Jordan-Lie-Banach algebra

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

# Contents
* table of contents
{: toc}

## Idea

A Jordan--Lie algebra is a [[quantum deformation]] of a [[Poisson algebra]], consisting of a single [[vector space]] with compatible structures of both a [[Jordan algebra]] and a [[Lie algebra]].  Secretly, this is the same thing as a [[complexification|complexified]] $*$-[[star-algebra|algebra]].  If we add an appropriate [[topological structure]], then we get a Jordan--Lie--[[Banach algebra]] (corresponding to complex Banach $*$-algebras), of which the $JLB$-algebras (corresponding to $C^*$-[[C-star algebra|algebras]]) and $JLBW$-algebras (corresponding to [[von Neumann algebras]]) are important special cases.

In traditional [[strict deformation quantization]], the outcome of [[quantization]] of [[Poisson algebras]] is regarded to be a non-commutative but associative complex $*$-[[star-algebra|algebra]] (such as a $C^*$-[[C-star-algebra|algebra]]).  But any such induces a Jordan--Lie algebra (consisting of its [[self-adjoint elements]]) by letting the Jordan product be the symmetrized product and the [[Lie bracket]] the [[commutator]] (times $\mathrm{i}/2$). There is a condition relating the [[associator]] of the Jordan product to the Lie bracket, which makes $JLB$-algebras effectively the same as $C^*$-algebras, the difference being that the single associative product is explicitly regarded as inducing the two products of a $JLB$-algebra. For more on this separation of the Lie-algebra and the Jordan-algebra aspect of quantization see at _[[order-theoretic structure in quantum mechanics]]_.

## Definitions and elementary properties

Let $K$ be a [[formally real field]] (in particular, $1/2 \in K$), and let $q \in K$ be a scalar.  (Most of what we say here works over more general fields or even [[commutative rings]], but there are arguments that the usual notion of [[Jordan algebra]] is inappropriate already for fields of characteristic $2$, so I don\'t want to pretend to more generality than we can justify.  Indeed, there does not yet seem to be any published literature on Jordan--Lie algebras that doesn\'t set $K$ to the field $\mathbb{R}$ of [[real numbers]].)  Since $K$ is a [[field]], $q$ is either zero or invertible, and some things will depend on which it is.

A __Jordan--Lie algebra__ (over $K$ with deformation constant $q$) consists of a $K$-[[vector space]] $A$ equipped with two [[bilinear operators]] $(-)\circ(-)$ and $[{-},{-}]$, respectively called the _[[Jordan product]]_ and the _[[Lie bracket]]_, satisfying the following identities:

* commutativity of the Jordan product: $x \circ y = y \circ x$;
* alternation (anticommutativity) of the Lie bracket: $[x, x] = 0$, or equivalently (using bilinearity and that $1/2 \in K$) $[x, y] = -[y, x]$;
* the Jacobi identity (self-derivation of the Lie bracket): $[x, [y, z]] = [[x, y], z] + [y, [x, z]]$, or equivalently (using anticommutativity) $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$;
* derivation of the Lie bracket over the Jordan product: $[x, y \circ z] = [x, y] \circ z + y \circ [x, z]$;
* the associator identity: $x \circ (y \circ z) = (x \circ y) \circ z + q^2 [y, [x, z]]$, or equivalently (using anticommutativity) $(x \circ y) \circ z - x \circ (y \circ z) = q^2 [[x, z], y]$.

Note that the last two axioms (and alternation) prove the Jordan identity: first,
$$ [x, x \circ x] = [x, x] \circ x + x \circ [x, x] = 0 ;$$
then (writing $x^2$ for $x \circ x$),
$$ x \circ (y \circ x^2) - (x \circ y) \circ x^2 = q^2 [y, [x, x^2]] = 0 .$$
Thus, $A$ is a [[Jordan algebra]] under the Jordan product, and of course $A$ is a [[Lie algebra]] under the Lie bracket.  (In a similar way, we can prove the Jacobi identity from the associator identity if $q$ is cancellable; but we state the Jacobi identity separately to cover when $q = 0$ as well.)

As usual for Jordan algebras, we write $x^n$ for an $n$-fold product of $x$ under the Jordan product; this is unambiguous since Jordan algebras are always power-associative.  (And of course, an $n$-fold Lie bracket is always zero for $n \gt 1$.)  Even more generally the un[[bias]]ed form of the Jordan identity is that multiplication by $x^m$ and by $x^n$ commute:
$$ (x^m \circ y) \circ x^n = x^m \circ (y \circ x^n) $$
(which may be proved by [[induction]] using only the usual Jordan identity).  We similarly have that powers of $x$ commute with each other under the bracket:
$$ [x^m, x^n] = 0 .$$
The proof is similar to the proof above that $[x, x^2] = 0$, by induction.  (Then you can use this to get a faster proof of the unbiased Jordan identity.)

A Jordan--Lie algebra is __unital__ if the Jordan product has   an [[identity element]] $1$:

* multiplicative identity: $1 \circ x = x$.

Then we may also interpret $x^0$ as $1$.  Note that $[x, 1] = 0$ too (a special case of $[x^m, x^n] = 0$), proved by examining $[x, 1 \circ 1]$.

If $K$ is a [[normed field]], then a __Jordan--Lie--Banach algebra__ consists of a $K$-[[Banach space]] $A$ equipped with a [[short linear operator|short]] [[bilinear operator]] (as the Jordan product) and a [[densely defined linear operator|densely defined]] bilinear operator (as the Lie bracket) making the domain of the Lie bracket into a Jordan--Lie algebra as above.

Typically, although the Lie bracket is allowed to be unbounded, it will be bounded when $q \gt 0$.  (This is a theorem for $JLB$-algebras below.)  Using continuity to prove all of the axioms, $A$ is also a [[Jordan–Banach algebra]] under the Jordan product.  A Jordan--Lie--Banach algebra is __unital__ if the Jordan algebra has a unit and ${\|1\|} \leq 1$ (in which case ${\|1\|} = 1$ if $A$ is nontrivial).  Note that Jordan--Lie--Banach algebras (besides the $JLB$-algebras below) don\'t seem to appear in the literature; this terminology is analogous to that of Jordan--Banach algebras.

A __$JLB$-algebra__ is a Jordan--Lie--Banach algebra satisfying the following additional identities:

* the $B$-identity: ${\|x\|^2} = {\|x^2\|}$ (compare the $B^*$-identity or $C^*$-identity of a $C^*$-[[C-star-algebra|algebra]]);
* positivity: ${\|x^2\|} \leq {\|x^2 + y^2\|}$.

Then $A$ is a [[JB-algebra]] under the Jordan product.

As with any $JB$-algebra, we don\'t need to explicitly state that the Jordan product is short (nor that ${\|1\|} \leq 1$ in the unital case), as this can be proved using the $B$-identity and the [[polarization identities]].  (Conversely, if we do state that the product is short, then we only need the $\leq$ half of the $B$-identity in addition, or we can even combine it with positivity as ${\|x\|^2} \leq {\|x^2 + y^2\|}$.)  We can also prove (by induction on $\lceil\log_2 n\rceil$) that ${\|x^n\|} = {\|x\|^n}$ (except for $n = 0$ in the trivial algebra).

When $q \gt 0$, we should also be able to prove that Lie bracket is bounded by $1/q$, but I don\'t see the proof.  (A naive look at the associator identity suggests a bound of $\sqrt{2}/q$, but it\'s not clear to me whether $[[x,z],y]$ is sufficently general to prove this.)  Then extending by continuity, we may assume that the Lie bracket is defined everywhere.

Positivity generalizes to any number of terms on either side (as long as the right-hand side has more, of course).  (However, I don\'t see how to prove this without going through the [[functional calculus]] to prove that every sum of squares has a [[square root]].)  Thus, a $JB$-algebra is [[formally real algebra|formally real]]: if $\sum_i x_i^2 = 0$, then each $x_i = 0$ (because the norms lie in a formally real field).

Following the established terminology for $JB$-algebras, a __$JLC$-algebra__ may be defined as a *concrete* $JLB$-algebra, that is one equipped with a faithful representation (see [below](#representations)) on a [[Hilbert space]], or equivalently a real [[subalgebra]] of $SA(H)$ (see [below](#examples)) that is closed in the [[operator topology]].  However, like $C^*$-algebras (but unlike general $JB$-algebras), every $JLB$-algebra may be so represented and this term seems not to be used.

Finally, a __$JLBW$-algebra__ is a unital $JLB$-algebra whose underlying Banach space is equipped with a (necessarily unique) [[predual]], and a __$JLW$-algebra__ is further equipped with a faithful representation (or equivalently is a $JC$-algebra which includes the identity and is closed in the [[weak operator topology]]).  Again, every abstract $JLBW$-algebra may be made concrete, so we really only need one term in practice, although neither seems to be established in the literature yet.

## Relation to Poisson algebras {#poisson}

If $q = 0$, then the Jordan product becomes associative; in fact, a Jordan--Lie algebra with $q = 0$ is precisely a commutative [[Poisson algebra]].  On the other hand, if $q$ is invertible, then dividing the Lie bracket by $q$ gives a Jordan--Lie algebra with $q = 1$, so the only thing that really matters about $q$ (at least over a field) is whether or not it is zero.  In this way, a Jordan--Lie algebra may be seen as a [[quantum deformation]] of a Poisson algebra.  (In the [[physics|physical]] interpretation, $q$ here is the [[Dirac constant]] $\hbar$.)

Similarly, a Jordan--Banach algebra with $q = 0$ is a commutative Banach algebra equipped with a Poisson bracket (typically unbounded), and similarly for the more specialized versions; then the $q \ne 0$ case is a quantum deformation of such Poisson--Banach algebras.  We can see from this why the Lie bracket cannot be expected to be defined on an entire $JLB$-algebra when $q = 0$, since Poisson brackets in typical examples of Poisson algebras are unbounded in the obvious norms.

## Relation to $C^*$-algebras {#cstar}

For a $JLB$-algebra with $q \gt 0$, the Jordan product and Lie bracket are respectively the real-symmetrized and imaginary-antisymmetrized parts of an associative operation on the [[complexification]] of $A$, defining a complex $C^*$-[[C-star-algebra|algebra]]; and every $C^*$-algebra likewise defines a $JLB$-algebra for any $q \gt 0$ consisting of its [[Hermitian operator|Hermitian]] elements.  (With $q = 0$, we can still get a $C^*$-algebra out of the $JLB$-algebra, and we can still recover the original $JB$-algebra as the self-adjoint elements, but trying to recover the original Lie bracket just results in $0/0$.)  This also works on the purely algebraic level, relating Jordan--Lie algebras to complex $*$-algebras, and on the measurable level, relating $JLBW$-algebras to [[von Neumann algebras]].

Specifically, starting with a Jordan--Lie algebra or $JLB$-algebra $A$ (possibly unital) with deformation constant $q$ over a formally real commutative ring $K$, we write $A \oplus A$ formally as $A + \mathrm{i} A$, on which we define the following operations:

* addition: $(a + \mathrm{i} b) + (c + \mathrm{i} d) \coloneqq (a + c) + \mathrm{i} (b + d)$,
* opposite: $-(a + \mathrm{i} b) \coloneqq (-a) + \mathrm{i} (-b)$ (which we write as $-a - \mathrm{i} b$ as usual),
* zero: $0 \coloneqq 0 + \mathrm{i} 0$,
* scalar multiplication: $(x + \mathrm{i} y) (a + \mathrm{i} b) = (x a - y b) + \mathrm{i} (x b + y a)$ (for $x + \mathrm{i} y$ a scalar in $K[\mathrm{i}]$),
* involution: $(a + \mathrm{i} b)^* \coloneqq a + \mathrm{i} (-b) = a - \mathrm{i} b$,
* multiplication: $(a + \mathrm{i} b) (c + \mathrm{i} d) \coloneqq (a \circ c + q [a, d] + q [b, c] - b \circ d) + \mathrm{i} (-q [a, c] + a \circ d + b \circ c + q [b, d])$,
* unit (in the unital case): $1 \coloneqq 1 + \mathrm{i} 0$.

This makes $A + \mathrm{i} A$ into an associative $*$-[[star-algebra|algebra]] over $K[\mathrm{i}] \coloneqq K[\mathrm{x}]/\lang\!\lang\mathrm{x}^2 + 1\rang\!\rang$ (which is a proper extension of $K$, and a field if $K$ is, because $K$ is formally real).  If $A$ is a $JLB$-algebra, then $A + \mathrm{i} A$ becomes a $C^*$-algebra; if $A$ is a $JLBW$-algebra, then $A + \mathrm{i} A$ becomes a von Neumann algebra.

The intermediate Jordan--Lie--Banach case doesn\'t seem to work.  For one thing, if we allow the Lie bracket to be unbounded (except when $q = 0$ so that it is irrelevant), then we might not be able to define multiplication everywhere.  Even assuming that the Lie bracket is bounded (at least when $q \ne 0$), it\'s not clear how to define the norm in this case.  (For $C^*$-algebras, in contrast, there is no need to define the norm explicitly, since it is determined by the algebraic structure.)

Conversely, starting with a $*$-algebra $A$ (possibly unital, possibly a Banach $*$-algebra) over $K[\mathrm{i}]$ and an *invertible* scalar $q$, and assuming that $2$ is invertible in $K$, we form the subspace $sa(A) = \{x\colon A \;|\; x^* = x\}$, on which we define the following operations (under each of which $sa(A)$ is closed):

* addition, opposite, zero, scalar multiplication (by scalars in $K$), unit (in the unital case): by [[restriction]],
* Jordan product: $a \circ b \coloneqq \frac{1}{2} a b + \frac{1}{2} b a$,
* Lie bracket: $[a, b] \coloneqq \frac{1} {2 \mathrm{i} q} a b - \frac{1} {2 \mathrm{i} q} b a$.

In the Banach case, we may also define the norm by restriction.  (There is no difficulty in this direction for arbitrary Banach $*$-algebras.)

We can write the resulting Jordan--Lie algebra as $sa_q(A)$ or simply $sa(A)$ if $q = 1$.  If $A$ is a Banach $*$-algebra, then $sa_q(A)$ is a Jordan--Lie--Banach algebra (with the Lie bracket defined everywhere); if $A$ is a $C^*$-algebra, then $sa_q(A)$ is a $JLB$-algebra; if $A$ is a von Neumann algebra, then $sa_q(A)$ is a $JLBW$-algebra.

(If $K$ is a formally real *field*, then it has characteristic $0$, so $2$ must be invertible; but for more general rings, if $2$ is not invertible in $K$, then the formulas for the Jordan product and Lie bracket don\'t make sense, even when $q$ is invertible.  If you leave out the divisions by $2$, then you still get a Jordan--Lie algebra with deformation constant $q$, but it won\'t necessarily be unital when $A$ is, the Jordan product could have a norm as high as $2$ in the Banach case, and the $B$-identity will fail in the $C^*$-case, gaining an unwanted factor of $2$.  In any case, a round-trip between Jordan--Lie algebras and complex $*$-algebras requires a division by $2$ somewhere.)

This all defines a [[functor]] each way between the [[groupoids]] of $C^*$-algebras and $JLB$-algebras with fixed $q \ne 0$, which in fact form an [[equivalence of categories|adjoint equivalence]].  (At least it should be; I don\'t know if this has been worked out in detail.)  Since we have a notion of [[morphism]] (not just [[isomorphism]]) of $C^*$-algebras, we can transport this along the equivalence to get a notion of morphism of $JLB$-algebras (which I would expect to be a short linear map that preserves both products, but somebody should check this) and thus a [[category]] $JLB Alg$ equivalent to $C^* Alg$.

In particular, for fixed $q \ne 0$, the category of $JLB$-algebras with one value of $q$ is equivalent to the category of $JLB$-algebras with another value of $q$ (since both are equivalent to complex $C^*$-algebras), but we have only a functor from $JLB$-algebras with $q = 0$ to any of these other categories.  (And this is not a very interesting functor, since it simply throws away the original Lie bracket and replaces it with zero.)

Similarly, real $JLBW$-algebras (with fixed $q \ne 0$) are equivalent to complex $W^*$-[[W-star-algebra|algebras]].

## Examples {#examples}

Any commutative [[associative algebra]] is a trivial example of a Jordan--Lie algebra with the Lie bracket set to [[zero]].  Similarly, any commutative [[Banach algebra]] is a Jordan--Lie--Banach algebra, any commutative $C^*$-[[C-star-algebra|algebra]] with trivial involution is a $JLB$-algebra, and any commutative $W^*$-[[W-star-algebra|algebra]] with trivial involution is a $JLBW$-algebra.  Note that the value of $q$ is irrelevant in these examples.

More interestingly, given any $C^*$-[[C-star-algebra|algebra]] (*not* assumed commutative or with trivial involution), the [[self-adjoint elements]] form a $JLB$-algebra with $q = 1$; if the original $C^*$-algebra is a $W^*$-[[W-star-algebra|algebra]], then the self-adjoint elements form a $JLBW$-algebra.  In fact, *every* $JLB$-algebra (including every $JLBW$-algebra) with $q = 1$ has this form; more generally, every $JLB$-algebra with $q \ne 0$ has this form upon dividing the Lie bracket by $q$.  (See [above](#cstar)).

More specifically, given a [[Hilbert space]] $H$, the [[self-adjoint operators]] form a $JLBW$-algebra $SA(H)$ with $q = 1$.  In fact, *every* $JLB$-algebra (including every $JLBW$-algebra) with $q = 1$ is a subalgebra of some $SA(H)$; more generally, every $JLB$-algebra with $q \ne 0$ has this form upon dividing the Lie bracket by $q$.  (This is a combination of the previous paragraph and the theorem that every $C^*$-algebra may be made concrete.)

As for $q = 0$, every commutative [[Poisson algebra]] is a Jordan--Lie algebra with $q = 0$, and every Jordan--Lie algebra with $q = 0$ has this form.  This settles the purely algebraic case.

For the topological case with $q = 0$, fix a [[symplectic manifold]] (or [[Poisson manifold]]) $X$ and take the unital Banach [[algebra of bounded continuous functions]] on $X$, equipped with the [[Poisson bracket]].  Here, the Poisson bracket is unbounded and not defined everywhere.  (You could also use the Banach space of [[smooth functions]] whose derivatives of all orders are bounded, but the Poisson bracket is still unbounded and not defined everywhere, since there are bounded smooth functions with unbounded derivatives, unless $X$ is [[closed manifold|closed]].)  This is a $JLB$-algebra with $q = 0$.  (But not *every* $JLB$-algebra with $q = 0$ has this form.)

Let $V$ be a (real) [[vector space]] of finite dimension $n$, which we will think of as the [[configuration space]] of a [[classical system]], and let $A$ be the [[symmetric algebra]] on the dual [[phase space]] $V \otimes V^* \cong \mathbb{R}^{2n}$.  We normally think of $A$ as consisting of (commutative) [[polynomials]] in the variables $x^1, \ldots, x^n, p_1, \ldots, p_n$, but let us now think of them as *noncommutative* polynomials consisting only of terms in which the variables come in order.  Fixing a scalar $q$, define the Jordan product on $A$ so that $x^i \circ x^j = x^{\min(i,j)} x^{\max(i,j)}$, $p_i \circ p_j = p_{\min(i,j)} p_{\max(i,j)}$, and $x^i \circ p_j = x^i p_j + q \delta^i_j$; in other words, to remove $\circ$ from a product, we must add a term with a factor of $q$ whenever $x^i$ and $p_i$ appear on opposite sides of $\circ$.  Also, define the Lie bracket so that $[x^i, x^j] = 0$, $[p_i, p_j] = 0$, and $[x^i, p_j] = -\frac{1}{2} \delta^i_j$.  Then when $q = 0$, $A$ is the usual symmetric algebra of commutative polynomials again, while for $q \ne 0$, $A$ becomes $sa_q(W)$, where $W$ is the [[Weyl algebra]] $A + \mathrm{i} A$.

## Representations

In general, a [[representation]] of a Jordan--Lie algebra $A$ in a Jordan--Lie algebra $B$ is simply a [[homomorphism]] (in the usual sense of [[universal algebra]]) from $A$ to $B$.  But when $q \gt 0$, we often take $B$ to be an associative $*$-[[star-algebra|algebra]] and represent $A$ in it with a homomorphism to $sa(B)$, the Jordan--Lie algebra of self-adjoint elements of $B$ (see [above](#examples)).  When $A$ is a Jordan--Lie--Banach algebra, then we generally want $B$ to be the same, or to be a [[Banach *-algebra|Banach algebra]], and for the homomorphism to be [[short map|short]].  Similarly, when $A$ is unital, we generally want $B$ to be the same, and to have the homomorphism preserve the unit.  When $A$ is a $JLB$-algebra, then we generally want $B$ to be the same, or to be a $C^*$-[[C-star-algebra|algebra]]; and when $A$ is a $JLBW$-algebra, then we generally want $B$ to be the same or to be a $W^*$-[[W-star-algebra|algebra]].  (In these cases, the homomorphism is automatically short and automatically preserves any unit.)

Instead of a representation *in* $B$, we can also consider a representation *on* $H$, where $H$ is an [[inner product space]] over the [[complex numbers]] (or the complexification of $K$ more generally).  This is a representation in the $*$-algebra of [[linear operators]] on $H$, or equivalently a representation in the Jordan--Lie algebra of self-adjoint operators on $H$.  When $B$ is a Jordan--Lie--Banach algebra (especially a $JLB$-algebra or even a $JLBW$-algebra), then we may take $H$ to be a [[Hilbert space]] and look only at the [[bounded operators]].

## References

A definition is in section 1.1 of

* [[Hans Halvorson]] and [[Rob Clifton]] (1999). _Maximal Beable Subalgebras of Quantum-Mechanical Observables_. [PDF](http://philsci-archive.pitt.edu/65/1/beables.pdf).
  {#HC1999}

A brief remark is on page 80 of

* [[Sean Bates]] and [[Alan Weinstein]] (1995?). _Lectures on the geometry of quantization_. [PDF](http://www.math.berkeley.edu/~alanw/GofQ.pdf).
  {#BW1995}

[[!redirects Jordan-Lie algebra]]
[[!redirects Jordan-Lie algebras]]
[[!redirects Jordan–Lie algebra]]
[[!redirects Jordan–Lie algebras]]
[[!redirects Jordan--Lie algebra]]
[[!redirects Jordan--Lie algebras]]
[[!redirects JL algebra]]
[[!redirects JL algebras]]
[[!redirects JL-algebra]]
[[!redirects JL-algebras]]

[[!redirects Jordan-Lie-Banach algebra]]
[[!redirects Jordan-Lie-Banach algebras]]
[[!redirects Jordan–Lie–Banach algebra]]
[[!redirects Jordan–Lie–Banach algebras]]
[[!redirects Jordan--Lie--Banach algebra]]
[[!redirects Jordan--Lie--Banach algebras]]

[[!redirects JLB algebra]]
[[!redirects JLB algebras]]
[[!redirects JLB-algebra]]
[[!redirects JLB-algebras]]
[[!redirects JLC algebra]]
[[!redirects JLC algebras]]
[[!redirects JLC-algebra]]
[[!redirects JLC-algebras]]

[[!redirects JLBW algebra]]
[[!redirects JLBW algebras]]
[[!redirects JLBW-algebra]]
[[!redirects JLBW-algebras]]
[[!redirects JLW algebra]]
[[!redirects JLW algebras]]
[[!redirects JLW-algebra]]
[[!redirects JLW-algebras]]


Joren W. Brunekreef

* [institute page](https://www.uva.nl/en/profile/b/r/j.w.brunekreef/j.w.brunekreef.html?cb)

* [insitute page](https://www.ru.nl/en/people/brunekreef-j)

## Selected writing

On [[quantum circuit]]-compilation ([[Solovay-Kitaev theorem]]) in [[topological quantum computation]]:

* [[Joren W. Brunekreef]], *Topological Quantum Computation and Quantum Compilation*, Utrecht (2014) [[studenttheses.uu.nl:20.500.12932/17738](https://studenttheses.uu.nl/handle/20.500.12932/17738)]

category: people

[[!redirects Joren Brunekreef]]
[[!redirects J. W. Brunekreef]]


Jorge Picado

Jorge Picado is a professor at the University of Coimbra.

[Website](http://www.mat.uc.pt/~picado/).

## Selected writings

A textbook on [[locales]]:

* [[Jorge Picado]], [[Aleš Pultr]], _[[Frames and Locales]]. Topology without points_, Frontiers in Mathematics, Birkhäuser (2012).


Jorge Russo

* [webpage](https://www.perimeterinstitute.ca/people/jorge-russo)

## Related $n$Lab entries

* [[T-duality]]

category: people


Jorge Santos

* [webpage](http://www.damtp.cam.ac.uk/people/j.e.santos/)

## Selected writings

Discussion of [[thermodynamics]], [[phase transitions]] and [[thermal field theory]] of the BMN matrix model:

* Miguel S. Costa, Lauren Greenspan, Joao Penedones, [[Jorge Santos]], _Thermodynamics of the BMN matrix model at strong coupling_, JHEP 03 (2015) 069 ([arXiv:1411.5541](https://arxiv.org/abs/1411.5541))

On the [[cosmic censorship hypothesis]] and the [[weak gravity conjecture]]:

* {#CrisfordSantos17} Toby Crisford, [[Jorge Santos]], _Violating weak cosmic censorship in AdS$_4$_, Phys. Rev. Lett. 118, 181101 (2017) ([arXiv:1702.05490](https://arxiv.org/abs/1702.05490))

* {#HorowitzSantos19} [[Gary Horowitz]], [[Jorge Santos]], _Further evidence for the weak gravity - cosmic censorship connection_ ([arXiv:1901.11096](https://arxiv.org/abs/1901.11096))

## Related $n$Lab entries

* [[cosmic censorship hypothesis]]

* [[weak gravity conjecture]]

category: people


Jorge Zanelli

* [webpage](http://www.cchen.cl/index.php?option=com_content&task=view&id=691&Itemid=2)

## Selected writings

Introducing [[BTZ black holes]]:

* [[Máximo Bañados]], [[Claudio Teitelboim]], [[Jorge Zanelli]], _The Black Hole in Three Dimensional Space Time_, Phys. Rev. Lett. 69 (1992) 1849-1851 ([arXiv:hep-th/9204099](https://arxiv.org/abs/hep-th/9204099))

* [[Máximo Bañados]], [[Marc Henneaux]], [[Claudio Teitelboim]], [[Jorge Zanelli]],
_Geometry of the 2+1 Black Hole_, Phys. Rev. D48: 1506-1525, 1993 ([arXiv:gr-qc/9302012](https://arxiv.org/abs/gr-qc/9302012))

category: people


Jorrit Kruthoff

* [Institute page](https://sitp.stanford.edu/people/jorrit-kruthoff)

* [GoogleScholar page](https://scholar.google.com/citations?user=aJWdfR4AAAAJ&hl=en)

## Selected writings

On the [[K-theory classification of topological phases of matter]] for the case of [[topological crystalline insulators]] classified by [[equivariant K-theory]]:

* [[Jorrit Kruthoff]], [[Jan de Boer]], Jasper van Wezel, [[Charles L. Kane]], [[Robert-Jan Slager]], *Topological Classification of Crystalline Insulators through Band Structure Combinatorics*, Phys. Rev. X **7** (2017) 041069 $[$[doi:10.1103/PhysRevX.7.041069](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.041069)$]$

* [[Luuk Stehouwer]], [[Jan de Boer]], [[Jorrit Kruthoff]], [[Hessel Posthuma]], *Classification of crystalline topological insulators through K-theory*, Adv. Theor. Math. Phys, **25** 3 (2021) 723–775 [[arXiv:1811.02592](https://arxiv.org/abs/1811.02592), [doi:10.4310/ATMP.2021.v25.n3.a3](https://doi.org/10.4310/ATMP.2021.v25.n3.a3)]

category: people


Jose Gracia-Bondia

* [GoogleScholar page](https://scholar.google.co.cr/citations?user=mFszLq4AAAAJ&hl=en)

## Selected writings:

On the [[Moyal product]] as the [[polarization|polarized]] [[groupoid convolution algebra]] of the corresponding [[symplectic groupoid]], hence as an example of [[geometric quantization of symplectic groupoids]]:

* {#GBV} [[José Gracia-Bondia]], [[Joseph Varilly]], _From geometric quantization to Moyal quantization_, J. Math. Phys. 36 (1995) 2691-2701 ([arXiv:hep-th/9406170](http://arxiv.org/abs/hep-th/9406170))

On the [[Connes-Lott-Chamseddine-Barrett model]]:

* {#G-B19} [[José Gracia-Bondia]], _On Marshak's and Connes' views of chirality_, in: E. C. G. Sudarshan (ed.), _A Gift of Prophecy, _ Essays in Celebration of the Life of Robert Eugene Marshak, pp. 208-217, World Scientific  (1995)  ([arXiv:hep-th/9706200](https://arxiv.org/abs/hep-th/9706200), [doi:10.1142/9789812831408_0017](https://doi.org/10.1142/9789812831408_0017))

On [[noncommutative geometry]]:

* [[José Gracia-Bondia]], [[Joseph Varilly]], Héctor Figueroa, _Elements of noncommutative geometry_, Birkhäuser 2001. xviii+685 pp. ([doi:10.1007/978-1-4612-0005-5](https://link.springer.com/book/10.1007/978-1-4612-0005-5), [gBooks](http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false))

On [[renormalization]] of the [[Higgs field]]'s diphoton decay using [[causal perturbation theory]]:

* [[Pawel Duch]], [[Michael Duetsch]], [[Jose Gracia-Bondia]], _Diphoton decay of the higgs from the Epstein--Glaser viewpoint_ ([arXiv:2011.12675](https://arxiv.org/abs/2011.12675))

On [[perturbative quantum gravity]] via [[string-localized quantum fields]]:

* Christian Gaß, [[José M. Gracia-Bondía]], Karl-Henning Rehren, *Quantum general covariance* [[arXiv:2304.09843](https://arxiv.org/abs/2304.09843)]

## Related entries

* [[Moyal deformation quantization]]

* [[geometric quantization of symplectic groupoids]]

* [[renormalization]]

* [[Feynman diagram]]

* [[differential renormalization]]

* [[Connes-Lott-Chamseddine-Barrett model]]

category: people

[[!redirects José Gracia-Bondia]]
[[!redirects José Gracia-Bondía]]

[[!redirects José M. Gracia-Bondia]]
[[!redirects José M. Gracia-Bondía]]


Jose J. Fernandez-Melgarejo

* [personal page](https://webs.um.es/melgarejo/)

* [GoogleScholar page](https://scholar.google.co.jp/citations?user=dsKuZOkAAAAJ&hl=en)

## Selected writings

On [[exotic branes]] in [[string theory]]/[[M-theory]]:

* [[Jose J. Fernandez-Melgarejo]], [[Tetsuji Kimura]], [[Yuho Sakatani]], *Weaving the Exotic Web*, JHEP **2018** 72 (2018) [doi:10.1007/JHEP09(2018)072]

category: people

[[!redirects Jose Fernandez-Melgarejo]]


JoseBrox

Getting a ph.D. in Nonassociative Algebra and Ring Theory.


Josef Janyška

* [institure page](https://www.muni.cz/lide/1384-josef-janyska)

* [GoogleScholar page](https://scholar.google.com/citations?user=J8QibfEAAAAJ&hl=en)

## Selected writings

On [[jet bundles]] and [[natural bundles]]/[[natural operations]] in [[differential geometry]]:

* [[Demeter Krupka]], [[Josef Janyška]], Part 2 of: _Lectures on differential invariants_, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, [researchgate](https://www.researchgate.net/publication/36792711_Lectures_on_Differential_Invariants)]

## Related entries

* [[jet group]]

category: people

[[!redirects Josef Janyska]]


Josef Leutgeb

* [Institute page](http://www.itp.tuwien.ac.at/index.php/Homepage_Josef_Leutgeb)

## Selected wrirings

On the [[anomalous magnetic moment]] of the [[muon]] via [[holographic QCD]]:

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the anomalous magnetic moment of the muon_ ([arXiv:1912.01596](https://arxiv.org/abs/1912.01596))

* [[Josef Leutgeb]], [[Anton Rebhan]], _Axial vector transition form factors in holographic QCD and their contribution to the muon $g-2$_ ([arXiv:2012.06514](https://arxiv.org/abs/2012.06514))

* [[Josef Leutgeb]], [[Anton Rebhan]], *Hadronic light-by-light contribution to the muon $g-2$ from holographic QCD with massive pions* ([arXiv:2108.12345](https://arxiv.org/abs/2108.12345))

* [[Josef Leutgeb]], Jonas Mager, [[Anton Rebhan]], *Holographic QCD and the muon anomalous magnetic moment* ([arXiv:2110.07458](https://arxiv.org/abs/2110.07458))

category: people


Joseph A. Gallian

* [Wikipedia entry](https://en.wikipedia.org/wiki/Joseph_Gallian)

* [webpage](https://en.wikipedia.org/wiki/Joseph_Gallian)

## Selected writings

On basic [[algebra]] and [[group theory]]:

* [[Joseph A. Gallian]], *Contemporary Abstract Algebra*, Chapman and Hall/CRC (2020)  [[doi:10.1201/9781003142331](https://doi.org/10.1201/9781003142331), [webpage](https://en.wikipedia.org/wiki/Joseph_Gallian), [pdf](https://ict.iitk.ac.in/wp-content/uploads/CS203-Mathematics-for-Computer-Science-III-Gallian.pdf)]

category: people

[[!redirects Joseph Gallian]]


Joseph A. Zilber

Joseph Abraham Zilber defended his PhD thesis at Harvard University in 1963.
He was advised by [[Andrew Gleason]] and [[Raoul Bott]].

## Selected writings

Introducing [[simplicial sets]] and [[semi-simplicial sets]] (see the historical remarks [here](semi-simplicial+set#History)):

* {#EilenbergZilber50} [[Samuel Eilenberg]], [[Joseph Zilber]], *Semi-Simplicial Complexes and Singular Homology*, *Annals of Mathematics* vol. 51 no. 3 (1950), 499--513 ([jstor:1969364](https://www.jstor.org/stable/1969364))

Introducing the [[Eilenberg-Zilber theorem]]:

* [[Samuel Eilenberg]], [[Joseph Zilber]], _On Products of Complexes_, Amer. Jour. Math. 75 (1): 200–204, (1953) ([jstor:2372629](https://www.jstor.org/stable/2372629))

## Related $n$Lab entries

* [[Eilenberg-Zilber theorem]]

* [[Eilenberg-Zilber lemma]]

* [[Elkins-Zilber theorem]] (1976): two [[small categories]] are [[Morita equivalent]]
if and only if their [[idempotent completions]] are [[equivalent]].

[[!redirects Joseph Zilber]]
[[!redirects J. Zilber]]
[[!redirects J. A. Zilber]]


Joseph Atick

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=183209)

## Related $n$Lab pahes

* [[10d super Yang-Mills theory]]

* [[heterotic string theory]]

* [[torsion constraints in supergravity]]

category: people


Joseph Ayoub

* [website](http://user.math.uzh.ch/ayoub/)

## Selected writings

On [[Grothendieck's yoga of six operations]] in [[motivic homotopy theory]]:

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique_ PhD thesis, Paris [[pdf](http://www.math.uiuc.edu/K-theory/0761/THESE.pdf), [K:0761](https://faculty.math.illinois.edu/K-theory/0761)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I)*, Astérisque **314** (2007) [[numdam:AST_2007__314__R1_0](http://www.numdam.org/issues/AST_2007__314__R1_0)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II)*, Astérisque **315** (2007) [[numdam:AST_2007__315__1_0](http://www.numdam.org/issues/AST_2007__315__1_0)]

On [[Grothendieck's yoga of six operations]] for [[motives]] in [[rigid analytic geometry]]:

* [[Joseph Ayoub]], [[Martin Gallauer]], [[Alberto Vezzani]], *The six-functor formalism for rigid analytic motives*, Forum of Mathematics, Sigma **10** (2022) E61 [[doi:10.1017/fms.2022.55](https://doi.org/10.1017/fms.2022.55), [arXiv:2010.15004](https://arxiv.org/abs/2010.15004)]

## Related entries

* [[motive]]

* [[model structure on homotopical presheaves]]

category: people

[[!redirects Yoseph Ayoub]]
[[!redirects Josef Ayoub]]
[[!redirects Ayoub]]


Joseph Bernstein

__Joseph Bernstein__ is an Israeli mathematician, born and educated in Russia (his Russian name iss Иосиф Наумович Бернштейн, transliterated as Josif/Iosif Naumovič Bernštejn); his teacher there was [[Israel Gel'fand]].
His numerous works (especically with [[Alexander Beilinson]]) brought to proliferation of sheaf-theoretic and algebro-geometric methods in representation theory (D-modules, perverse sheaves, Beilinson-Bernstein localization theory), and are now cornerstones of [[geometric representation theory]]. With [[V. Lunts]] he introduced the appropriate definition of [[equivariant derived category]] (in general not equivalent to the (naive) derived category of equivariant sheaves).

* [wikipedia](http://en.wikipedia.org/wiki/Joseph_Bernstein)

* homepage at TelAviv: [web](http://www.math.tau.ac.il/~bernstei), publication [list](http://www.math.tau.ac.il/~bernstei/Publication_list/Publication_list.html)

* webpage at MathNet.ru: [Russian](http://www.mathnet.ru/php/person.phtml?option_lang=rus&personid=18576)_, [English](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=18576);
list of publications at mathnet.ru: [pdf](http://www.mathnet.ru/PersLogos/18576/Publication_list.pdf)

## Selected writings

Introducing the notion of [[perverse sheaves]] (and of [[t-structures]] on [[triangulated categories]]):

* [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]

See also:

* I. N. Bernšteĭn, I. M. Gelʹfand, V. A. Ponomarev,
Coxeter functors, and Gabriel's theorem. (Russian)
Uspehi Mat. Nauk __28__ (1973), no. 2(170), 19--33.

* J. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, _Алгебраические расслоения на $P^n$ и задачи линейной алгебры_, Функц. анализ и его прил., 12:3 (1978), 66–67, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2008&volume=12&year=1978&issue=3&fpage=66&what=fullt&option_lang=eng) (Russian); Engl. transl. _Algebraic bundles over $P^n$ and problems of linear algebra_, Funct. Anal. and its Appl. 1978, 12:3, 212–214

* A. Beilinson, J. Bernstein, _Localisations de $\mathfrak{g}$–modules_, C. R. Acad. Sci. Paris __292__ (1981), 15–18.

* J. Bernstein, [[V. Lunts]], _On nonholonomic irreducible D-modules_, Invent. Math. __94)), (1988), no. 2, 223–243.

* J. Bernstein, [[V. Lunts]], _Equivariant sheaves and functors, Springer Lecture Notes in Math. __1578__ (1994). [MR95k:55012](http://www.ams.org/mathscinet-getitem?mr=95k:55012)

* [[A. Beĭlinson]], J. Bernstein, _A proof of Jantzen conjectures_, I. M. Gelʹfand Seminar, 1--50, Adv. Soviet Math. __16__, Part 1, Amer. Math. Soc. 1993, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf)

* Joseph Bernstein's notes on [[D-module]]s: [pdf](http://www.math.uchicago.edu/~arinkin/langlands/Bernstein/Bernstein-dmod.pdf)

On the [[Langlands program]]:

*  {#BernsteinGelbart18} [[Joseph Bernstein]], [[Stephen Gelbart]] (eds.), _An Introduction to the Langlands Program_, Springer 2018 ([doi:10.1007/978-0-8176-8226-2](https://link.springer.com/book/10.1007/978-0-8176-8226-2))

[[!redirects J. Bernstein]]
[[!redirects I. N. Bernšteĭn]]


Joseph Chuang

* [Webpage](https://www.city.ac.uk/about/people/academics/joseph-chuang)

## Selected writings

* Joseph Chuang, [[Andrey Lazarev]], _L-infinity maps and twistings_, [arxiv/0912.1215](http://arxiv.org/abs/0912.1215);

* [[Joseph Chuang]], [[Andrey Lazarev]], _Abstract Hodge decomposition and minimal models for cyclic algebras_, [arxiv./0810.2393](http://arxiv.org/abs/0810.2393),

* Joseph Chuang, [[Andrey Lazarev]], _Feynman diagrams and minimal models for operadic algebras_, [arxiv/0802.3507](http://arxiv.org/abs/0802.3507),

* Joseph Chuang, [[Andrey Lazarev]], _Dual Feynman transform for modular operads_, [arxiv/0704.2561](http://arxiv.org/abs/0704.2561)

* Joseph Chuang, [[Julian Holstein]], [[Andrey Lazarev]]. _Homotopy theory of monoids and derived localization._ Journal of Homotopy and Related Structures. 16, pages 175–189 (2021) [doi:10.1007/s40062-021-0027](https://doi.org/10.1007/s40062-021-0027), and the corresponding [journal page](https://link.springer.com/article/10.1007/s40062-021-00276-6)

On the [[Riemann-Hilbert correspondence]]:

* Joseph Chuang, [[Julian Holstein]], [[Andrey Lazarev]] _Maurer–Cartan moduli and theorems of Riemann–Hilbert type_, Applied Categorical Structures **29** (2021) 685–728 [[doi:10.1007/s10485-021-09631-3](https://doi.org/10.1007/s10485-021-09631-3)]

category:people

[[!redirects J. Chuang]]


Joseph Conlon

* [webpage](http://www-thphys.physics.ox.ac.uk/people/JosephConlon/)

## related $n$Lab entries

* [[moduli stabilization]]

category: people


Joseph E. Avron

* [personal page](https://phsites.technion.ac.il/avron/)

* [GoogleScholar page](https://scholar.google.co.il/citations?user=epon4g8AAAAJ&hl=en)

## Selected writings

On the [[quantum adiabatic theorem]] with applications to [[condensed matter physics]]:

* [[Joseph E. Avron]], R. Seiler & L. G. Yaffe, *Adiabatic theorems and applications to the quantum Hall effect*, Communications in Mathematical Physics **110** (1987) 33–49 [[doi:10.1007/BF01209015](https://doi.org/10.1007/BF01209015)]

* [[Joseph E. Avron]], A. Raveh, B. Zur, *Adiabatic quantum transport in multiply connected systems*, Rev. Mod. Phys. **60** (1988) 873-915 [[doi:10.1103/RevModPhys.60.873](https://doi.org/10.1103/RevModPhys.60.873)]

category: people

[[!redirects Joseph Avron]]
[[!redirects J. E. Avron]]


Joseph Fourier

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Joseph_Fourier)

## Related entries

* [[Fourier transform]]

* [[harmonic analysis]]

[[!redirects Jean-Baptiste Joseph Fourier]]

category: people


Joseph Goguen

[[computer science|Computer scientist]] and applied [[category theory|category theorist]], 28th June 1941 – 3rd July 2006. His theoretical work includes the algebraic theory of abstract data types, initial model semantics, [[institution|institutions]], order sorted algebra, parameterized programming, database integration, hidden algebra, [[sheaf semantics of concurrent interacting objects|sheaf semantics of interacting objects]], and algebraic semiotics. Very important to computer science was his rôle in founding algebraic specification, including abstract data types, modules and module composition, and the OBJ language family. Goguen wrote [[A Categorical Manifesto|_A Categorical Manifesto_]], which tried to encourage the use of category theory in computer science.

## Links ##

[ Goguen's Web site at UCSD ]( http://cseweb.ucsd.edu/~goguen/ )

[ Wikipedia entry for _Joseph Goguen_ ]( http://en.wikipedia.org/wiki/Joseph_Goguen )

[ _In memoriam_ ]( http://homepage.mac.com/fiadeiroj/jose/IFIP-WG1.3/goguen.html ). By the [ IFIP Working Group WG1.3 on Foundations of System Specification ]( http://homepage.mac.com/fiadeiroj/jose/IFIP-WG1.3/index.html ).

## Selected writings

Introducing [[indexed categories]] as a tool of [[categorical semantics]] in [[computer science]], with fundamental results such as the existence and construction of [(co)limits in Grothendieck constructions](Grothendieck+construction#CoLimitsInAGrothendieckConstruction):

* {#TBG91} [[Andrzej Tarlecki]], [[Rod M. Burstall]], [[Joseph A. Goguen]], *Some fundamental algebraic tools for the semantics of computation: Part 3. Indexed categories*, Theoretical Computer Science **91** 2 (1991) 239-264 [doi:10.1016/0304-3975(91)90085-G]

category: people

[[!redirects Joseph A. Goguen]]


Joseph Hundley

* [MO homepage](http://mathoverflow.net/users/21252/joseph-hundley)

## related $n$Lab entries

* [[L-function]]

* [[automorphic representation]]

* [[adelic integration]]

category: people


Joseph J. Rotman

* [Wikipedia entry](https://en.wikipedia.org/wiki/Joseph_J._Rotman)

## Selected writings

On [[algebraic topology]]:

* [[Joseph J. Rotman]], *An Introduction to Algebraic Topology*, Graduate Texts in Mathematics **119** (1988) $[$[doi:10.1007/978-1-4612-4576-6](https://doi.org/10.1007/978-1-4612-4576-6)$]$

On [[group theory]]:

* [[Joseph J. Rotman]], *An Introduction to the Theory of Groups*, Springer (1995) [[doi:10.1007/978-1-4612-4176-8](https://doi.org/10.1007/978-1-4612-4176-8), [pdf](https://eclass.uoa.gr/modules/document/file.php/MATH676/Rotman%20An%20introduction%20to%20the%20theory%20of%20groups.pdf)]

## Related entries

* [[finitely presented group]]

category: people

[[!redirects Joseph Rotman]]


Joseph Krasil'shchik

Joseph Krasil'shchik (Иосиф Семёнович Красильщик) is a mathematician at the Independent University of Moscow
and the Russian State University for the Humanities.
He got his PhD degree in 1978 and habilitated in 1997.

* [webpage](https://diffiety.mccme.ru/curvita/isk.htm)

## related $n$Lab entries

* [[diffiety]]

* [[variational bicomplex]]

* [[differential operator]]

* [[jet bundle]]

* [[conserved current]]

category: people


Joseph Liouville

[[!redirects Liouville]]

* [Wikipedia entry](http://en.wikipedia.org/wiki/Liouville)

category: people


Joseph Lipman

* [webpage](http://www.math.purdue.edu/~lipman/)

## Related entries

* [[Grothendieck duality]]

category: people


Joseph M. Renes

* [personal page](https://people.phys.ethz.ch/~renes/)

## Selected writings

On [[quantum information theory]]:

* [[Joseph M. Renes]], *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)] De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]

category: people

[[!redirects Joseph Renes]]


Joseph Malkoun

* [personal page](https://josephmalkoun.com/)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=165709)

## Selected writings

On the [[Atiyah-Sutcliffe conjecture]]:

* [[Joseph Malkoun]], *Root Systems and the Atiyah-Sutcliffe Problem*, Journal of Mathematical Physics 60, 101702 (2019) ([arXiv:1903.00325](https://arxiv.org/abs/1903.00325))

* [[Joseph Malkoun]], _The Atiyah-Sutcliffe determinant_ ([arXiv:1903.05957](https://arxiv.org/abs/1903.05957))

category: people


Joseph Neisendorfer

* [webpage](https://people.math.rochester.edu/faculty/jnei/)

## related $n$Lab entries

* [[localization of a ring]]

* [[rational homotopy theory]], [[nilpotent topological space]]

category: people

[[!redirects Joseph A. Neisendorfer]]


Joseph Polchinski

* [webpage](http://www.physics.ucsb.edu/people/academic/joseph-polchinski)

* [wikipedia entry](http://en.wikipedia.org/wiki/Joseph_Polchinski)

* _Memoirs_ ([arXiv:1708.09093](https://arxiv.org/abs/1708.09093))

* [[Raphael Bousso]], [[Fernando Quevedo]], [[Steven Weinberg]]:

  _Joseph Polchinski: A Biographical Memoir_

  ([arXiv:2002.02371](https://arxiv.org/abs/2002.02371))

## Selected writings

Introducing [[Polchinski's flow equation]] in [[renormalization]]:

* {#Polchinski84} [[Joseph Polchinski]], _Renormalization and effective Lagrangians_ , Nuclear Phys. B B231, 1984 ([pdf](http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf))

Introducing [[orientifolds]]:

* {#DaiLinPolchinski89} Jin Dai, R.G. Leigh, [[Joseph Polchinski]], p. 12 of: _New Connections Between String Theories_, Mod. Phys. Lett. A4 (1989) 2073-2083 ([spire:25758](http://inspirehep.net/record/25758), [pdf scan](https://lib-extopc.kek.jp/preprints/PDF/1989/8905/8905564.pdf))

Introducing [[D-branes]]:

* {#Polchinski95} [[Joseph Polchinski]], _Dirichlet-Branes and Ramond-Ramond Charges_, Phys. Rev. Lett. 75:4724-4727, 1995 ([arXiv:hep-th/9510017](https://arxiv.org/abs/hep-th/9510017))

  from p.7:

> although it appears that we have modified the type II theory by adding something new to it, we are now arguing that these objects are actually intrinsic to any nonperturbative formulation of the type II theory; presumably one should think of them as an alternate representation of the black $p$-branes

General review includes

* [[Joseph Polchinski]], _TASI Lectures on D-Branes_ ([arXiv:hep-th/9611050](https://arxiv.org/abs/hep-th/9611050))

On [[orientifolds]], [[RR-field tadpole cancellation]] and orientifolds of [[K3]]:

* [[Eric Gimon]], [[Joseph Polchinski]], _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54: 1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

  (specifically for [[D5-D9 brane bound states]] [[duality between heterotic and type I string theory|dual]] to [[small instantons]] in [[heterotic string theory]])

On [[AdS/CFT duality]]:

* [[Gary Horowitz]], [[Joseph Polchinski]], _Gauge/gravity duality_, [gr-qc/0602037](http://arxiv.org/abs/gr-qc/0602037)

* [[Joseph Polchinski]], _Introduction to Gauge/Gravity Duality_ ([arXiv:1010.6134](https://arxiv.org/abs/1010.6134))

Textbook on [[string theory]]:

* [[Joseph Polchinski]], _[[String theory]]_, Cambridge Monographs on Mathematical Physics

## Related $n$Lab entries

* [[Polchinski's flow equation]]

* [[string theory]]

* [[D-brane]]

* [[orientifold]], [[tadpole cancellation]]

* [[landscape of string theory vacua]]

[[!redirects Joe Polchinski]]

category: people


Joseph S. Verducci

* [Institute page](https://stat.osu.edu/people/verducci.1)

## Selected writings

On [[geometric group theory]] on the [[symmetric group]] ([[Cayley distance]], [[Kendall distance]], [[Mallows kernel]], etc.):

* [[Michael A. Fligner]], [[Joseph S. Verducci]], Section 4 of: *Distance Based Ranking Models*, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 ([jstor:2345433](https://www.jstor.org/stable/2345433))

* [[Michael A. Fligner]], [[Joseph S. Verducci]] (eds.), *Probability Models and Statistical Analyses for Ranking Data*, Lecture Notes in Statistics **80**, Springer 1993 ([doi:10.1007/978-1-4612-2738-0](https://link.springer.com/book/10.1007/978-1-4612-2738-0))

category: people

[[!redirects Joseph Verducci]]
[[!redirects J. S. Verducci]]


Joseph Silverman

* [webpage](http://www.math.brown.edu/~jhs/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Joseph_H._Silverman)

## related $n$Lab entries

* [[elliptic curve]]

* [[moduli stack of elliptic curves]]

category: people


Joseph Varilly

* [webpage](http://www.cimm.ucr.ac.cr/varilly/)

## Selected writings

On the [[Moyal product]] as the [[polarization|polarized]] [[groupoid convolution algebra]] of the corresponding [[symplectic groupoid]], hence as an example of [[geometric quantization of symplectic groupoids]]:

* {#GBV} [[José Gracia-Bondia]], [[Joseph Varilly]], _From geometric quantization to Moyal quantization_, J. Math. Phys. 36 (1995) 2691-2701 ([arXiv:hep-th/9406170](http://arxiv.org/abs/hep-th/9406170))

On [[noncommutative geometry]]:

* [[José Gracia-Bondia]], [[Joseph Varilly]], Héctor Figueroa, _Elements of noncommutative geometry_, Birkhäuser 2001. xviii+685 pp. ([doi:10.1007/978-1-4612-0005-5](https://link.springer.com/book/10.1007/978-1-4612-0005-5), [gBooks](http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false))

Discussion of the [[spectral triple|spectral]] [[Riemannian geometry]] of the [[fuzzy 2-sphere]]:

* [[Francesco D'Andrea]], [[Fedele Lizzi]], [[Joseph Várilly]], _Metric Properties of the Fuzzy Sphere_, Lett. Math. Phys.103 (2013), 183-205 ([arXiv:1209.0108](https://arxiv.org/abs/1209.0108))

## Related $n$Lab entries

* [[Moyal deformation quantization]]

* [[geometric quantization of symplectic groupoids]]

category: people

[[!redirects Joseph Várilly]]


Joseph Wolf

* [webpage](https://math.berkeley.edu/people/faculty/joseph-wolf)

## Selected writings

On [[symmetric space|symmetric]] [[quaternion-Kähler manifolds]] ([[Wolf spaces]]):

* [[Joseph Wolf|Joseph A. Wolf]], _Complex homogeneous contact manifolds and quaternionic symmetric spaces_, Journal of Math. and Mech., 14 (1965), p. 166 ([jstor:24901319](https://www.jstor.org/stable/24901319))

On [[Clifford-Klein space forms]] such as [[spherical space forms]]:

* {#Wolf74} [[Joseph Wolf]], _Spaces of constant curvature_, Publish or Perish, Boston, Third ed., 1974 ([doi:10.1090/chel/372](https://doi.org/10.1090/chel/372))

## Related $n$Lab entries

* [[group action on an n-sphere]]

category: people

[[!redirects Joseph A. Wolf]]


Joseph Y. Halpern

* [personal page](https://www.cs.cornell.edu/home/halpern/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Joseph_Halpern)

## Selected writings

On [[S5 modal logic]] as [[epistemic logic]]:

* [[Joseph Y. Halpern]], [[Yoram Moses]], *A guide to completeness and complexity for modal logics of knowledge and belief*, Artificial Intelligence **54** 3 (1992) 319-379

* {#FaginHalpernMosesVardi95} Ronald Fagin, [[Joseph Y. Halpern]], [[Yoram Moses]], Moshe Y. Vardi, *Reasoning About Knowledge*, The MIT Press (1995) $[$[ISBN:9780262562003](https://mitpress.mit.edu/9780262562003/reasoning-about-knowledge/)$]$

* [[Joseph Y. Halpern]], *Should knowledge entail belief?*, Journal of Philosophical Logic **25** 5 (1996) 483-494  $[$[jstor:30226583](https://www.jstor.org/stable/30226583), [philpapers:HALSKE](https://philpapers.org/rec/HALSKE)$]$

[[!redirects Joseph Halpern]]

category: people


Joseph-Louis Lagrange

* [Wikipedia entry](https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange)

## Selected writings

Introducing [[Lagrange's theorem]] in [[group theory]]:

* [[Joseph-Louis Lagrange]], *Réflexions sur la résolution algébrique des équations*,  Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, années 1770 et 1771, Œuvres complètes, tome 3, 205-421 ([mathdoc](http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__3_205_0))

for the history see:

* Richard L. Roth, *A History of Lagrange's Theorem on Groups*, Mathematics Magazine **74** 2 (2001) 99-108 ([doi:10.2307/2690624](https://doi.org/10.2307/2690624), [jstor:2690624](https://www.jstor.org/stable/2690624))

## Related $n$Lab entries

* [[Lagrange's theorem]]

* [[Lagrange multiplier]]

* [[Lagrangian density]]

* [[Euler-Lagrange equation]]

category: people

[[!redirects Joseph-Louis de Lagrange]]

[[!redirects Lagrange]]


Josephson effect

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

\tableofcontents

## Idea

(...)

## Related concepts

* [[superconductor]]

## References

### General

See also:

* Wikipedia, _[Josephson effect](http://en.wikipedia.org/wiki/Josephson_effect)_

* Wikipedia, *[Pi Josephson junction](https://en.wikipedia.org/wiki/Pi_Josephson_junction)*

A review of the history of the discovery is here:

* Donald MacDonald, _The Nobel Laureate versus The Graduate Student_, Physics Today, July 2001 ([pdf](http://www.physics.umd.edu/courses/Phys798S/anlage/Phys798SAnlageSpring06/Josephson%20Physics%20Today%205153075.pdf))

[[!include superconducting qbits -- references]]

[[!redirects Josephson's effect]]

[[!redirects Josephson junction]]
[[!redirects Josephson junctions]]


Josh Cork

* [webpage](https://physicalsciences.leeds.ac.uk/pgr/806/josh-cork)

## Selected writings

On [[calorons]]:

* {#Cork18a} [[Josh Cork]], _Calorons, symmetry, and the soliton trinity_, PhD thesis, University of Leeds 2018 ([web](http://etheses.whiterose.ac.uk/22097/))

On [[Skyrmions]]:

* {#Cork18b} [[Josh Cork]], _Skyrmions from calorons_, J. High Energ. Phys. (2018) 2018: 137 ([arXiv:1810.04143](https://arxiv.org/abs/1810.04143))

On gauged [[skyrmions]] with more realistic binding energies (via [[equivariant de Rham cohomology]] and maybe some kind of [[equivariant Hopf degree theorem]]):

* [[Josh Cork]], [[Derek Harland]], [[Thomas Winyard]], *A model for gauged skyrmions with low binding energies* ([arXiv:2109.06886](https://arxiv.org/abs/2109.06886))

* [[Josh Cork]], [[Derek Harland]], *Geometry of Gauged Skyrmions*, SIGMA **19** o71 (2023) [[doi:10.3842/SIGMA.2023.071](https://doi.org/10.3842/SIGMA.2023.071), [arXiv:2303.02623](https://arxiv.org/abs/2303.02623)]

On [[Skyrmions]] via [[Yang-Mills instantons]]:

* [[Josh Cork]], [[Chris Halcrow]], *ADHM skyrmions* ([arXiv:2110.15190](https://arxiv.org/abs/2110.15190))

## Related $n$Lab entries

* [[Skyrmion]]

* [[caloron]]

* [[instanton]]

* [[soliton]]

category: people


Josh Guffin

* [InSpire page](https://inspirehep.net/authors/1045269)

## Selected writings

On [[quantum sheaf cohomology]]:

* [[Ron Donagi]], [[Josh Guffin]], [[Sheldon Katz]], [[Eric Sharpe]], *A Mathematical Theory of Quantum Sheaf Cohomology*, Asian J. Math. **18** (2014) 387-418 [[arXiv:1110.3751](https://arxiv.org/abs/1110.3751), [doi:10.4310/AJM.2014.v18.n3.a1](https://doi.org/10.4310/AJM.2014.v18.n3.a1)]

category: people


Joshua Erlich

* [webpage](https://www.wm.edu/as/physics/people/fulltimefaculty/erlich_j.php)

## Selected writings

On [[holographic QCD]]:

* {#Erlich09} [[Joshua Erlich]], *How Well Does AdS/QCD Describe QCD?*,	Int. J. Mod. Phys.A **25** (2010) 411-421 [[arXiv:0908.0312](https://arxiv.org/abs/0908.0312), [doi:10.1142/S0217751X10048718](https://doi.org/10.1142/S0217751X10048718)]

* {#Erlich14} [[Joshua Erlich]], *An Introduction to Holographic QCD for Nonspecialists*, Contemporary Physics, **56** 2 (2015) [[arXiv:1407.5002](https://arxiv.org/abs/1407.5002), [doi:10.1080/00107514.2014.942079](https://doi.org/10.1080/00107514.2014.942079)]

On [[holographic light front QCD]]:

* [[Stanley Brodsky]], [[Guy de Teramond]],  [[Hans Günter Dosch]], [[Joshua Erlich]],
_Light-Front Holographic QCD and Emerging Confinement_, Physics Reports Volume 584, 8 July 2015, Pages 1-105 ([arXiv:1407.8131](https://arxiv.org/abs/1407.8131))

category: people


Joshua Friedman

* [webpage](http://www.math.sunysb.edu/~joshua/)

## related $n$Lab entries

* [[Selberg trace formula]]

* [[Selberg zeta function]]

category: people


Joshua Moerman

* [webpage](https://joshuamoerman.nl)

## Selected writings

On [[rational homotopy theory]] via a [[model structure on dgc-algebras]]:

* [[Joshua Moerman]], _Rational Homotopy Theory_, 2015 ([pdf](https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf), [[MoermanRationalHomotopyTheory.pdf:file]])

category: people


Joshua Schwartz

category: people


Joshua Tan

I'm a PhD student at Oxford, studying under Samson Abramsky and Bob Coecke. My [website](http://www.joshuatan.com/research/).


Joshua Zak

* [personal page](https://phsites.technion.ac.il/zak/)

## Selected writings

On [[Berry phases]] along 1-[[cycles]] of a [[Brillouin torus]] ("Zak phases"):

* {#Zak89} [[Joshua Zak]], *Berry’s phase for energy bands in solids*, Phys. Rev. Lett. **62** (1989) 2747 ([doi:10.1103/PhysRevLett.62.2747](https://doi.org/10.1103/PhysRevLett.62.2747))

category: people


Josip Plemelj

__Josip Plemelj__ was a Slovenian mathematician (1873-1967) with main results in the theory of analytic functions. He is known, among the rest, for the Plemelj-Sochocki formulae (also spelled Sokhotski-Plemelj formulae) about the [[principal value]].

* [wikipedia](https://en.wikipedia.org/wiki/Josip_Plemelj)
* biography at [MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Plemelj)
* biography in Slovenian at [Slovenski biografski leksikon](https://www.slovenska-biografija.si/oseba/sbi433627)

category: people


Josué Remedios

Josué Remedios is a Spanish mathematican based in the Departamento de Matemática Fundamental, Universidad de La Laguna, 38271 La Laguna, Spain.  He has worked with [[Antonio Cegarra]] on the total complex construction for [[bisimplicial sets]].

## Selected writings

On the [[diagonal of a bisimplicial set|diagonal]] of [[bisimplicial sets]]:

* [[Antonio Cegarra]], [[Josué Remedios]], _The relationship between the diagonal and the bar constructions on a bisimplicial set_, Topology and its applications, volume 153 (1)  (2005) ([pdf](http://www.ugr.es/~acegarra/Paperspdfs/TRBDWC.pdf), [doi:10.1016/j.topol.2004.12.003](https://doi.org/10.1016/j.topol.2004.12.003))

* [[Antonio Cegarra]], [[Josué Remedios]], _The behaviour of the $\overline{W}$-construction on the homotopy theory of bisimplicial sets_, Manuscripta Mathematica, volume 124 (4) Springer (2007)

On [[double groupoids]] and [[homotopy 2-types]]:

* [[Antonio Martínez Cegarra]], Benjamın A. Heredia, [[Josué Remedios]],  _Double groupoids and homotopy 2-types_, Appl. Categ. Struct. 20, No. 4, 323-378 (2012), see also [arXiv:1003.3820](http://arxiv.org/abs/1003.3820).

category: people


José de Azcárraga

José Adolfo de Azcárraga is full university professor of Theoretical Physics, IFIC member and emeritus professor at the University of Valencia, and president of the Spanish Royal Society of Physics (RSEF) ([web](http://www.uv.es/uvweb/college/en/news-release/-emeritus-professor-university-valencia-jose-adolfo-de-azcarraga-takes-responsabilityas-new-president-royal-spanish-society-physics-1285846070123/Noticia.html?id=1285890922554))

* [cv pdf](http://www.uv.es/~azcarrag/CV%20%20summary.pdf)

* [personal webpage](https://sites.google.com/a/azcarraga.es/j-a-de-azcarraga/)

## Selected writings

On [[Lie groups]], [[Lie algebras]], [[Lie algebra cohomology]] and applications to [[physics]]:

* [[José de Azcárraga]], [[José M. Izquierdo]], *[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]*, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [[doi:10.1017/CBO9780511599897](https://doi.org/10.1017/CBO9780511599897)]

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *An introduction to some novel applications of Lie algebra cohomology and physics*, RACSAM **95** 2 (2001) 225-248 [[arXiv:physics/9803046](http://arxiv.org/abs/physics/9803046), [eudml:40921](https://eudml.org/doc/40921)]

On higher [[Nambu brackets]]/[[higher Poisson structure]]:

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *On the generalizations of Poisson structures*, J. Phys. A **30** (1997) L607-L616 [[arXiv:hep-th/9703019](http://arxiv.org/abs/hep-th/9703019), [doi:10.1088/0305-4470/30/18/001](https://doi.org/10.1088/0305-4470/30/18/001)]

On [[n-Lie algebras]]:

* [[José de Azcárraga]], J. C. Perez Bueno, _Higher-order simple Lie algebras_, Commun. Math. Phys. **184** (1997) 669-681 [[arXiv:hep-th/9605213](http://arxiv.org/abs/hep-th/9605213), [doi:10.1007/s002200050079](https://doi.org/10.1007/s002200050079)]

* [[José de Azcárraga]], [[José M. Izquierdo]], *$n$-Ary algebras: a review with applications*, J. Phys. A: Math. Theor. **43** (2010) 293001 [[doi:10.1088/1751-8113/43/29/293001](https://doi.org/10.1088/1751-8113/43/29/293001)]

On [[extended super Minkowski spacetimes]] in the language of the  [[D'Auria-Fre formulation of supergravity]] ("FDA"s) and the [[brane scan]]:

* C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

## Related entries

* [[Green-Schwarz sigma-model]]

* [[extended super Minkowski spacetime]]

* [[gauged WZW model]]

* [[n-Lie algebra]]

* [[classical anomaly]]

* [[higher Poisson structure]]

* [[1d WZW model]]

category: people

[[!redirects Jose de Azcarraga]]

[[!redirects J. de Azcárraga]]
[[!redirects J. de Azcarraga]]


José Edelstein

* [webpage](http://www-fp.usc.es/~edels/)

## related $n$Lab entries

* [[vacuum amplitude]]

category: people

[[!redirects Jose Edelstein]]


José Figueroa-O’Farrill

* [webpage](http://www.maths.ed.ac.uk/~jmf/)

## Selected writings

On the [[BRST complex]] (with focus on its [[cochain cohomology]]):

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], *The cohomology of BRST complexes* (1988) [[pdf](https://www.maths.ed.ac.uk/~jmf/Research/PVBLICATIONS/homol.pdf), [[FigueroaKimuar-BRSTCohomology.pdf:file]]]

and application to the [[NSR string|NSR]] [[superstring]]:

* [[José Figueroa-O'Farrill]], [[Takashi Kimura]], _The BRST cohomology of the NSR string: vanishing and ``no-ghost'' theorems_, Comm. Math. Phys. **124** 1 (1989) 105-132. [[euclid:cmp/1104179078](http://projecteuclid.org/euclid.cmp/1104179078)]

On [[Lie algebra weight systems]] arising from [[super Lie algebras]]:

* {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014))

On [[M2-brane 3-algebras]] in the [[BLG model]] for coincident [[M2-branes]]:

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], _Metric Lie 3-algebras in Bagger-Lambert theory_, JHEP 0808 : 045, 2008 ([arXiv:0806.3242](https://arxiv.org/abs/0806.3242))

On the equivalence of [[M2-brane 3-algebras]] to [[Lie algebra representations]] over [[metric Lie algebras]]:

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

On the [[ADE classification]] of [[black brane|black]] [[BPS state|BPS]]-[[M2-brane]] solutions to [[D=11 supergravity]] (via [[spherical space forms]] of the [[7-sphere]]):

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* {#MedeirosFigueroa10} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

On [[Spencer cohomology]] applied to the classification of [[BPS state|BPS solutions]] of [[D=11 supergravity]]:

* {#FFS17} [[José Figueroa-O’Farrill]], Andrea Santi, _Spencer cohomology and eleven-dimensional supergravity_, Comm. Math. Phys. 349 (2017), 627-660 ([arXiv:1511.08737](https://arxiv.org/abs/1511.08737))

On [[torsion of a G-structure|torsion of G-structures]] for kinematical groups:

* [[José Figueroa-O'Farrill]], _On the intrinsic torsion of spacetime structures_ ([arXiv:2009.01948](https://arxiv.org/abs/2009.01948))

On [[spin geometry]]:

* {#FigueroaFarrill10} [[José Figueroa-O'Farrill]], _[PG course on Spin Geometry](https://empg.maths.ed.ac.uk/Activities/Spin/)_ lecture 7: _Riemannian holonomy_, 2010 ([pdf](https://empg.maths.ed.ac.uk/Activities/Spin/Lecture7.pdf))

On [[supersymmetry]] and [[Spencer cohomology]] in [[D=5 supergravity|D=5]]:

* Andrew Beckett, [[José Figueroa-O'Farrill]], *Killing superalgebras for lorentzian five-manifolds* ([arxiv:2105.05775](https://arxiv.org/abs/2105.05775))

## Related $n$Lab entries

* [[exceptional Lie algebra]], [[exceptional Lie group]], [[F4]]

* [[spin representation]], [[Majorana spinor]], [[charge conjugation matrix]]

* [[M2-brane]], [[M5-brane]], [[NS5-brane]]

* [[spherical space form]], [[group action on n-spheres]]

* [[4-sphere]], [[7-sphere]]

* [[near horizon geometry]]

* [[ADE classification]]

* [[BLG model]], [[ABJM model]]

* [[11-dimensional supergravity]]

* [[Freund-Rubin compactification]]

* [[superisometry group]]

* [[symplectic reduction]]

* [[BRST complex]]

* [[n-Lie algebra]]

* [[L-infinity algebra]], [[L-infinity algebras in physics]]

* [[Clifford algebra]]

* [[spinning string]]

* [[electric-magnetic duality]]

* [[gauge theory]], [[Yang-Mills theory]]

* [[moment map]]

* [[super Cartan geometry]]

* [[geometric quantization]]

* [[quaternion-Kähler manifold]]

* [[geometry of physics -- fundamental super p-branes]]

category: people

[[!redirects José Figueroa-O'Farrill]]

[[!redirects José Miguel Figueroa-O'Farrill]]

[[!redirects Figueroa-O’Farrill]]

[[!redirects Jose Figueroa-O'Farrill]]


José Gómez-Torrecillas

__José Gómez-Torrecillas__ is an algebraist from Spain.

* [personal page](https://www.ugr.es/~gomezj/)

## Selected writings

On [[Frobenius functors]]:

* {#CGN98} F. Castaño Iglesias, [[José Gómez-Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

See also:

* [[Tomasz Brzeziński]], [[Laiachi El Kaoutit]], J. Gómez-Torrecillas, _The bicategories of corings_, J. Pure & Appl. Alg. __205__ 3 (2006) 510--541 [[doi:10.1016/j.jpaa.2005.07.013](https://doi.org/10.1016/j.jpaa.2005.07.013) [math.RA/0408042](https://arxiv.org/abs/math/0408042)]

* [[Gabriella Böhm]], [[José Gómez-Torrecillas]], _Bilinear factorization of algebras_, Bull. Belg. Math. Soc. Simon Stevin 20(2): 221--244 (may 2013) [doi](https://doi.org/10.36045/bbms/1369316541)

* José Gómez-Torrecillas, _Separable functors in [[corings]]_, Int. J. of Math. and Math. Sci. __30__:4 (2002) 203--225, [doi](https://doi.org/10.1155/S016117120201270X), [pdf](http://www.emis.de/journals/HOA/IJMMS/Volume30_4/225.pdf)

* [[Alessandro Ardizzoni]], José Gómez-Torrecillas, [[Claudia Menini]], _Monadic decompositions and classical Lie theory_,  Appl Categor Struct __23__, 93--105 (2015) [arXiv:1203.2881](https://arxiv.org/abs/1203.2881), [doi](https://doi.org/10.1007/s10485-013-9326-7)

* F. Castaño Iglesias, José Gómez-Torrecillas, C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__:10, 4879--4900 (1998) [doi](https://doi.org/10.1080/00927879908826735)

category: people

[[!redirects José Gómez Torrecillas]]
[[!redirects Jose Gomez-Torrecillas]]
[[!redirects J. Gómez-Torrecillas]]


José Ignacio Burgos

* [webpage](http://www.icmat.es/miembros/burgos/)

category: people

[[!redirects Jose Ignacio Burgos]]

[[!redirects José Ignacio Burgos Gil]]
[[!redirects Jose Ignacio Burgos Gil]]


José M. Izquierdo

* [InSpire page](https://inspirehep.net/authors/1058867)

## Selected writings

On [[Lie groups]], [[Lie algebras]], [[Lie algebra cohomology]] and applications to [[physics]]:

* [[José de Azcárraga]], [[José M. Izquierdo]], *[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]*, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [[doi:10.1017/CBO9780511599897](https://doi.org/10.1017/CBO9780511599897)]

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *An introduction to some novel applications of Lie algebra cohomology and physics*, RACSAM **95** 2 (2001) 225-248 [[arXiv:physics/9803046](http://arxiv.org/abs/physics/9803046), [eudml:40921](https://eudml.org/doc/40921)]

On [[M2-M5-brane bound states]]:

* {#ILPT95} [[José M. Izquierdo]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], *Dyonic Membranes*, Nucl. Phys. B **460** (1996) 560-578 [[arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177), doi:10.1016/0550-3213(95)00606-0]

On higher [[Nambu brackets]]/[[higher Poisson structure]]:

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *On the generalizations of Poisson structures*, J. Phys. A **30** (1997) L607-L616 [[arXiv:hep-th/9703019](http://arxiv.org/abs/hep-th/9703019), [doi:10.1088/0305-4470/30/18/001](https://doi.org/10.1088/0305-4470/30/18/001)]

On [[n-Lie algebras]]:

* [[José de Azcárraga]], [[José M. Izquierdo]], *$n$-Ary algebras: a review with applications*, J. Phys. A: Math. Theor. **43** (2010) 293001 [[doi:10.1088/1751-8113/43/29/293001](https://doi.org/10.1088/1751-8113/43/29/293001)]

On the [[M-theory super Lie algebra]] in relation to the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$:

* {#FernandezIzquierdoOlmo15} J.J. Fernandez, [[José M. Izquierdo]], M.A. del Olmo, _Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators_, Nuclear Physics B **897** (2015) 87-97 [[arXiv:1504.05946](http://arxiv.org/abs/1504.05946)]

On [[extended super Minkowski spacetimes]] in the language of the  [[D'Auria-Fre formulation of supergravity]] ("FDA"s) and the [[brane scan]]:

* C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

* [[José de Azcárraga]], [[Jerome Gauntlett]], J. M. Izquierdo, [[Paul Townsend]], _Topological extensions of the supersymmetry algebra for extended objects_, Phys. Rev. Lett. __63__ (1989) 2443 ([spire](https://inspirehep.net/record/26393?ln=en))

category: people

[[!redirects Jose M. Izquierdo]]

[[!redirects José Izquierdo]]
[[!redirects Jose Izquierdo]]


José M. M. Senovilla

* [personal page](http://tp.lc.ehu.es/seno.html)

## Selected writings

On the [[Penrose singularity theorems]]:

* [[José M. M. Senovilla]], *A critical appraisal of the singularity theorems* ([arXiv:2108.07296](https://arxiv.org/abs/2108.07296))

category: people

[[!redirects José Senovilla]]
[[!redirects Jose Senovilla]]
[[!redirects J. M. M. Senovilla]]


José M. Moreno-Fernández

* [GoogleScholar page](https://scholar.google.com/citations?user=DZiEgu4AAAAJ)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=236068)

## Selected writings

On [[Whitehead L-infinity algebras|Whitehead $L_\infty$-algebras]] in [[rational homotopy theory]]:

* [[Francisco Belchí]], [[Urtzi Buijs]], [[José M. Moreno-Fernández]], [[Aniceto Murillo]], _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, **520** (2017) 16-31 [[arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008)]

Lifting of the theorem [Milnor & Moore, 1965 (Appendix)](Pontrjagin+ring#MilnorMoore65), equipping the [[rational numbers|rational]] [[Pontrjagin algebra]] of a [[loop space]] with [[A-infinity algebra|$A_\infty$-algebra]] structure and identifying it with the universal envelope of the [[Whitehead L-infinity algebra|Whitehead $L_\infty$-algebra]]:

* [[José M. Moreno-Fernández]], *The Milnor-Moore theorem for $L_\infty$-algebras in rational homotopy theory*, Mathematische Zeitschrift **300** (2022) 2147–2165  [[arXiv:1904.12530](https://arxiv.org/abs/1904.12530), [doi:10.1007/s00209-021-02838-z](https://doi.org/10.1007/s00209-021-02838-z)]

category: people

[[!redirects Jose M. Moreno-Fernandez]]

[[!redirects José Manuel Moreno-Fernández]]
[[!redirects Jose Manuel Moreno-Fernandez]]


José María Cantarero López

[[!redirects Jose Cantarero]]

* [personal page](http://personal.cimat.mx:8181/~cantarero/)

## Selected writings

On [[equivariant K-theory]] and [[twisted equivariant K-theory]]:

* {#Cantarero09} [[Jose Cantarero]], _Equivariant K-theory, groupoids and proper actions_, Thesis 2009 ([ubctheses:1.0068026](https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0068026), [[CantareroEquivariantKTheory.pdf:file]])

* {#Cantarero12} [[Jose Cantarero]], _Equivariant K-theory, groupoids and proper actions_, Journal of K-Theory, Volume 9, Issue 3  June 2012, pp. 475 - 501  ([arXiv:0803.3244](https://arxiv.org/abs/0803.3244), [doi:10.1017/is011011005jkt173](https://doi.org/10.1017/is011011005jkt173))

  > (short version of [Cantarero 09](#Cantarero09))

category: people


José Meseguer

[[!redirects Meseguer]]
José Meseguer is Professor of Computer Science at UIUC and leads the Formal Methods and Declarative Languages Laboratory there. He has worked on concurrency and categorical models corresponding to [[Petri nets]].

* [Webpage](https://formal.cs.illinois.edu/meseguer/)

category: people


José Moyal

* [Wikipedia entry](http://en.wikipedia.org/wiki/Jos%C3%A9_Enrique_Moyal)

## Related entries

* [[Moyal quantization]]

category: people

[[!redirects Jose Moyal]]
[[!redirects Moyal]]
[[!redirects J. Moyal]]
[[!redirects J. E. Moyal]]


José Sanjurjo

José Sanjurjo is a Spanish topologist working on dynamical systems and their study using notions derived from [[shape theory]]. He is at the Faculty of Mathematics, UCM, Madrid.

[home page](http://www.mat.ucm.es/imi/People/Rodriguez_Sanjurjo_JoseManuel.htm)

category: people

[[!redirects J. M. R. Sanjurjo]]


José Seade

* [Institute page](https://www.matem.unam.mx/fsd/jseade)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=49505)

* [French Wikipedia entry](https://fr.wikipedia.org/wiki/Jos%C3%A9_Seade)

## Selected writings

On [[twistor fibrations]]:

* {#SeadeVerjovsky02} [[José Seade]], [[Alberto Verjovsky]], _Higher dimensional complex Kleinian groups_,  Math Ann 322, 279–300 (2002) ([doi:10.1007/s002080100247](https://doi.org/10.1007/s002080100247))

* Angel Cano, Juan Pablo Navarrete, [[José Seade]], Section 10.1 in: _Kleinian Groups and Twistor Theory_, In: _Complex Kleinian Groups_, Progress in Mathematics, vol 303. Birkhäuser 2013 ([doi:10.1007/978-3-0348-0481-3_10](https://doi.org/10.1007/978-3-0348-0481-3_10))

and also on the equivariant [[Arnold-Kuiper-Massey theorem]]:

* Le, [[José Seade]], [[Alberto Verjovsky]], Section 4 of: _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t.  49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))

category: people


José Zapata

* [institute page](http://www.matmor.unam.mx/~zapata/)

* [GoogleScholar page](https://scholar.google.com/citations?user=4V8kW2EAAAAJ&hl=en)

## Selected writings

On [[multisymplectic geometry|multisymplectic]] [[boundary field theory]]:

* [[José Zapata]], _Multisymplectic effective General Boundary Field Theory_ (2013) [[pdf](http://relativity.phys.lsu.edu/ilqgs/zapata090313.pdf), [[Zapata-MultisymplecticBoundary.pdf:file]]]

On [[higher gauge theory|higher]] [[lattice gauge theory]]:

* [[Juan Orendain]], [[José A. Zapata]], *Higher homotopy and lattice gauge fields* [[arXiv:2311.02363](https://arxiv.org/abs/2311.02363)]

category: people

[[!redirects José A. Zapata]]

[[!redirects Jose Zapata]]
[[!redirects Jose A. Zapata]]


José Ádem

* [Wikipedia entry](http://en.wikipedia.org/wiki/José_Ádem)

## Related $n$Lab entries

* [[Adem relations]]

category: people

[[!redirects Jose Adem]]
[[!redirects José Adem]]

[[!redirects Adem]]


Jouanolou cover

__Jouanolou cover__ of a projective $n$-space by quadrics has been introduced by [[Jean-Pierre Jouanolou]] for the purposes of [[algebraic K-theory]]

* J.-P. Jouanolou, _Une suite exacte de Mayer-Vietoris en K-th´eorie alg´ebrique_, Algebraic K-theory, I (Seattle, 1972), 293–316, Lecture Notes in Mathematics __341__, Springer (1973).

* MathOverflow: [The Jouanolou trick](http://mathoverflow.net/questions/7374/the-jouanolou-trick)

* [[Aravind Asok]], _The Jouanolou-Thomason homotopy lemma_, [pdf](http://www-bcf.usc.edu/~asok/notes/Jouanolou.pdf)

This is also used in [[Michel Van den Bergh]]'s proof that every projective variety over complexes is a quiver variety (see [blog](http://www.neverendingbooks.org/index.php/quiver-grassmannians-can-be-anything.html#comment-23413))

There is a noncommutative version of a Jouanolou cover for noncommutative projective space in a paper of Rosenberg and Konstevich.

* [[M. Kontsevich]], A. [[Rosenberg]], _Noncommutative smooth spaces_, The Gelfand Mathematical Seminars, 1996--1999,  85--108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; ([arXiv:math/9812158](http://arxiv.org/abs/math/9812158))


Jouko Mickelsson

* [webpage](http://www.theophys.kth.se/~jouko/)

## Selected writings

On [[twisted differential K-theory]]:

* [[Alan Carey]], [[Jouko Mickelsson]], [[Bai-Ling Wang]], _Differential Twisted K-theory and Applications_, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 ([arXiv:0708.3114](https://arxiv.org/abs/0708.3114), [doi:10.1016/j.geomphys.2009.02.002](https://doi.org/10.1016/j.geomphys.2009.02.002))

category: people


Jouko Väänänen

* [webpage](http://www.math.helsinki.fi/logic/people/jouko.vaananen/)

category: people


The Journal of Homotopy and Related Structures is a journal specialising in homotopy theory.

* [list of articles 2006-11](http://tcms.org.ge/Journals/JHRS/public.htm)

* [SpringerLink for articles 2012-](http://link.springer.com/journal/40062)


Joyal delta-ring

\tableofcontents

## Idea

The notion of a $\lambda$-[[lambda ring|ring]] $R$ is closely tied to classical [[Adams operations]] $\psi^n \colon R \to R$, modeled by [[monoid]] [[homomorphisms]] from the multiplicative monoid $(\mathbb{N}_+, \cdot)$ to the monoid of [[ring homomorphisms]] on the [[underlying]] [[commutative ring]] $R$ and their compositions. As $\mathbb{N}_+$ is freely generated by primes $p$, the Adams operations $\psi^p$ play a distinguished role. Classically, they are "[[Frobenius lifts]]", i.e., $\psi^p(x)$ agrees with $x^p$ modulo $p$: the Frobenius map $x \mapsto x^p$ which defines a ring endomorphism on $R/p R$ lifts to the homomorphism $\psi^p\colon R \to R$.

The notion of $\delta$-ring, originally due to [[André Joyal]] and developed independently by [[Alexandru Buium]] and others, axiomatizes the formal properties that hold for the "[[arithmetic derivative]]"

$$\delta(x) = \frac{\psi^p(x) - x^p}{p}.$$

## Definition

Fix a prime number $p$ and a power $q = p^r$ ($r \geq 0$).

\begin{definition} A $\delta$-ring is a commutative ring equipped with a unary operation $\delta: A \to A$ satisfying the following identities:

1. $\delta(x+y) = \delta(y) + \delta(y) - \sum_{i=1}^{q-1} \frac1{p}\binom{q}{i} x^i y^{q-i}$;

1. $\delta(x y) = x^q\delta(y) + \delta(x)y^q + p\delta(x)\delta(y)$, $\delta(1) = 0$.

We say that $A$ is a $\delta$-ring with respect to the pair $(p, q)$.
\end{definition}

These conditions imply that the unary operation $f(x) = x^q + p\delta(x)$ is an endomorphism of the ring $A$:

$$f(x+y) = f(x) + f(y); $$

$$f(x y) = f(x)f(y)$$

$$f(1) = 1$$
We say that $f$ is the Frobenius endomorphism associated with $\delta$.

Conversely, let $f$ be an endomorphism of a commutative ring $A$ without $p$-torsion. Suppose that for all $x \in A$

$$f(x) \equiv x^q \mod p A.$$
We obtain a structure of $\delta$-ring on $A$ defined as follows:

$$\delta(x) = \frac1{p}(f(x) - x^q).$$

## Examples

(...)


Joyal locus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On the one hand, [[infinity-groupoids|$\infty$-groupoids]] form an [[(infinity,1)-topos|$(\infty,1)$-topos]] ($Grpd_\infty$), while their [[stabilization]] -- plain [[spectra]] -- form instead a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]; on the other hand, the collection of (ie.: the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]] on) *[[parameterized spectra]]*, parameterized over [[infinity-groupoids|$\infty$-groupoids]], forms again an [[(infinity,1)-topos|$(\infty,1)$-topos]]: the "[[tangent (infinity,1)-topos|tangent $(\infty,1)$-topos]]" $T Grpd_\infty$.

This observation is originally due to [Biedermann (2007)](#Biedermann07), noted down in [Joyal (2008) §35](#Joyal08).

In search for a term to capture this curious phenomenon more generally, [Joyal 2015](#Joyal15) proposed to call a ([[pointed (infinity,1)-category|pointed]]) [[presentable (infinity,1)-category|presentable $(\infty,1)$-category]] an "$\infty$-locus" if the collection of its $Grpd_\infty$-parameterized objects (namely [[(infinity,1)-functors|$(\infty,1)$-functors]] $Grpd_\infty \to \mathcal{C}$) forms an [[(infinity,1)-topos|$(\infty,1)$-topos]].

Notice that the terminology "locus" here is unrelated to the common use of *[[locus]]* in mathematics. Compare the earlier proposal in [Joyal 2008](#Joyal08) to say "[[logos]]" for *[[(infinity,1)-category|$(\infty,1)$-category]]*, which is similarly ideosyncratic.

More generally, one may consider "loci" given by [[(infinity,2)-sheaves|$(\infty,2)$-sheaves]] on any [[(infinity,1)-topos|$(\infty,1)$-topos]] ([Hoyois 2019](#Hoyois19)).

## Examples

In the above motivating example, $\mathcal{C} =$ [[Spectra]]. In fact, every [[presentable (infinity,1)-category|presentable]] [[stable (infinity,1)-category|stable $(\infty,1)$-category]] is a Joyal $\infty$-locus (essentially by the [stable Giraud theorem](stable+infinity-category#StableGiraud), cf. [Hoyois 2019, p. 1 and Ex. 7](#Hoyois19)).
But, for instance, not just [[spectra]] but already [[prespectra]] and in fact just [[pointed homotopy types]] form a Joyal $\infty$-locus, in this sense.

If one drops the requirement that $\mathcal{C}$ be [[pointed (infinity,1)-category|pointed]], then every [[(infinity,1)-topos|$(\infty,1)$-topos]] is a Joyal $\infty$-locus (cf. [Hoyois 2019, Ex. 6](#Hoyois19)).

For the case over more general [[base (infinity,1)-toposes|base $(\infty,1)$-toposes]]: [[sheaves of spectra]] over an [[(infinity,1)-site|$(\infty,1)$-site]] $\mathcal{S}$ may be [[parameterized spectrum|parameterized]] over objects of the [[infinity-stack (infinity,1)-topos|$\infty$-stack $(\infinity,1)$-topos]] $Sh_\infty(\mathcal{S})$ and the collection of these parameterized sheaves of spectra forms the [[tangent (infinity,1)-topos|tangent $(\infty,1)$-topos]] $T Sh_\infty(\mathcal{S})$. Analogous statements hold more generally for [[n-excisive (infinity,1)-functors|$n$-excisive $(\infty,1)$-functors]] into any [[(infinity,1)-topos|$(\infty,1)$-topos]] (see [there](https://ncatlab.org/nlab/show/n-excisive+functor#nExcisiveFunctorsFormInfinityTopos)).

## Related concepts

* [[doubly monoidal (∞,1)-topos]]

* [[doubly closed monoidal category]]

## References

The terminology was proposed in:

* {#Joyal15} [[André Joyal]] at [58:00](https://youtu.be/qmCh9KwrQq8?t=3476), [59:52](https://youtu.be/qmCh9KwrQq8?t=3592) in: *New variations on the notion of topos*, talk at IHES (2015) [[video](https://www.youtube.com/watch?v=qmCh9KwrQq8)]

apparently motivated by the archetypical example of the [[tangent (infinity,1)-topos|tangent]] [[(infinity,1)-topos|$(\infty,1)$-topos]] of [[parameterized spectra]], previously noticed in

* {#Joyal08} [[André Joyal]], §35 of: _Notes on Logoi_ (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf), [[JoyalOnLogoi2008.pdf:file]]]

and further highlighted in

* [[Mathieu Anel]], [[André Joyal]], §4.2.3 in: *Topo-logie*, in *[[New Spaces for Mathematics and Physics]]*, Cambridge University Press (2021) 155-257 [[doi:10.1017/9781108854429.007](https://doi.org/10.1017/9781108854429.007), [pdf](http://mathieu.anel.free.fr/mat/doc/Anel-Joyal-Topo-logie.pdf)]

which goes back to

* {#Biedermann07} [[Georg Biedermann]], private communication with Joyal (2007)

Dedicated discussion of the notion is in:

* {#Hoyois19} [[Marc Hoyois]], *Topoi of parametrized objects*, Theory and Applications of Categories, **34** 9 (2019) 243-248 [[arXiv:1611.02267](https://arxiv.org/abs/1611.02267), [tac:34-09](http://www.tac.mta.ca/tac/volumes/34/9/34-09abs.html)]

[[!redirects Joyal loci]]

[[!redirects Joyal infinity locus]]
[[!redirects Joyal infinity loci]]

[[!redirects Joyal infinity-locus]]
[[!redirects Joyal infinity-loci]]

[[!redirects Joyal ∞ locus]]
[[!redirects Joyal ∞ loci]]

[[!redirects Joyal ∞-locus]]
[[!redirects Joyal ∞-loci]]


Joyal-Tierney calculus

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

# Contents
* table of contents
{: toc}

## Idea

In the context of [[factorization systems]] such as they appear notably in [[enriched model category]] one frequently needs to handle iterated [[lifting problems]]. In the appendix of ([Joyal--Tierney, 06](#JoyalTierney)) a symbolic calculus is introduced to facilitate these computations.

A central point of it is to have the statement of prop. \ref{IteratedLifting} below be easily expressible in terms of "division on both sides"-operations.

## The calculus
 {#TheCalculus}

### Lifting

Let $\mathcal{E}$ be a [[category]] ([[locally small category|locally small]]).

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

For $f, g \in Mor(\mathcal{E})$, write

$$
  f \,⧄\, g
$$

if $f$ has the [[left lifting property]] against $g$, or equivalently if $g$ has the [[right lifting property]] against $f$.

For $S \in \mathcal{E}$ an [[object]], write

$$
  f \,⧄\, S
$$

to indicate that for the morphism $f : X \to Y$ the induced [[hom set]] morphism

$$
  \mathcal{E}(f, S) : \mathcal{E}(Y,S) \to \mathcal{E}(X,S)
$$

is surjective, dually for

$$
  S \,⧄\, f
  \,.
$$

In the case that $\mathcal{E}$ has a [[terminal object]] $*$ we have equivalently

$$
  f \,⧄\, S
  \;\;\Leftrightarrow\;\;
  f \,⧄\, (S \to *)
$$

and if $\mathcal{E}$ has an [[initial object]] $\emptyset$ we have equivalently

$$
  S \,⧄\, f
  \;\;\Leftrightarrow \;\;
  (\emptyset \to S) \,⧄\, f
  \,.
$$

Accordingly, for $Q \subset Mor(\mathcal{E})$ write
${}^{⧄}Q$ and $Q^{⧄}$ for the class of morphisms with left or right lifting property against all elements of $Q$, respectively.

=--

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

If $(L \dashv R) : \mathcal{E} \to \mathcal{F}$ is a pair of [[adjoint functors]], then

$$
  f \,⧄\, R(g)
  \;\;
  \Leftrightarrow
  \;\;
  L(f) \,⧄\, g
$$

=--

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

A pair of classes of morphisms $(L,R)$ in $\mathcal{E}$ is a _[[weak factorization system]]_ precisely if

1. every morphism in $\mathcal{E}$ factors as the [[composition]] of a morphism in $L$ followed by a morphism in $R$;

1. $R = L^{⧄}$ and $L = {}^{⧄} R$.

=--

### Tensoring

Let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ be three categories.

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

A [[functor]]

$$
  \otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3
$$

1. is called _divisible on the left_ if for every $A \in \mathcal{E}_1$ the functor $A \otimes (-)$ has a [[right adjoint]], to be denoted

   $$
     A \backslash (-) : \mathcal{E}_3 \to \mathcal{E}_2
     \,;
   $$

1. is called _divisible on the right_ if for every $A \in \mathcal{E}_2$ the functor $(-) \otimes A$ has a [[right adjoint]], to be denoted

   $$
     (-)/ A : \mathcal{E}_3 \to \mathcal{E}_1
     \,;
   $$

=--

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

If $\otimes$ is divisble on both sides, then there are [[natural isomorphisms]]  between the collections of morphisms

$$
  A \otimes B \to X
$$

and

$$
  B \to A\backslash X
$$

and

$$
  A \to X / B
  \,.
$$

=--

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

For every $f \in Mor(\mathcal{E}_1)$, $g \in Mor(\mathcal{E}_2)$ and $X \in \mathcal{E}_3$ we have

$$
  f \,⧄\, (X/g)
  \;\;
  \Leftrightarrow
  \;\;
  g \,⧄\, (f \backslash X)
  \,.
$$

=--

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

If $\mathcal{E}$ is a [[closed monoidal category|closed]] [[symmetric monoidal category]], then its [[tensor product]] functor $\otimes : \mathcal{E} \times \mathcal{E} \to \mathcal{E}$ is divisible on both sides, the two divisions coincide and are given by the [[internal hom]] $[-,-] : \mathcal{E}^{op} \times \mathcal{E} \to \mathcal{E}$

$$
  X/A \simeq [A,X] \simeq A\backslash X
  \,.
$$

=--

### Pushout-tensoring

Let now $\mathcal{E}_3$ have [[finite limits|finite]] [[colimits]] and let $\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$ be a [[functor]].

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

for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_2$, write

$$
  A \otimes Y
    \overset
      {A \otimes X}
      {\amalg}
  B \otimes X
  \longrightarrow
  B \otimes Y
$$

for the induced _[[pushout-product]]_ morphism, the canonical morphism out of the [[pushout]] induced from the commutativity of the diagram

$$
  \array{
    A \otimes X &\to& B \otimes X
    \\
    \downarrow && \downarrow
    \\
    A \otimes Y &\to& B \otimes Y
  }
  \,.
$$

=--

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

The pushout-product extends to a functor

$$
  \bar \otimes : \mathcal{E}_1^I \times \mathcal{E}_2^I \to \mathcal{E}_3^I
  \,,
$$

where $C^I$ denotes the [[arrow category]] of $C$.
=--

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

If in the above situation $\mathcal{E}_1$ and $\mathcal{E}_2$ have [[finite limits]] and $\otimes$ is divisble on both sides, def. \ref{Divisibility}, then also $\bar{\otimes}$ is divisible on both sides:

1. for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_3$, the left quotient is

   $$
     f \bar \backslash g
     \;\colon\;
     B \backslash X \to B \backslash Y \times_{A \backslash Y} A \backslash X
     \,;
   $$

1. for $f : S \to T$ in $\mathcal{E}_2$ and $g : X \to Y$ in $\mathcal{E}_3$, the right quotient is

   $$
      g \bar / f
      \;\colon\;
      X / T \to Y / T \times_{Y / S} X / S
     \,;
   $$

=--

A key statement now is the following, characterizing the [[right lifting property]] again [[pushout product]] morphisms:

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

In the above situation, let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ have all [[finite limit|finite]] [[limits]] and [[colimits]]. For all $u \in Mor(\mathcal{E}_1)$, $v \in Mor(\mathcal{E}_2)$, $f \in Mor(\mathcal{E}_3)$ we have

$$
  (u \bar \otimes v) \,⧄\, f
  \;\;
  \Leftrightarrow
  \;\;
  u \,⧄\, f \bar /v
  \;\;
  \Leftrightarrow
  \;\;
  v \,⧄\, u \bar \backslash f
  \,.
$$

=--

## Applications

### Reedy theory

Let $\mathcal{E}$ be a [[model category]]. Write $\Delta$ for the [[simplex category]] and [[sSet]] for the category of [[simplicial sets]]. In the [[Reedy model structure]] on the [[presheaf category]] $[\Delta^{op}, \mathcal{E}]$ the following constructions are central.

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

Write

$$
  \Box : sSet \times \mathcal{E} \to [\Delta^{op}, \mathcal{E}]
$$

for the functor given by

$$
  (S \Box X) : n \mapsto S_n \cdot X
  \,.
$$

Write

$$
  \otimes : [\Delta, Set] \times [\Delta^{op}, \mathcal{E}] \to\mathcal{E}
$$

for the functor given by the [[coend]]

$$
  S \otimes X = \int^{n \in \Delta} S_n \cdot X_n
  \,.
$$

=--

(Here on the right we have the canonical [[tensoring]] of $\mathcal{E}$ over [[Set]], where $S_n \cdot X \simeq \coprod_{s \in S_n} X$.)

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

The functor $\Box$ is divisible on both sides.

Let $X \in [\Delta^{op}, sSet]$. Then

* the object $\partial \Delta[n] \backslash X$ is the _[[Reedy model structure|matching object]]_ of $X$ at stage $n$;

* the morphism $(\partial \Delta[n] \hookrightarrow \Delta[n]) \backslash X$ is the canonical morphism from $X_n$ into the $n$-matching object.

Let $f : X \to Y$ be a morphism in $[\Delta^{op}, sSet]$. Then

* the [[Reedy model structure|relative matching morphism]] of $f$ at stage $n$ is

  $$
    (\partial \Delta[n] \hookrightarrow \Delta[n]) \bar \backslash f
    \,;
  $$

* the object $(\partial \Delta^c) \otimes X$ is the _latching object_ at stage $n$;

* the morphism $(\partial \Delta^c \to \Delta)\otimes X$ is the canonical morphism out of the latching object into $X_n$;

* the morphism $(\partial \Delta^c \to \Delta) \bar \otimes f$ is the _relative latching morphism_ of $f$.

=--

## Related concepts

* [[pushout-product axiom]]

* [[pullback-power]]

* [[Quillen negation]]/[[weak orthogonal class]]

## References

* {#JoyalTierney} [[André Joyal]], [[Myles Tierney]], _Quasi-categories vs Segal spaces_ ([arXiv:0607820](http://arxiv.org/abs/math/0607820))

[[!redirects Joyal-Tierney calculus]]
[[!redirects Joyal–Tierney calculus]]
[[!redirects Joyal--Tierney calculus]]


Jozef Steenbrink

* [webpage](http://www.math.ru.nl/~steenbrink/)

## Related entries

* [[Hodge theory]], [[Hodge structure]]

category: people


Joás Venâncio

## Selected writings

On [[SL(2,H)]] and [[spin representations]] of [[Spin(5,1)]] (see also _[[supersymmetry and division algebras]]_):

* [[Joás Venâncio]], [[Carlos Batista]], _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_, Adv. Appl. Clifford Algebras **31** 71 (2021) ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296), [doi:10.1007/s00006-021-01172-1](https://doi.org/10.1007/s00006-021-01172-1))

category: people


João Faria Martins

[[!redirects Joao Faria Martins]]

* [website](http://www1.maths.leeds.ac.uk/~pmtjfa/index.html)

## Selected writings

On the relation between the [[Turaev-Viro model]] and the [[Crane-Yetter model]]:

* {#BarrettGarciaIslasMartins04} [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))

On [[spin networks]],

* [[João Faria Martins]], [[Aleksandar Mikovic]], _Invariants of spin networks embedded in three-manifolds_.
 Comm. Math. Phys. (2006) 279, pp. 381-399,
([arXiv:gr-qc/0612137](https://arxiv.org/pdf/gr-qc/0612137.pdf), [doi:10.1007/s00220-008-0422-8](https://doi.org/10.1007/s00220-008-0422-8))

On the [[Yetter model|Yetter invariant]],

* [[João Faria Martins]] and  [[Tim Porter]], _On Yetter's invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups_, Theory and Applications of Categories, Vol. 18, 2007, No. 4, pp 118-150.
([TAC](http://www.tac.mta.ca/tac/volumes/18/4/18-04abs.html))

On  [[2-crossed modules]], [[2-crossed complexes]] and the maps and homotopies between such,

* [[Joao Faria Martins]], _The fundamental 2-crossed complex of a reduced CW-complex_,  Homology Homotopy Appl. 13(2): 129-157 (2011) (web [pdf] (https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-2/The-fundamental-2-crossed-complex-of-a-reduced-CW-complex/hha/1335806746.full).)

* [[Björn Gohla]] and João Faria Martins, _Pointed homotopy and pointed lax homotopy of 2-crossed module maps_, Advances in Mathematics, (248), (2013), Pages 986 - 1049, [doi](https://doi.org/10.1016/j.aim.2013.08.020)

On [[motion groupoids]] and [[mapping class groups]]:

* [[Fiona Torzewska]], [[João Faria Martins]], [[Paul Purdon Martin]], *Motion groupoids and mapping class groupoids*, Comm. Math. Phys. **402** (2023) 1621-1705 [[arXiv:2103.10377](https://arxiv.org/abs/2103.10377), [doi:10.1007/s00220-023-04755-0](https://doi.org/10.1007/s00220-023-04755-0)]

On Quinn's finite [[homotopy field theory]]:

* [[João Faria Martins]], [[Timothy Porter]], *A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids* [[arXiv:2301.02491](https://arxiv.org/abs/2301.02491)]

## Related $n$Lab entries

* [[path n-groupoid]]

* [[principal 2-bundle]]

* [[3-groupoid of Lie 3-algebra valued forms]]

* [[Yetter model]]

category: people


João Pedro P. dos Santos

* [webpage](http://www.math.jussieu.fr/~dos-santos/)

category: people