categorification

Contents

Definition

Horizontal categorification or Oidification describes the process by which

1. a concept is realized to be equivalent to a certain type of category 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

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.

• 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 finally the exception from the rule: a many-object monoid is not called a monoidoid – but is called a category! :-)

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

Further discussion

Related $n$-Café-discussion is in

Revised on March 6, 2012 09:11:06 by Urs Schreiber (82.113.98.74)