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Stream: learning: reading & references

Topic: the (un)popularity of double categories

Notification Bot (Jan 01 2025 at 12:36):

This topic was moved here from #learning: questions > the (un)popularity of double categories by Morgan Rogers (he/him).

Tim Hosgood (Jan 02 2025 at 16:06):

slightly more general, but Andrée Ehresmann describes the popularity (and lack of) of internal and structured categories in her notes to Charles Ehresmann's Collected Works:

Benabou used internal categories in the late sixties (unpublished) and certainly helped to propagate them. Several Theses and papers written near us are wholly or partially devoted to structured or internal categories, e.g. Bourn [13], Conduche [22], Kempf [60], Langbaum [63], Lellahi [69] , Vaugelade [97] (without mentioning those on examples).

However internal categories, so universally used to-day, seem to be really of interest to other schools only in the seventies (Gray [39] , Diaconescu [26], ...). Though Grothendieck mentions the simplicial object associated to a category in [42], he prefers to work with the associated fibration (called a category object ) for avoiding pullbacks.

Comment 55.2, on /63/

John Baez (Jan 02 2025 at 17:06):

Thanks!

David Corfield (Jan 04 2025 at 10:42):

There seems to be growing interest in [[orthogonal factorization systems]] meet double categories. E.g.,

• Miloslav Štěpán, Factorization systems and double categories [arXiv:2305.06714]

and for $\infty$-categories:

• Branko Juran, On orthogonal factorization systems and double categories [arXiv:2501.01363]

Neither of these papers seems to suggest any great history to this connection. Is that right? It doesn't seem so great a step, and then to the well-studied area of distributive laws.

Peter Arndt (Jan 06 2025 at 02:16):

Since you only mentioned Grandis and Paré as early advocates of double categories: There was also Ronnie Brown, although mostly with doube groupoids. His inspiration was the double groupoid of maps of squares into a topological spaces up to homotopy (more precisely maps $(I^2, \partial I^2, \text{corners}) \to (X, B, A)$ for space triples $X \supseteq B \supseteq A$, up to triple homotopy). He proved in 1976, with Chris Spencer that double groupoids with connection with a single object are equivalent to crossed modules (introduced by Whitehead in the 40s in topology).

Peter Arndt (Jan 06 2025 at 02:18):

For my master thesis long ago I gave a pedestrian proof of the many object version of this result.

Peter Arndt (Jan 06 2025 at 02:21):

Apparently Lie double groupoids first appeared in J. Pradines, Géometrie différentielle au-dessus d'un groupoı̈de, C. R. Acad. Sci. Paris Sér. A, 266 (1967), pp. 1194-1196, see this, for more hints, but I don't think they became popular.

John Baez (Jan 06 2025 at 06:38):

Thanks, yes, I should not forget Ronnnie!