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left cancellative category

Contents

Contents

Idea

The concept of a (left/right) cancellative category is the generalization of the concept of (left/right) cancellative monoid from monoids to categories.

Definition

In category theory, “left cancellative” is a synonym for all arrows are monic. Thus the typical way for left cancellative categories to be constructed to take a category CC and then restrict to a class of monomorphisms closed under composition, such as all monomomorphisms, or regular monomorphisms if the category is regular, etc.

In fact every left cancellative CC arises this way (in the tautological sense of applying this consideration to CC itself): a category 𝒞\mathcal{C} being left cancellative means all its morphisms are monos.

Equivalently, for arbitrary morphisms f,h 0,h 1f,h_0,h_1 of 𝒞\mathcal{C}, if fh 0=fh 1f\circ h_0=f\circ h_1, then h 0=h 1h_0=h_1.

Examples

Tautological examples:

Non-tautological 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

References

  • M. V. Lawson and A. R. Wallis, A categorical description of Bass-Serre theory (arXiv:1304.6854v5)

  • M. V. Lawson, “Ordered Groupoids and Left Cancellative Categories” Semigroup Forum, Volume 68, Issue 3, (2004), 458–-476

Last revised on May 23, 2021 at 13:53:16. See the history of this page for a list of all contributions to it.