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Snippets

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{{Disambiguation needed}}

{{anchor|Measure spaces}}

A tuple|triple is called a {{visible anchor|measure space}}.

transitive functional dependent[clarify]

{{contradict-other-multiple|Permutation|Lehmer code|Factorial number system|date=March 2013}}

Strong nouns

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Fem. A declension

plain ā[1]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[V 1] = 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)

  1. ^ a b Proto-Germanic ref
  1. ^ a b Cleasby strong nouns

Strong verbs

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Morpho tables

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

Inflection (-suff) & umlaut (ę←a→ǫ) in a Norse strong verb
  Inf -a Imp Pr P -andi
Pa P
N -it M -inn F -in
ek þú þat vér þér þau
Ind Pr T ę ę-r ǫ-um -ið -a
Pa T -t ǫ-um ǫ-uð ǫ-u
Subj Pr T -a -ir -i -im -ið -i
Pa T ę-a ę-ir ę-i ę-im ę-ið ę-i

-ra verbs

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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

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Image - Morpho stem map

  Infl. Root/Infl.
-um ę -r
Noun dat pl
Adj dat pl, masc dat sg
Verb Ind 1st plur

Articles needing diagrams

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List of opposite categories

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See: Presheaf (category theory), Opposite category, Dagger category, Spectrum of a ring#Functoriality, Chu space, Categorical algebra#Dual, Pontryagin duality#Categorical considerations, Category of relations, Grassmanian Algebras w/ Superpoints

Piecing things together

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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 is a generator: If f and g are different, then there is an element , such that f(x)≠g(x). Hence the map 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."