nLab cancellative category

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 $\mathcal{C}$ is left cancellative if all morphisms in $\mathcal{C}$ are monomorphisms (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $f\circ h_0=f\circ h_1$, then $h_0=h_1$). $\mathcal{C}$ is right cancellative if all morphisms in $\mathcal{C}$ are epimorphisms (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $h_0 \circ f=h_1\circ f$, then $h_0=h_1$). $\mathcal{C}$ is cancellative if it is both left cancellative and right cancellative.

The following was observed by Zhen Lin on MathOverflow.

Proposition. The following are equivalent for a category $\mathcal{C}$.

• Every morphism in $\mathcal{C}$ is a monomorphism.
• Every slice category $\mathcal{C}/X$ is a preorder.
• There exists a preorder $\mathcal{D}$ and a surjective discrete fibration $\mathcal{D} \to \mathcal{C}$.
• (Assuming $\mathcal{C}$ is small.) The presheaf topos $[\mathcal{C}^{\text{op}}, \text{Set}]$ is an étendue, i.e. there exists a presheaf $D : \mathcal{C}^{\text{op}} \to \text{Set}$ such that the unique morphism $D \to 1$ is an epimorphism and the slice $[\mathcal{C}^{\text{op}}, \text{Set}]_{/ D}$ is a localic topos.
• $\mathcal{C}$ is the codescent category of a preordered groupoid whose domain and codomain maps are discrete fibrations.

Examples

Cancellative categories

• any cancellative monoid, regarded as a category with a single object

• any groupoid

• any poset, or indeed any preorder

Left cancellative categories

• the category Field of fields (with ring homomorphisms as the morphisms)

• the category of nontrivial vector spaces (over the field of real numbers or complex numbers) equipped with positive definite inner products

Related concepts

• cancellative monoid

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

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set/pseudo-equivalence relation
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory
quantale	quantaloid