# What is the history of the notion of subdivision of categories?

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my research and that I therefore almost do not have any motivation but sheer curiosity.)

According to Peter May, there was a "folklore" notion of categorical subdivision in the 1960's. I learnt about it by Matias del Hoyo's paper cited by Roman Bruckner in his comment to Peter May's answer. If I am mot mistaken, this notion had appeared in Anderson's paper "Fibrations and Geometric Realizations" as well as a paper authored by Dwyer and Kan, "Function complexes for diagrams of simplicial sets".

Who introduced the notion of subdivision of a (small) category? Are there any early references other than the two aforementioned papers?

Del Hoyo claims that performing the subdivision of a small category amounts to taking the nerve, applying Kan's simplicial subdivision functor, and coming back in $Cat$ by applying nerve's left-adjoint. Unfortunately, he does not prove this result. (I have discussed about this fact with him recently. If my memory serves me right, among other things, he proves that Anderson's and Dwyer-Kan's notions are equivalent.) Georges Maltsiniotis has given a rough proof of the verification to me. I was at that time unable to find any published proof. Even if it is an easy "folklore" result, I think it would be useful to have a proof publicly available somewhere.

Is there a published proof of the fact that this categorical subdivision is merely the composition of three well-known functors as above? Was it also "common knowledge" in the 1960's?

Finally, I cannot help asking a question which had come to my mind at that time, but to which I have not devoted much consideration since then. Using higher categorical nerves, there is an "obvious" definition of what could be analogs of this construction for higher categories. Therefore:

Have "higher analogs" of this categorical subdivision been studied?

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Sorry to blow my own horn, but I am teaching things related to this in the REU I run at University of Chicago, and I'm writing a book that will include an exposition of subdivisions of categories and some neat combinatorial relationships (due to students, not me) between that and other notions, certainly including the factorization you mention (which is probably the best definition of the subdivision of a category). Note that as a composite of left and right adjoints, this is not a categorically well-behaved construction. –  Peter May Jul 27 '12 at 17:45
Many thanks to Peter May for blowing such an interesting horn. I am eagerly waiting for this book to appear. I have been wondering for a while why neither Dwyer-Kan nor Thomason mention the fact that the subdivision is merely that composite. I would accept this comment as an answer if I could. (P.S. I had to look for the meaning of "REU". It means "Research Experience for Undergraduates".) –  Jonathan Chiche Jul 29 '12 at 3:29
This is not exactly an answer to your last question, but you may be interested in the work of Barwick and Kan on "relative categories" and "$n$-relative categories" -- I think they use a notion of "relative subdivision". –  Mike Shulman Jul 30 '12 at 2:23
Thanks, Mike. I was not aware of that, yet I was planning to study that work anyway! –  Jonathan Chiche Jul 31 '12 at 2:09