# Classifying the endofunctors of the category $\Delta$ of finite linear orders

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

• 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.) Aug 18, 2022 at 9:14
• I deleted two misguided comments with an incorrect definition of an endofunctor $[0<..<n]\to [0<...<2n]$. Aug 20, 2022 at 14:22