terminal object $1$ is [[connected object|connected]] and [[projective object|projective]] (since this means precisely that $\Gamma = \hom(1, -)$ preserves colimits, and therefore has a right adjoint by virtue of an [[adjoint functor theorem]]). Another term for this: we say $1$ is _[[tiny object|tiny]]_ ([[atomic object|atomic]]). Notice the similarity to the concept of _[[amazing right adjoint]]_ (the difference being that this is a right adjoint not to the external but to the [[internal hom]] out of 1.)

=--

Note also that in an infinitary-[[extensive category]] an object is connected as soon as $\hom(X,-)$ preserves *binary* coproducts (see [[connected object]]).  Moreover, a coproduct-preserving functor between toposes preserves coequalizers as soon as it preserves epimorphisms, since any coequalizer can be constructed as the quotient of an equivalence relation generated using images and countable coproducts and quotients of equivalence relations are effective.  Thus, we can say that a Grothendieck topos is local iff $\hom(1,-)$ preserves binary coproducts and epis.  Moreover, a cospan $A \to C \leftarrow B$ in a topos is a coproduct diagram iff $A$ and $B$ are disjoint monos whose union is all of $C$; thus $\hom(1,-)$ preserves binary coproducts as soon as it preserves the initial object and binary unions.  This leads to the following equivalent form of "locality" that makes sense even for elementary toposes:

+-- {: .num_defn}
###### Definition
An [[elementary topos]] $E$ is *local* if the terminal object $1$ is

* nonempty: $1\ncong 0$,

* connected: $1 = p \vee q$ implies $1 = p$ or $1 = q$, and

* projective: every epi $U \to 1$ admits a section $t: 1 \to U$.
=--

+-- {: .num_remark}
###### Remark
Some authors have instead used the term "local" to refer just to the condition that $1$ is connected; note this is equivalent to $Sub_E(1)$ being a [[local ring|local rig]]. In his [thesis](#Awodey), Steve Awodey upgraded this, saying that a topos is *hyperlocal* if both connectivity and projectivity are satisfied. As argued above, for a Grothendieck topos this is slightly *weaker* than locality, although the difference is only in the inclusion of the trivial topos (for if $1\cong 0$ then $E\simeq 1$).  Note that a local elementary topos, as defined above, is [[well-pointed topos|constructively well-pointed]] if and only if $1$ is additionally a [[generator]].
=--

+-- {: .num_example}
###### Example
The [[free topos]] is a local elementary topos.
=--

## Properties

### Equivalent characterizations
 {#EquivalentCharacterizations}

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

A geometric morphism $f : E \to S$ is local precisely if

1. there exists a geometric morphism $c : S \to E$ such that $f \circ c \simeq id$;

1. for every other geometric morphism $g : G \to S$ the composite $c\circ g$ is an [[initial object]] in the hom-category $Topos_{/S}(g,f)$ of the [[slice category|slice]] [[2-category]]  of [[Topos]] over $S$.

=--

This is ([Johnstone, theorem 3.6.1 vi)](#Johnstone)).

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

In particular this means that the category of [[topos points]] of a local topos has a [[contractible]] [[nerve]].

=--

### General

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

The [[global section]] geometric morphism of any local $\mathcal{S}$-topos (over a [[base topos]] $\mathcal{S}$) is a [[Grothendieck fibration]] and a [[Grothendieck opfibration]].

=--

This appears in ([Shulman](#Shulman)).

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

The [[Freyd cover]] of a [[topos]] is a [[local topos]], and in fact freely so. Every local topos is a [[retract]] of a Freyd cover.

=--

This appears as ([Johnstone, lemma C3.6.4](#Johnstone)).

### Homotopy dimension
 {#HomotopyDimension}

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

In a local [[sheaf topos]] over [[Set]], every [[inhabited object]] is globally inhabited:

every object $X$ for which the unique morphism $X \to *$ to the [[terminal object]] is an [[epimorphism]] has a [[global point]] $* \to X$.

=--

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

Since in a local topos the [[global section]] functor $\Gamma$ is a [[left adjoint]], it preserves [[epimorphisms]]. Since it is a right adjoint it preserves the terminal object. Therefore $\Gamma(X) \to \Gamma(*) \simeq *$ is an epimorphism in [[Set]], hence a [[surjection]], meaning that $\Gamma(X)$ is [[inhabited]]. Since $\Gamma(X) \simeq Hom(*,X)$ (see [[global section geometric morphism]]), the claim follows.

=--

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

In a [[topos]] every [[epimorphism]] is an [[effective epimorphism]]. Therefore $X \to *$ being an epi means that $X$ is a [[(-1)-connected]] object. Therefore the above statement says in terms of [[(infinity,1)-category theory]] that a non-trivial local topos has [[homotopy dimension]] 0.

The same is true for any [[local (infinity,1)-topos]].

=--

### Concrete sheaves

Every local topos $\Gamma : E \to S$ comes with a notion of [[concrete sheaves]], a [[reflective subcategory]] $Conc_\Gamma(E) \hookrightarrow E$ which factors the topos inclusion of $S$:

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

and is a [[quasitopos]]. See [[concrete sheaf]] for details.

### Homotopy equivalence

Since a local geometric morphism has a [[left adjoint]] in the [[2-category]] [[Topos]], it is necessarily a [[homotopy equivalence of toposes]].

## Elementary Axiomatization
 {#ElementaryAxiomatization}

For any local topos $\Gamma \colon \mathcal{E} \to \mathcal{S}$, the base topos $\mathcal{S}$ is equivalent to the [[category of sheaves]] for a [[Lawvere-Tierney topology]] $j$ on $\mathcal{E}$.  A sound and complete elementary [[axiomatization]] of local maps of (bounded) toposes can be given in terms of properties of topos $E$ and topology $j$ ([AwodeyBirkedal](#AwodeyBirkedal))

We discuss first

* [the setup](#SetupForAxiomatization)

and then

* [the axioms](#TheAxioms)

themselves.

### The setup
 {#SetupForAxiomatization}

Let $\mathcal{E}$ be an [[elementary topos]] equipped with a [[Lawvere-Tierney topology]] $j : \Omega \to \Omega$.

Write $V \mapsto \sharp  V$ for the $j$-[[closure operation]] on [[subobjects]] $V \hookrightarrow X$, the _[[sharp modality]]_

$$
  \array{
    V &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    \sharp  V &\to& &\to& {*}
    \\
    \downarrow && \downarrow  &&  \downarrow
    \\
    X &\stackrel{\chi_V}{\to}& \Omega
    &\stackrel{j}{\to}&
    \Omega
  }
$$

Write

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

for the [[reflective subcategory]] [[category of sheaves|of j-sheaves]].

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

We say that $j$ is an **[[essential geometric morphism|essential]] topology** if for all objects $X$ the closure operation $\sharp  : Sub(X) \to Sub(X)$ on [[posets of subobjects]] has a [[left adjoint]] $\flat \dashv \sharp $:

$$
  (U \hookrightarrow \sharp  V)
   \Leftrightarrow
  (\flat U \hookrightarrow V)
  \,.
$$

=--

This appears under the term "principal" in ([Awodey-Birkedal, def. 2.1](#AwodeyBirkedal)).

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

We use the notation "$\flat$" and "$\sharp $" oppositely to the use on p.14 of [Awodey-Birkedal](#AwodeyBirkedal). Our convention is such that it harmonizes with the terminology at _[[cohesive topos]]_ and _[[cohesive (infinity,1)-topos]]_, where it makes interpretational sense to pronounce "$\flat$" as "flat".

=--

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

The left adjoints $\flat : Sub(X) \to Sub(X)$ for all $X \in \mathcal{E}$
extend to a functor $\flat : \mathcal{E} \to \mathcal{E}$ on all of $\mathcal{E}$.

=--

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

(...)

=--

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

A [[Lawvere-Tierney topology]] $j$ is [essential](#EssentialTopology), $(\flat \dashv \sharp )$, precisely if for all objects $X$ there exists a least $\sharp $-[[dense subobject]] $U_X \hookrightarrow X$.

=--

This appears as ([AwodeyBirkedal, lemma 2.3](#AwodeyBirkedal)).

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

By the discussion at [[category of sheaves]] we have that $\sharp $ is given by the composite

$$
  \sharp  : \mathcal{E} \stackrel{\Gamma}{\to} Sh_j(\mathcal{E}) \stackrel{coDisc}{\hookrightarrow} \mathcal{E}
  \,,
$$

where the first morphism is [[sheafification]] and the second is [[full and faithful functor|full and faithful]].

If now the [[left adjoint]] $\flat$ exists, it follows that this comes from a left adjoint $Disc$ to $\Gamma$ as

$$
  \flat
   :
  \mathcal{E}
   \stackrel{\overset{Disc}{\hookleftarrow}}{\underset{\Gamma}
{\to}}
  Sh_j(\mathcal{E})
  \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{coDisc}{\hookrightarrow}}
  \mathcal{E}
  :
  \sharp
  \,.
$$

Therefore the $(Disc \dashv \Gamma)$-[[counit of an adjunction|counit]] provides morphisms

$$
  (\flat X \to X) = (\flat X \to U_x \hookrightarrow X)
  \,,
$$

whose [[image]] factorization $U_x \hookrightarrow X$
we claim provides the least dense subobjects.

To show that $U_X$ is dense it is sufficient to show
that

$$
  \sharp (\flat X \to X)
  =
  (coDisc \Gamma Disc \Gamma \to coDisc \Gamma X)
$$

is an [[isomorphism]].

Composing this morphism with $CoDisc$ of the $(Disc \dashv\Gamma)$-unit on $\Gamma X$ (which is an [[isomorphism]] since $Disc$ is a [[full and faithful functor]] by the discussion at _[fully faithful adjoint triples](adjoint+triple#FullyFaithFulAdjointTriples)_) and using the $(Disc \dashv \Gamma)$ [[triangle identity]] we have

$$
  coDisc(
    \Gamma X
      \stackrel{\simeq}{\to}
    \Gamma Disc \Gamma X
      \stackrel{}{\to}
    \Gamma X
  )
  =
  coDisc ( \Gamma X \stackrel{Id}{\to} \Gamma X)
  \,.
$$

Using that also $coDisc$ is full and faithful and then [[2-out-of-3]] for [[isomorphisms]] it follows that $coDisc \Gamma Disc \Gamma X \stackrel{\simeq}{\to} coDisc \Gamma X$ hence

$$
  \sharp  \flat X \stackrel{\simeq}{\to} \sharp  X
$$

is indeed an isomorphism.

Moreover, by one of the equivalent characterizations of [[reflective subcategories]] we have (...)

=--

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

An object $X \in \mathcal{E}$ is called **discrete** if
for all $\Gamma$-[[local isomorphisms]] $f : A \to B$ the induced morphism

$$
  \mathcal{E}(X,A) \to \mathcal{E}(X,B)
$$

is an [[isomorphism]] (of sets, hence a bijection).

=--

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

An object $X \in \mathcal{E}$ is called **o-discrete** if $\flat X \simeq X$.

=--

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

Every discrete object is o-discrete.

=--

### The axioms

 {#TheAxioms}

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

On $\mathcal{E}$ a [[elementary topos]] with [[Lawvere-Tierney topology]] $j$ consider the following axioms.

* **Axiom 1**. $j$ is [essential](#EssentialTopology).

* **Axiom 2 a**. There is an object $G \in \mathcal{E}$ such that every object is a [[subquotient]] of the [[product]] of a [discrete object](#DiscreteObjects) with $G$.

* **Axiom 2 b**. With $G$ as above, there is a discrete object $G'$ and an [[epimorphism]] $G' \to \flat G$.

* **Axiom 3**. For all [discrete objects](#DiscreteObjects) $D$, if the [[internal hom]] $[X,D]$ is  [o-discrete](#ODiscreteObjects), then $X$ is also discrete.

* **Axiom 4**. [Discrete objects](#DiscreteObjects) are closed under binary [[products]].

=--

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

These axioms characterize local geometric morphisms $\mathcal{E} \to Sh_j(\mathcal{E}) \simeq D_j(\mathcal{E})$.

If $G$ is fixed to be the terminal object (in which case Axiom 2 b becomes empty), then they characterize local and [[localic geometric morphisms]].

=--

This is ([Awodey-Birkedal, theorem 3.1](#AwodeyBirkedal)) together with the discussion around remark 3.7.

## Examples

### Easy examples
 {#EasyExamples}

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

If $C$ is a [[small category]] with a [[terminal object]] $* \in C$, then the [[presheaf topos]] $[C^{op},Set]$ is a local topos.

=--

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

Notice that $Set \simeq [*,Set]$ is the presheaf topos over the point category, the category with a single object and a single morphism. Therefore the constant presheaf functor

$$
  Const : Set \to [C^{op}, Set]
$$

can be thought of as sending a set $S \in Set$, hence a functor $S : * \to Set$ to the composite functor

$$
  Const(S) \colon C \to * \stackrel{S}{\to} Set
  \,.
$$

Notice that in the presence of a terminal object in $C$, $Const$ is a [[full and faithful functor]]: a [[natural transformation]] $Const(S_1) \to Const(S_2)$ has components

$$
  \array{
    S_1 = & Const(S_1)(*) &\stackrel{f}{\to}& Const(S_2)(*) & = S_2
    \\
    & \downarrow^{\mathrm{id}} && \downarrow^{\mathrm{id}}
    \\
    & Const(S_1)(U) &\to& Const(S_2)(U)
  }
$$

where the vertical morphisms are $Const(U \to *)$, the point being that they exist for every $U \in C$ given the presence of the terminal object. It follows that such a natural transformation is given by any and one and the same [[function]] $f \colon S_1 \to S_2$.

The functor $Const$ has a [[left adjoint]] and a [[right adjoint]], and these are -- essentially by definition -- the [[colimit]] and the [[limit]] operations

$$
  (\underset{\rightarrow}{\lim} \vdash Const \vdash \underset{\leftarrow}{\lim})
$$

which send a presheaf/functor $F \colon C^{op} \to Set$ to its [[colimit]] $\underset{\rightarrow}{\lim} F \in Set$ or [[limit]] $\underset{\leftarrow}{\lim} F \in Set$, respectively.

Since adjoints are essentially unique, it follows that the [[global section]] functor $\Gamma \colon [C^{op}, Set]$ is given by taking the limit, $\Gamma \simeq \underset{\leftarrow}{\lim}$.

Observe that the [[terminal object]] $* \in C$ is the [[initial object]] in the [[opposite category]] $C^{op}$. But the limit over a [[diagram]] with initial object is given simply by evaluation at that object, and so we have for any $F \in [C^{op}, Set]$ that

$$
  \begin{aligned}
    \Gamma(F) & \simeq \underset{\leftarrow}{\lim} F
    \\
    & \simeq F(*) \in Set
  \end{aligned}
$$

hence that the global section functor is simply given by evaluating a presheaf on the terminal object of $C$.

Limits and colimit _in_ a presheaf category $[C^{op}, Set]$ are computed objectwise over $C$ (see at _[[limits and colimits by example]]_). Therefore evaluation at any object in $C$ preserves in limits and colimits, and in particular evaluation at the terminal object does. Therefore $\Gamma$ preserves all colimits. Hence by the [[adjoint functor theorem]] it has a further [[right adjoint]] $CoDisc$.

We can compute it explicitly by the  [[Yoneda lemma]] and using the defining Hom-isomorphism of adjoints to be the functor $CoDisc \colon Set \to [C^{op}, Set]$ such that for $S \in Set$ the presheaf $CoDisc(S)$ is given over $U \in C$ by

$$
  \begin{aligned}
   CoDisc(S)(U) & \underoverset{Yoneda}{\simeq}{\to} Hom_{[C^{op}, Set]}(U,CoDisc(S))
    \\
    & \underoverset{adjunction}{\simeq}{\to} Hom_{Set}(\Gamma(U), S)
    \,.
  \end{aligned}
$$

So in conclusion we have an [[adjoint triple]] $(Const \vdash \Gamma \vdash CoDisc)$ where $Const$ is a [[full and faithful functor]]. By the discussion at _[fully faithful adjoint triples](adjoint+triple#FullyFaithFulAdjointTriples)_ it follows then that also $CoDisc$ is full and faithful.

=--

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

The converse to prop. \ref{PresheafToposOverSiteWithTerminal} is true if $C$ is [[Cauchy complete category|Cauchy complete]].

=--

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

If $X$ is a [[topological space]], or more generally a [[locale]], then $Sh(X)$ is local (over Set) iff $X$ has a [[focal point]] $x$, i.e. a point whose only [[neighborhood]] is the whole space. In this case, the extra right adjoint $f^! : Set \to Sh(X)$ to the global sections functor $f_* : Sh(X) \to Set$ is given by the functor which computes the stalk at $x$. This can also be given without reference to $x$, by the formula
$$ \Gamma(U, f^!(M)) = Hom(\{\star|U=X\}, M) \cong
  \begin{cases} M, & U = X, \\ \{\star\}, & U \neq X, \end{cases} $$
for sets $M$ and open subsets $U \subseteq X$.

=--

### Sheaves on a local site

For $C$ a [[local site]], the [[category of sheaves]] $Sh(C)$ is a local topos over $Set$.

For instance [[CartSp]] is a local site. Objects in $Sh(C)$ are [[generalized smooth spaces]] such as [[diffeological space]]s.
The further right adjoint

$$
  Codisc : Set \to Sh(CartSp)
$$

is the functor that sends a set to the [[diffeological space]] on that set with _[[codiscrete space|codiscrete]]_ smooth structure (every map of sets is smooth).

### Relative Realizability

Let $A$ be a partial combinatory algebra and let $A\sharp\subseteq A$ be
a sub partial combinatory algebra of $A$. Then there is a
(localic) local geometric morphism from the [[relative realizability topos]]
$\mathrm{RT}(A,A\sharp)$ to the standard [[realizability topos]]
$\mathrm{RT}(A\sharp)$.

### Localization

Let $LocTopos$ denote the 2-category of local [[Grothendieck toposes]] (over Set) with all geometric morphisms between them.  Let $PTopos$ denote the 2-category whose objects are [[pointed topos|pointed toposes]] (i.e. (Grothendieck) toposes $E$ equipped with a geometric morphism $s\colon Set\to E$), and whose morphisms are pairs $(f,\alpha)$ such that $f\colon E\to E'$ is a geometric morphism and $\alpha\colon s'\to f s$ is a (not necessarily invertible) geometric transformation.

Note that if $E$ is a local topos with global sections geometric morphism $e^*\dashv e_*$, then the adjunction $e_*\dashv e^!$ is also a geometric morphism $Set\to E$.  In this way we have a functor $LocTopos \to PTopos$, which is a full embedding, and turns out to have a [[right adjoint]]: this right adjoint is called the **localization** of a pointed topos at its specified point.  For example:

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

If $C$ is a [[small category]] and $U$ is an object of $C$, then the localization of the [[presheaf topos]] $[C^{op},Set]$ at the point induced by $U\colon 1\to C$ can be identified with the [[presheaf topos]] $[(C/U)^{op},Set]$ over the [[over category]] of $C$ over $U$. By the general properties of [[over topos]]es, this is equivalently the over-topos $PSh(C)/U$ (where $U$ is regarded in $PSh(C)$ by the [[Yoneda embedding]]).

=--

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

If $X=Spec(A)$ is the [[Zariski spectrum]] of a [[commutative ring]] $A$, and $P\subset A$ is a prime ideal of $A$ (i.e. a point of $X$), then the localization of $Sh(X)$ at $P\colon 1\to X$ can be identified with $Sh(Spec(A_P))$, where $A_P$ denotes the [[localization]] of $A$ at $P$.  Of course, this is the origin of the terminology.

=--

A similar construction is possible for [[bounded geometric morphism|bounded]] toposes over any base (not just Set).

### Local over-toposes
  {#LocalOverTopoes}

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

For $\mathcal{E}$ a [[Grothendieck topos]] and $X \in \mathcal{E}$ an object, the [[over topos]] $\mathcal{E}/X$ is local if $X$ is a [[tiny object]] ([[atomic object]]).

=--

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

We check that the [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E}/X \to Set$ preserves [[colimit]]s. It is given by the [[hom-functor]] out of the [[terminal object]] of $\mathcal{E}/X$, which is $(X \stackrel{Id}{\to} X)$:

$$
  \Gamma : (A \stackrel{f}{\to} X) \mapsto Hom_{\mathcal{E}/X}(Id_X, f)
  \,.
$$

The [[hom-set]]s in the [[over category]] are [[fiber]]s of the hom-sets in $\mathcal{E}$: we have a [[pullback]] diagram

$$
  \array{
      Hom_{\mathcal{E}/X}(Id_X, (A \to X)) &\to&
       Hom_{\mathcal{E}}(X,A)
      \\
      \downarrow && \downarrow^{f_*}
      \\
      * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X)
  }
  \,.
$$

Moreover, overserve that [[colimit]]s in the [[over category]] are computed in $\mathcal{E}$.

$$
  {\lim_{\to}}_i (A_i \stackrel{f_i}{\to} X)
  \simeq
  ({\lim_\to}_i A_i) \to X
  \,.
$$

If $X$ is a [[tiny object]] then by definition we have

$$
  Hom_{\mathcal{E}}(X, {\lim}_i A_i)
  \simeq
  {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i)
  \,,
$$

Inserting all this into the above pullback gives the pullback

$$
  \array{
      Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X)) &\to&
       {\lim_\to}_i Hom_{\mathcal{E}}(X, A_i)
      \\
      \downarrow && \downarrow^{f_*}
      \\
      * &\stackrel{Id_x}{\to}& Hom_{\mathcal{E}}(X,X)
  }
  \,.
$$

By [[universal colimits]] in the topos [[Set]], this pullback of a colimit is the colimit of the separate pullbacks, so that

$$
  \Gamma({\lim_\to}_i (A_i \to X)))
  \simeq
  Hom_{\mathcal{E}/X}(Id_X, {\lim_\to}_i (A_i \to X))
   \simeq
  {\lim_\to}_i Hom_{\mathcal{E}/X}(Id_X,(A_i \to X))
  \simeq
  {\lim_\to}_i \Gamma(A_i \to X)
  \,.
$$

So $\Gamma$ does commute with colimits if $X$ is tiny. By the [[adjoint functor theorem]] then the [[right adjoint]] $\nabla : Set \to \mathcal{E}/X$ does exist and so $\mathcal{E}/X$ is a local topos.

=--

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

As a special case this reproduces the [above statement](#LocalizationOfPresheafTopos) that slices $PSh(C)/j(U)$ of presheaf toposes over objects in the image of the [[Yoneda embedding]] are local: every [[representable functor]] is [[tiny object|tiny]] (see there).

=--

### Gros toposes

Let $A$ be a [[commutative ring]] (such as $\mathbb{Z}$ or a [[field]]), let $Spec A$ be the [[prime spectrum]] of $A$, and let $\mathcal{Z}_A$ be the [[big and little toposes|big]] [[Zariski site|Zariski topos]] for $A$ (i.e. the [[classifying topos]] for [[local ring|local $A$-algebras]]). For each element $a$ of $A$, we have an open subset $D (a) = \{ \mathfrak{p} \in Spec A : a \notin \mathfrak{p} \}$, and these open subsets constitute a basis for the topology on $Spec A$. The full subcategory of the frame of open subsets of $Spec A$ spanned by these basic open subsets admits a contravariant [[full and faithful functor|full embedding]] in the category of finitely-presented $A$-algebras via the functor $D (a) \mapsto A [a^{-1}]$ (the well-definedness of this functor requires a non-trivial check!), and this functor moreover has the cover lifting property, so induces a local geometric morphism $\mathcal{Z}_A \to Sh (Spec A)$.

## Related concepts

* [[essential geometric morphism]]

* [[locally connected topos]] / [[locally ∞-connected (∞,1)-topos]]

  * [[connected topos]] / [[∞-connected (∞,1)-topos]]

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* **local topos** / [[local (∞,1)-topos]]

  * [[Pi modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]], [[locally ∞-connected site]]

* [[connected site]]

* [[local site]]

* [[cohesive site]], [[(∞,1)-cohesive site]]

## References

Standard references include

* [[Peter Johnstone]], [[Ieke Moerdijk]], _Local maps of toposes_  Proc. London Math. Soc.  (1989)   s3-58  (2):  281-305.  ([pdf](http://plms.oxfordjournals.org/content/s3-58/2/281.full.pdf+html))

and Chapter C3.6 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

A completely [[internal logic|internal]] characterization of local toposes is discussed in

* [[Steve Awodey]], [[Lars Birkedal]], _Elementary axioms for local maps of toposes_, Journal of Pure and Applied Algebra, 177(3):215-230, (2003) ([ps](http://www.itu.dk/people/birkedal/papers/elealm.ps.gz), [[AwodeyBirkedalLocalTopos.pdf:file]])
 {#AwodeyBirkedal}

This is based on part 2 of

* [[Lars Birkedal]], _Developing Theories of Types and
Computability via Realizability_ PhD Thesis ([pdf](https://cs.au.dk/~birke/papers/devttc.pdf))

Free local constructions are considered in

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

Notions of local topos, with a view to logical completeness theorems, appear in Steve Awodey's thesis:

* [[Steve Awodey]], *Logic in topoi: functorial semantics for higher-order logic*, Thesis, University of Chicago (1997).
 {#Awodey}

[[!redirects local geometric morphisms]]
[[!redirects local topos]]
[[!redirects local topoi]]
[[!redirects local toposes]]


local homeomorphism

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

#Contents#
* table of contents
{:toc}

## Idea

A [[continuous map]] $f : X \to Y$ between [[topological spaces]] is called a _local homeomorphism_ if restricted to a [[neighbourhood]] of every point in its [[domain]] it becomes a [[homeomorphism]] onto its image which is required
to be open.

One also says that this exhibits $X$ as an [[étale space]] over $Y$.

Notice that, despite the similarity of terms, local homeomorphisms are, in general, _not_ [[local isomorphisms]] in any natural way. See the [examples](#Examples) below.

## Definition

\begin{definition}
A *local homeomorphism*  is a [[continuous map]] $p \colon E \to B$ between [[topological spaces]] (a [[morphism]] in [[Top]]) such that

* for every [[element]] $e \in E$, there is an [[open subset]] $U \ni e$ whose [[image]] $p_*(U)$ is open in $B$ and the restriction of $p$ to $U$ is a [[homeomorphism]] $p|_U \colon U \to p_*(U)$,

or equivalently

* for every $e \in E$, there is a [[neighbourhood]] $U$ of $e$ such that the image $p_*(U)$ is a neighbourhood of $p(e)$ and $p|_U: U \to p_*(U)$ is a homeomorphism.

\end{definition}

See also at *[[étale space]]*.

## Examples
 {#Examples}

For $X$ any [[topological space]] and  for $S$ any [[set]] regarded as a [[discrete space]], the [[projection]]

$$
  X \times S \to X
$$

is a local homeomorphism.

For $\{U_i \to Y\}$ an [[open cover]], let

$$
  X := \coprod_i U_i
$$

be the [[disjoint union]] space of all the pathches. Equipped with the canonical projection

$$
  \coprod_i U_i \to Y
$$

this is a local homeomorphism.

In general, for every [[sheaf]] $A$ of sets on $Y$; there is a local homeomorphism $X \to Y$ such that over any open $U \hookrightarrow X$ the set $A(U)$ is naturally identified with the set of [[sections]] of $Y \to X$. See [[étale space]] for more on this.

## Properties
 {#Properties}

\begin{proposition}\label{LocalHomeosAreOpenMaps}
  A [[local homeomorphism]] is an [[open map]].
\end{proposition}
\begin{proof}
  Let $f \colon X \to Y$ be a [[local homeomorphism]] and $U \subset X$ an [[open subset]]. We need to see that the [[image]] $f(U) \subset Y$ is an open subset of $Y$.
For this we may equivalently show that each $y \in f(U)$ has an [[open neighbourhood]] inside $f(U)$.

But since any function is [[surjection|surjective]] onto its [[image]], there exists $x \in U$ with $f(x) = y$.
By local homeomorphy of $f$, this $x \in X$ has an  [[open neighbourhood]] $U_x \subset X$ with $f_{|U_x} \colon U_x \to f(U_x)$ a [[homeomorphism]]. Since $U \cap U_x \subset U_x$ is an [[open neighbourhood]] of $x$ in $U_x$, the homeomorphy of $f_{|U_x}$ implies that $f(U \cap U_x) \subset f(U)$ is an open neighbourhood of $f(x) = y$.
\end{proof}

## Related concepts

* [[étale map]], [[étale morphism]]

* [[locally homeomorphic geometric morphism]]

[[!redirects local homeomorphisms]]


Local homotopy theory

This page is about the book

* [[John F. Jardine]]

  \linebreak

  **Local homotopy theory**

  \linebreak

  Springer Monographs in Mathematics (2015)

  [doi:10.1007/978-1-4939-2300-7](https://doi.org/10.1007/978-1-4939-2300-7)

on the abstract [[homotopy theory]] ([[model category]] theory) of [[presheaves]] of [[homotopy types]] and [[stable homotopy types]] ([[model structures on simplicial presheaves]], model structures on [[sheaves of spectra]], [[Brown-Gersten property]], etc.) with an eye towards [[motivic homotopy theory]].

Brief exposition in:

* [[John F. Jardine]], *Motivic spaces and the motivic stable category* [[pdf](http://www.aimath.org/WWN/motivesdessins/motivic.pdf), [[Jardine-MotivicSpaces.pdf:file]]]

category: reference

[[!redirects Local Homotopy Theory]]


local integral domain

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

\tableofcontents

## Definition

A **local integral domain** is a [[commutative ring]] which is both a [[local ring]] and a [[integral domain]], hence a commutative ring such that:

* The ring is nontrivial $0 \neq 1$;

* if the sum of two elements $x + y$ is equal to $1$, then $x$ is [[invertible]] or $y$ is invertible;

* If the product of two elements $x \cdot y$ is equal to $0$, then $x = 0$ or $y = 0$.

## Examples

* Every [[field]] is a local integral domain which is also a [[Artinian ring]].

* Given a [[field]] $K$, the ring of [[power series]] $K[[x]]$ is a local integral domain which is not a [[field]].

* In general, given a local integral domain $R$, the ring of [[power series]] $R[[x]]$ is a local integral domain.

* Every [[discrete valuation ring]] is a local integral domain with a [[Dedekind-Hasse norm]].

* Every [[finite set|finite]] local integral domain is a [[finite field]].

## Properties

Unlike the [[theory]] of [[Heyting fields]], the theory of local integral domains is a [[coherent theory]]. It is the condition that a local integral domain is an [[Artinian ring]] that fails in [[coherent logic]], since it relies on either negation to refer to non-invertible elements (see [[Artinian local ring]]), proper ideals/maximal ideals, or the [[natural numbers]] to refer to the [[descending chain condition]]. Even if one has the natural numbers around to define a [[dependent sequence]] of [[monomorphisms]] for the [[descending chain condition]], such as working internally in an [[arithmetic pretopos]], it is not necessarily the case that every such [[Artinian local ring]] has a maximal ideal or that [[Nakayama's lemma]] holds in [[coherent logic]].

While every [[ring homomorphism]] between fields is an [[injection]], it is not the case that every ring homomorphism between local integral domains is an injection. The ring homomorphism which takes a local integral domain to its [[residue field]] is an injection if and only if the local integral domain is already a field.

## See also

* [[ordered local integral domain]]

* [[discrete valuation ring]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

[[!redirects local integral domain]]
[[!redirects local integral domains]]

[[!redirects local ring without zero divisors]]
[[!redirects local rings without zero divisors]]

[[!redirects local ring without zero-divisors]]
[[!redirects local rings without zero-divisors]]


local isomorphism

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

#Contents#
* table of contents
{:toc}

## Idea

A _local isomorphism_ in a [[presheaf]] category $PSh(S)$ is a morphism that becomes an [[isomorphism]] after passing to  [[sheaf|sheaves]] with respect to a given [[Grothendieck topology]] on $S$.

The collection of all local isomorphisms not only determines the [[Grothendieck topology]] but is precisely the collection of morphisms that are inverted when passing to sheaves. Hence local isomorphisms serve to understand [[sheaf|sheaves]] and [[sheafification]] in terms of the passage to a [[homotopy category]] of $PSh(S)$.

This is a particular case of the notion of [[reflective factorization system]], applied to the sheafification reflector.  It is discussed in more detail at [[category of sheaves]]:

In terms of the discussion at [[geometric embedding]], local isomorphisms in $PSh(S)$ are precisely the [[calculus of fractions|multiplicative system]] $W$ that is sent to isomorphisms by the [[sheafification]] functor

$$
  \bar{(-)} : PSh(S) \to Sh(S)
$$

which is left [[exact functor|exact]] [[left adjoint]] to the [[full and faithful functor|full and faithful]] inclusion

$$
  Sh(S) \hookrightarrow PSh(S)
  \,.
$$

## Axioms

A system of **local isomorphism**s on $PSh(S)$ is any collection of morphisms satisfying

1. local isomorphisms are a system of [[category with weak equivalences|weak equivalences]] (i.e. every [[isomorphism]] is a local isomorphism and they satisfy [[2-out-of-3]]);

2. a morphism $Y\to X$ is a local isomorphism if and only if its [[pullback]]
$$
  \array{
    U \times_X Y &\to& Y
    \\
    {}^{\mathllap{loc iso}}\downarrow
    &&
    \downarrow^{\mathrlap{\Leftrightarrow loc iso}}
    \\
    U &\to& X
  }
$$
along any morphism $U \to X$, where $U$ is [[representable presheaf|representable]], is a local isomorphism.

## Relation to Grothendieck topologies

Systems of local isomorphisms on $PSh(S)$ are equivalent to [[Grothendieck topology|Grothendieck topologies]] on $S$.

The following indicates how choices of systems of local isomorphisms are equivalent to choices of systems of [[local epimorphisms]]. The claim follows by the discussion at [[local epimorphism]].

### Local epimorphisms from local isomorphisms

A system of [[local epimorphisms]] is defined from a system of local isomorphisms by declaring that $f : Y \to X$ is a [[local epimorphism]] precisely if $im(f) \to X$ is a local isomorphism.

### Local isomorphisms from local epimorphisms

Given a [[Grothendieck topology]] in terms of a system of [[local epimorphisms]], a system of local isomorphisms is constructed as follows.

A **local monomorphism** with respect to this topology is a morphism $f : A \to B$ in $[S^{op}, Set]$ such that the canonical morphism $A \to A \times_B A$ is a [[local epimorphism]].

A **local isomorphism** with respect to a Grothendieck topology is a morphism in $[S^{op}, Set]$ that is both a [[local epimorphism]] as well as a local monomorphism in the above sense.

## Relation to Lawvere-Tierney topologies

Recall that [[Grothendieck topology|Grothendieck topologies]] on a [[small category]] $S$ are in bijection with [[Lawvere-Tierney topology|Lawvere-Tierney-topologies]] on $PSh(S)$ and that [[sheafification]] with respect to a [[Lawvere-Tierney topology]] is encoded in terms of monomorphisms in $PSh(S)$ which are _[[dense monomorphism|dense]]_ with respect to the [[Lawvere-Tierney topology]].

We have:

the [[dense monomorphisms]] are precisely the local isomorphisms which are also ordinary [[monomorphisms]].

## Properties

* Local isomorphisms admit a left saturated [[calculus of fractions]].

### Sheafification

The [[sheafification]] functor which sends a [[presheaf]] $F$ to its weakly
equivalent [[sheaf]] $\bar F$ can be realized using a [[colimit]] over local
isomorphisms. See there.

### Characterization and relation to sieves

Often one concentrates on the local isomorphisms whose codomain is a [[representable functor|representable presheaf]], i.e. those of the form

$$
  A \to Y(U)
  \,,
$$

where $U$ is an object in $S$ and $Y$ is the [[Yoneda embedding]]. These come from covering [[sieves]] of a [[Grothendieck topology]] on $S$: for $U \in S$ and $\{V_i \to U\}_i$ a covering [[sieve]] on $U$, the coresponding local isomorphism is the presheaf which is the [[image]] of the joint injection map

$$
  \sqcup_i Y(V_i) \to Y(U)
  \,.
$$

Using the fact that morphisms in a presheaf category are [[strict morphisms]], so that [[image]] and [[coimage]] coincide, it is useful, with an eye towards generalizations from [[sheaves]] to [[stacks]] and [[∞-stacks]] (see in particular [[descent for simplicial presheaves]]), to say this equivalently in terms of the [[coimage]]: the local isomorphism corresponding to the covering [[sieve]] $\{V_i \to U\}$ is

$$
  colim (
     (\sqcup_i Y(V_i))\times_{Y(U)}
     (\sqcup_i Y(V_i))
     \stackrel{\to}{\to}
     (\sqcup_i Y(V_i))
  )
  \to
  Y(U)
$$

Notice that in general these are not _all_ the local isomorphism with representable codomain (more generally these are [[hypercovers]], where
$(\sqcup_i Y(V_i))\times_{Y(U)} (\sqcup_i Y(V_i))$ is replaced in turn by one of its covers).

(...)

Notice that local isomorphism with codomain a representable already induce general local isomorphisms using the fact that every presheaf is a colimit of representables (the [[co-Yoneda lemma]]) and that local isomorphisms/sieves are stable under [[pullback]]:

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

If $A \in PSh(S)$ is a [[local object]] with respect to local isomorphisms whose codomain is a representable, then every morphism $X \to Y$ of presheaves such that for every representble $U$ and every morphism $U \to Y$ the pullback
$X \times_Y U  \to U$ is a local isomorphism, the canonical morphism
$$
  Hom(Y,A) \to Hom(X,A)
$$
is an isomorphism.
=--

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

We may first rewrite trivially
$$
  X \simeq X \times_Y Y
$$
and then use the [[co-Yoneda lemma]] to write (suppressing notationally the Yoneda embedding)
$$
  Y \simeq colim_{U \to Y} U
$$
and hence rewrite $(X \to Y)$ as
$$
  X \times_Y (\colim_{U \to Y} U) \to colim_{U \to Y} U
  \,.
$$

Then using that colimits of presheaves are [[commutativity of limits and colimits|stable under base change]] this is
$$
  (\colim_{U \to Y}(X \times_Y U)) \to colim_{U \to Y} U
  \,.
$$

Recall that by assumption the components $X \times_Y U \to U$ of this are local isomorphisms. Hence
$$
  (Hom(Y,A) \to Hom(X,A))
  =
  lim_{U \to Y}
  Hom(U, A) \to
  \lim_{U  \to Y}
  Hom(X \times_Y U, A)
$$
is a limit over isomorphisms, hence an isomorphism.
=--

## References

This is in section 16.2 of

* Kashiwara-Schapira, _[[Categories and Sheaves]]_ .

See in particular exercise 16.5 there for the characterization of [[Grothendieck topology|Grothendieck topologies]] in terms of local isomorphisms.

[[!redirects local isomorphisms]]