nLab
differential algebroid

Contents

Context

Enriched category theory

Algebra

Categorification

Contents

Idea

A differential algebroid is an oidification of the concept of differential algebra.

Definitions

Let KK be a commutative ring and CC be a KK-linear category. Then CC is a KK-differential algebroid if for every hom-KK-module Mor(a,b)Mor(a,b) there is a KK-linear morphism d Mor(a,b):Mor(a,b)Mor(a,b)d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b) such that for all objects a,b,cOb(C)a,b,c\in Ob(C), morphisms f:Mor(a,b)f:Mor(a,b) and g:Mor(b,c)g:Mor(b,c), and KK-linear morphisms d Mor(a,b):Mor(a,b)Mor(a,b)d_{Mor(a,b)}:Mor(a,b) \to Mor(a,b), d Mor(b,c):Mor(b,c)Mor(b,c)d_{Mor(b,c)}:Mor(b,c) \to Mor(b,c), and d Mor(a,c):Mor(a,c)Mor(a,c)d_{Mor(a,c)}:Mor(a,c) \to Mor(a,c), a generalised Leibniz rule is satisfied:

d Mor(a,c)(gf)=d Mor(b,c)(g)f+gd Mor(a,b)(f). d_{Mor(a,c)}(g \circ f) = d_{Mor(b,c)}(g) \circ f + g \circ d_{Mor(a,b)}(f) \,.

If all three objects are the same, this reduces down to the Leibniz rule for a derivation.

Examples

algebraic structureoidification
truth valuetransitive relation
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
(left,right) cancellative monoid(left,right) cancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
differential ring?differential ringoid?
nonassociative algebralinear magmoid
nonassociative unital algebraunital linear magmoid
nonunital algebralinear semicategory
associative unital algebralinear category
C-star algebraC-star category
differential algebradifferential algebroid
flexible algebraflexible linear magmoid
alternative algebraalternative linear magmoid
Lie algebraLie algebroid
strict monoidal categorystrict 2-category
strict 2-groupstrict 2-groupoid
monoidal poset?2-poset
monoidal groupoid?(2,1)-category
monoidal category2-category/bicategory
2-group2-groupoid/bigroupoid

Last revised on May 23, 2021 at 17:10:36. See the history of this page for a list of all contributions to it.