Is there a theory of endofunctors of the category $\Delta$ of finite linear orders ? Can they be classified ? Is there a reference on this ?

Can one classify endofunctors $T:\Delta\to\Delta$ which give rise to sequences in $sC:=\operatorname{Func}(\Delta^\text{op},C)$ $$X_\bullet\to X_\bullet\circ T \to X_\bullet\times X_\bullet$$ (maybe with some nice properties making $X\circ T$ a path object etc.)?

The motivation for the question is that these endofunctors are sometimes used to define simplicial analogues of path spaces. I am particularly interested in the "edgewise subdivision" endofunctor, as explained in Dmitri Pavlov's answer. I quote two comments about this endofunctor.

This endofunctor is basically a special case of the simplicial join construction. See kerodon.net/tag/016K and also ncatlab.org/nlab/show/join+of+simplicial+sets. — Dmitri Pavlov

The endofunctor of simplicial sets induced by precomposition with [x2] on \Delta is the case r=2 of the r-fold "edgewise subdivision". See section 1 of Bokstedt, Hsiang, Madsen, Invent. Math. 111 (465-540) 1993, or section 6.2.1 of Dundas-Goodwillie-McCarthy, "The local structure of algebraic K-theory". A conceptual precursor is given in appendix 1 of Segal, Invent. Math. 21 (213-221) 1973, with the idea being credited to Quillen. — John Rognes


1 Answer 1


See Edgewise subdivision and simple maps by Knut Berg (supervised by me), Generalized edgewise subdivisions by Katerina Velcheva (supervised by Clark Barwick) and the earlier MathOverflow question What are the endofunctors on the simplex category?

  • $\begingroup$ Sorry, I am confused. Is $sd(X)$ a path object, in the sense that there is a sequence of maps $X\to sd(X)\to X\times X)$ with appropriate properties ? The thesis does not mention this explicitly, and Theorem 7.2.1. rather seems to contradict it ( $\S7.2$ (The natural map $e : sd \implies id$ is unique), Theorem 7.2.1. The natural transformation $e : sd \implies id$ in sSet is unique.) $\endgroup$
    – user420620
    Aug 18, 2022 at 9:14
  • $\begingroup$ I deleted two misguided comments with an incorrect definition of an endofunctor $[0<..<n]\to [0<...<2n]$. $\endgroup$
    – user420620
    Aug 20, 2022 at 14:22

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