Categorification

definition annotation example

Categorification takes a known structure and adds structure so that what were properties become structures, in such a way that the original is recovered by forgetting the new structure. “This is rather vague; let’s give an example.”

Sources: 7 Sketches §4.4.1–4.4.2 (Remark 4.46–4.47, 4.53), §4.6; [BD98; CY96].

• $\mathbb{N}$ to $\mathbf{FinSet}$: replace $5,3,8$ by five-, three- and eight-element sets, $+$ by disjoint union $\bigsqcup$, and the property $5+3=8$ by the structure of an isomorphism $\underline{5}\bigsqcup \underline{3}\cong \underline{8}$; the analogue of the equation is still true, and “we can be more precise about how things relate”. Choosing a good categorification “is part of the art of mathematics… its success is often empirical”.
• Preorders to categories: the brute property “there exists a morphism $a\to b$” ($P(a,b)=\mathsf{true}$) becomes “here is a set of morphisms $a\to b$” — hom-Booleans become hom-sets; preorders are $\mathbf{Bool}$-categories and categories are $\mathbf{Set}$-categories (Enriched Category).
• Wiring diagrams (§4.4.2): for preorders, boxes are anonymous $\le$; for categories the boxes carry morphism names, and the coherence laws (associativity, unitality) are “precisely what is needed to lengthen and shorten wires without ambiguity”: $f;(g;h)=(f;g);h$ means the composite box is unambiguous, and identities may be inserted or discarded. Combining with the multi-port boxes of monoidal preorders gives symmetric monoidal categories.
• Monoidal preorders to monoidal categories: the equation $(p\otimes q)\otimes r=p\otimes (q\otimes r)$ becomes the associator isomorphism ${\alpha }_{s,t,u}:(S\times T)\times U\cong S\times (T\times U)$ — “just a matter of bookkeeping” — but “new brute stuff emerges, and tends to be more complex”: the $\alpha$‘s must satisfy coherence conditions (pentagon, triangle). “The only way out of this morass is to add infinitely much structure, which leads to $\infty$-categories.” Mac Lane’s coherence theorem lets us pretend monoidal categories are strict (Remark 4.46–4.47).
• $\mathcal{V}$-categories (Remark 4.53): the properties $I\le \mathcal{X}(x,x)$ and $\mathcal{X}(x,y)\otimes \mathcal{X}(y,z)\le \mathcal{X}(x,z)$ become structures: chosen identity elements ${\mathrm{id}}_x:I\to \mathcal{X}(x,x)$ and composition morphisms. In a preorder “we do not need to choose, we just need to make sure they exist”.
• Categories to bicategories: composition associative only up to isomorphism (profunctors, cospans); symmetric monoidal bicategories categorify symmetric monoidal categories (§4.6). Kittenlab Lecture 15: “we choose not to walk that route today”.