nLab cancellative category

Contents

Contents

Idea

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

Definition

In category theory, a category π’ž\mathcal{C} is left cancellative if all morphisms in π’ž\mathcal{C} are monomorphisms (for arbitrary morphisms f,h 0,h 1f,h_0,h_1 of π’ž\mathcal{C}, if f∘h 0=f∘h 1f\circ h_0=f\circ h_1, then h 0=h 1h_0=h_1). π’ž\mathcal{C} is right cancellative if all morphisms in π’ž\mathcal{C} are epimorphisms (for arbitrary morphisms f,h 0,h 1f,h_0,h_1 of π’ž\mathcal{C}, if h 0∘f=h 1∘fh_0 \circ f=h_1\circ f, then h 0=h 1h_0=h_1). π’ž\mathcal{C} is cancellative if it is both left cancellative and right cancellative.

Examples

Cancellative categories

Left cancellative categories

References

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

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

algebraic structureoidification
truth valuepreorder
magmamagmoid
unital magmaunital magmoid
quasigroupquasigroupoid
looploopoid
semigroupsemicategory
monoidcategory
anti-involutive monoiddagger category
associative quasigroupassociative quasigroupoid
groupgroupoid
flexible magmaflexible magmoid
alternative magmaalternative magmoid
absorption monoidabsorption category
cancellative monoidcancellative category
rigCMon-enriched category
nonunital ringAb-enriched semicategory
nonassociative ringAb-enriched unital magmoid
ringringoid
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
monoidal poset2-poset
strict monoidal groupoid?strict (2,1)-category
strict 2-groupstrict 2-groupoid
strict monoidal categorystrict 2-category
monoidal groupoid(2,1)-category
2-group2-groupoid/bigroupoid
monoidal category2-category/bicategory

Last revised on May 20, 2022 at 20:03:59. See the history of this page for a list of all contributions to it.