walking anti-shibboleth

the other day, probably via some more recent links, i bumped into this, where tom leinster described the idea of categorified universal properties as a category theory "shibboleth", something like an unconsciously acquired exclusionary trait; and where there was also some discussion of the category-theoretic usage of "walking" semi-popularized by john baez and myself.

(it's probably possible to find online a bit of my version of the history of "walking" if you look very hard, but perhaps i should give more of it at some point.)

in the discussion leinster did suggest that the shibbolethic exclusivity of the idea is unfortunate, and mike shulman did mention the centralness of the idea to the "doctrines of algebraic geometry" program. nevertheless there's something ironic (or worse) here to me: that an idea seen from within one culture as a shibboleth of it should in disguised form be a crucial shibboleth of another culture, and the two cultures (in this case category theory and algebraic geometry) go on miscommunicating with each other in such close quarters.

(i'm not planning to get into whether this is actually a common pattern with shibboleths.)

i should try to find out where it was that i first ran across an algebraic geometer saying something like "projective n-space is the classifying space for line bundles generated by n+1 sections", and to remember the process by which i gradually realized that what they were really saying was that "in the world of symmetric monoidal finitely cocomplete algebroids, the category of coherent sheaves over projective n-space is the walking example of a line object embedded in the direct sum of n+1 copies of the unit object"; unifying the mundane characterization of projective n-space as "the space of lines through the origin in affine [n+1]-space" with the more sophisticated characterization in terms of line bundles via the category-theoretic (and/or 2-category-theoretic) analysis of "variation" in terms of "variable elements" aka "generalized elements" (and/or "variable objects" aka "generalized objects").

i did at one point make somewhat of an effort to explain the "doctrines of algebraic geometry" philosophy to mike shulman, but i felt that i didn't get the real point of it across. i felt that he thought i was just trying to explain the idea of categorified universal properties, and moreover doing a bad job of it by belaboring only very simple examples in comparison to the much more sophisticated ones that he's used to; whereas what i was actually trying to explain was that if category-theorists will condescend to think about the very simple examples of categorified universal properties that algebraic geometers have used to build the edifice of algebraic geometry, they'll discover and understand the spectacular things that algebraic geometers have accomplished with such simple examples. (and if they think about how to extend the work of algebraic geometers using more sophisticated examples then they may discover something new and interesting.)

up until fairly recently i didn't have a clue as to what algebraic geometers were really doing; namely, exploiting categorified universal properties in amazingly powerful ways. once i stumbled upon that realization, the aspects of algebraic geometry that had previously seemed mysterious to me became practically transparent, to the point where shulman (quite falsely i'm sure) accused me of having more of a background in algebraic geometry than he did.

one of the alternate titles of the "doctrines of algebraic geometry" program is "algebraic geometry for category theorists". category theory sometimes shows up in algebraic geometry in conscious, mundane, complicated ways, but if category-theorists can instead tune in to to the unconscious, spectacular, simple ways then they're uniquely positioned to appreciate what the algebraic geometers are secretly really doing. this is what i think i failed to get across to shulman.