nLab horizontal categorification

Idea

Horizontal categorification or Oidification describes the process by which

1. a concept is realized to be equivalent to a certain type of category or magmoid with a single object;

2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

Remarks

• This is to be contrasted with vertical categorification.

• It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.

• It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.

Examples

• The horizontal categorification of groups are groupoids: categories in which every morphism is invertible.

• A horizontal categorification of algebras are algebroids: enriched categories in the category of vector spaces.

• A horizontal categorification of rings are ringoids: enriched categories over the category of abelian groups. (blog)

• A horizontal categorification of $C^*$-algebras hence ought to be known as $C^*$–algebroids but is usually known as C*-categories.

• Since, by the Gelfand-Naimark theorem, C-star algebras are dual to topological spaces, Paolo Bertozzini et. al proposed to define spaceoids to be entities dual to $C^*$-categories (blog).

• And an exception to the rule: a many-object monoid is not called a monoidoid – but is called a category!

  On the other hand, if the category is $R$-linear (enriched in R Mod) then it may again be referred to as an algebroid over $R$.

• A horizontal categorification of the notion of monads is that of categories enriched in a bicategory.

• A horizontal categorification of a single-sorted Lawvere theory is a multisorted Lawvere theory. Analogously an operad categorifies to a many-colored operad.

• In type theory/programming language theory, horizontal categorification is analogous to introducing a type distinction: an untyped language is a special case of a typed language where there is exactly one type. This fits with the categorical semantics: untyped lambda calculus has models in Lawvere theories, whereas (simply) typed lambda calculus has models in Cartesian multicategories.

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
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

Further discussion

Related $n$-Café-discussion is in

• John Baez, What is categorification?

• John Baez, Ringoids