User talk:LokiClock/Sandbox

Snippets
Disambiguation needed

A tuple|triple $$\left(X, \Sigma, \mu\right)$$ is called a  .

Strong nouns
Fem. A declension

plain ā 3.3.2 - gjǫf -ing, -ung (in plural — i-decl. in sg.)

-j- (Gothic jō cog.s sibja-sif (this stem in pl. only), mawi-mær, and þiwi-þýr) fit hęl Frigg ey

-r = Gothic feminines in -is = stem(brúðr,bruþs,*brudhiz) assum. iyā stems i-umlauted Dat.&Acc. sg., as well for most roots ęrmr from armr heiðr heiðar heiði heiði kýr and other contracted roots -uðr/-unnr, -unn (e.g. Iðunn), -dís

-v- (assum. wā) ǫr(var) stǫð(var) bǫð(var) gǫt(var)



Morpho tables
Note the reversal of mood and tense in the column hierarchy.

-ra verbs
Sweet gives Pa T S sló, P slógu, Pa P slęginn for slá.

Unsourced past tense forms very likely, given the declension's name and the fact that the first form of a strong verb used to specify the conjugation in C-V entries is most often the past 1st-person sg. (e.g. bera)

Reflexives of contracted -ra can be constructed from part B of the entry for slá.

Table of inflections
Image - Morpho stem map

Articles needing diagrams

 * Cauchy–Riemann equations - orthogonality
 * Plate trick - cone between shoulder's (identity) and hand's rotation as a homotopy

List of opposite categories
See: Presheaf (category theory), Opposite category, Dagger category, Spectrum of a ring, Chu space, Categorical algebra, Pontryagin duality, Category of relations, Grassmanian Algebras w/ Superpoints

Piecing things together
Monoidal category:


 * 1) R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗R  serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has:


 * K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit.
 * Ab, the category of abelian groups, with the group of integers Z serving as the unit.

Generator (category theory):


 * In the category of abelian groups, the group of integers $$\mathbf Z$$ is a generator: If f and g are different, then there is an element $$x \in X$$, such that f(x)≠g(x). Hence the map $$\mathbf Z \rightarrow X,  n \mapsto n \cdot x$$  suffices.

Group scheme:

"There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine formal groups." "Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length."