I gave *Fourier-Mukai transform* a bit of an Idea-section. It overlaps substantially with the Definition section now, but I thought one needs to say the simple basic idea clearly in words first. Also added a few more pointers to literature.

I finally started *linear equation*. But am too tired now to really do it justice…

there had been no references at *Hilbert space*, I have added the following, focusing on the origin and application in quantum mechanics:

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John von Neumann,

Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.George Mackey,

The Mathematical Foundations of Quamtum MechanicsA Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963E. Prugovecki,

Quantum mechanics in Hilbert Space. Academic Press, 1971.

the first paragraphs at *topological vector space* seem odd to me.

I’d think it should start out saying that a topological vector space is a vector space *over a topological field* $k$, such that etc.. Then the following remark presently in the entry, about the internalization using the discrete topology is moot.

I had had need to link to and hence create some trivial entries, such as *real structure* and *antilinear map*. Didn’t find time yet to fill anything non-stubby into *quaternionic structure*.

I see that (from long, long time ago) one section of the entry *graded vector space* defines “pre-graded” to mean $\mathbb{Z}$-graded and “graded” to be $\mathbb{N}$-graded.

I am not sure if that is a good terminology, mainly because it seems not to be common. I came here from the entry *dg-Lie algebra*, wondering what that entry might actually mean by a “pre-graded” Lie algebra. (I should have commented on this long ago, of course).

brief entry *superoperator*, just for completeness

I’d be trying to write out a more detailed exposition of how fiber integration in twisted generalized cohomology/twisted Umkehr maps are axiomaized in linear homotopy-type theory.

To start with I produced a dictionary table, for inclusion in relevant entries:

]]>Created a little entry *Vect(X)* (to go along with *Vect*) and used the occasion to give *distributive monoidal category* the Examples-section that it was missing and similarly touched the Examples-section at *rig category*.

At *sesquilinear form* I have added the following general definition:

let $A$ be a not-necessarily commutative star-algebra. Let $V$ be a left $A$-module with $A$-linear dual denoted $V^\ast$. Then a sesquilinear form on $V$ is simply an element in the tensor product

$V^\ast \otimes_A V^\ast \,,$where we use the only possible way to regard the left $V$-module as a right $V$-module: by the star-involution.

I am wondering if there is anywhere some discussion as to how far one may push dg-algebra over $A$ this way, specifically for the cases where $A$ is a normed division algebra such that the quaternions or the octonions.

For instance which structure do we need on $A$ to make sense of the Grassmann algebra $\wedge^\bullet_A V^\ast$ of $V^\ast$ this way?

This is motivated by the following:

At *spin representation* I once put a remark that one may obtain the $N = 1$ super-translation Lie algebra simply by starting with the super-point $\mathbb{R}^{0\vert 2}$, regarded as an abelian super Lie algebra, and then forming the central extension by $\mathbb{R}^3$ which is classified by the cocycle $d \theta_i \wedge d \theta_j \in \wedge^2 (\mathbb{R}^2)^\ast$ $(1 \leq i \leq j \leq 2)$, with $\theta_i$ and $\theta_j$ the two canonical odd-graded coordinates on $\mathbb{R}^{0\vert 2}$.

Yesterday with John Huerta we were brainstorming about how to best formulate this such that the statement goes through verbatim for the other real normed division algebras to yield the super-translation Lie algebra alsoin dimensions 4,6 and 10.

With sesquilinear forms as above it is obvious: Let $\mathbb{K}$ any of the four real normed division algebras, consider the superpoint $\mathbb{K}^{0\vert 2}$ and then form the central extension of super Lie algebras classified by the sesquilinear forms $d \theta_i \otimes_{\mathbb{K}} d \theta_j$ ($1 \leq i \leq j \leq 2$).

These forms being sesquilinear expresses nothing but the spinor pairing of the susy algebra that Baez-Huerta (as reviewed here ) write as $(\psi,\phi)\mapsto \psi \phi^\dagger + \phi \psi^\dagger$.

]]>I gave the entry *super vector space* some expositional background and a more detailed (pedantic) definition.

At *Fréchet space* I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of $\mathbb{R}^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$. And I touched the description of this example in the main text, now here.

I have added to *locally convex topological vector space* the standard alternative characterization of continuity of linear functionals by a bound for one of the seminorms: here

(proof and/or more canonical reference should still be added).

]]>I spelled out the elementary definitions, relations and examples at *Kähler vector space* and *Hermitian space*.

This started out as a section that I added to *Kähler manifold*.

needed to be able to point to *direction of a vector*.

needed to be able to point to *wave vector*, so I created a bare minimum entry

created *inner product of vector bundles* with the construction over paracompact Hausdorff spaces

I have expanded the list of references at *supersymmetry and division algebras*.

I had splitt-off *quadratic refinement* from *quadratic form* and expanded slightly

It seems to me that the category of bundles over some base space, with morphisms the differential operators on spaces of sections of these bundles, is equivalently the co-Kleisli category of the Jet bundle comonad. Is this known? (It seems to be a different statement than that discussed by Blute-Cockett-Seely, as far as I see.)

More in detail, consider differential cohesion with infinitesimal shape modality $\Im$. For a given base space $X$, write

$i \colon X \longrightarrow \Im X$for the $X$-component of the unit of the $\Im$-monad. Then the operation of forming jet bundles is the comonad given by the base change adjoint triple $(i_! \dashv i^\ast \dashv i_\ast)$:

$Jet \coloneqq i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \to \mathbf{H}_{/X} \,.$Now, it is a standard fact that given two bundles $E_1, E_2$ over $X$, then differential operators

$D \colon \Gamma(E_1) \to \Gamma(E_2)$are equivalently bundle maps

$\tilde D \;\colon\; Jet(E_1) \longrightarrow E_2 \,,$where the equivalence is given by

$D(\phi) = \tilde D \circ j^\infty(\phi)$with $j^\infty \phi \in \Gamma(Jet(E_1))$ the jet bundle prolongation of $\phi$.

Combining this with the information that $Jet$ is a comonad, we have the impulse to say that $Jet(E_1) \to E_2$ is to be regarded as a morphism in its co-Kleisli category, and hence that under the above equivalence the composition of differential operators $D_2 \circ D_1$ corresponds to the composite

$Jet(E_1) \to Jet (Jet(E_1)) \stackrel{Jet(\tilde D_1)}{\longrightarrow} Jet(E_2) \stackrel{\tilde D_2}{\longrightarrow} E_3 \,,$where the first map is the co-product of the $Jet$ co-monad.

And it seems to me that this is actually true.

]]>added to *differential operator* the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the co-Kleisli category of the Jet bundle comonad.

just to clean up entries, I have given *torsion module* its own entry (the keyword used to have non-overlapping discussion at *torsion subgroup* and at *torsion approximation*)

created a simple table *(co)isotropic subspaces - table* for inclusion in other entries, just so as to usefully cross-link the relevant entries