horizontal categorification



Category theory





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.


  • 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.


David Roberts: How do Lie algebroids fit into this framework?

Urs Schreiber: at a rough level it is clear that the base space of a Lie algebroid has to be regardedas the “space of objects”. Certainly a Lie algebroid over a point is precisely a Lie algebra.

But for a more precise statement one needs a more conceptual way to think of Lie algebroids. I am claiming at ∞-Lie algebroid that there is a way to regard Lie algebroids precisely as certain kinds of synthetically smooth groupoids, namely those all whose morphisms have “infinitesimal extension” in some sense. In such a picture Lie algebroids are on the same footing as Lie groupoids and are precisely the many-object version of Lie algebras = infinitesimal Lie groups.

Mike: Is there (and do we want there to be) a general rule about whether an X-oid means an internal category whose one-object version is an X, or an enriched category whose one-object version is an X? The examples above seem to be taking the “internal” side, but the Cafe discussion on “ringoids” was about the “enriched” version; the two are very different! And “dg-algebroid” was suggested for dg-category, which is an enriched oidification of a dg-algebra, but a Hopf algebroid is an internal oidification of a Hopf algebra.

Urs: good point. We should say this at the beginning and split the list of examples in two sorts

algebraic structureoidification
truth valuetransitive relation
unital magmaunital magmoid
associative quasigroupassociative quasigroupoid
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
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

Further discussion

Related nn-Café-discussion is in

Last revised on May 22, 2021 at 23:49:08. See the history of this page for a list of all contributions to it.