I am looking for a reference explicitly defining simplicial homotopy in terms of endofunctors of $\Delta$, and developing homotopy theory in this terms. The following is a particular question.

Is it possible to rephrase the following definition of simplicial homotopy in e.g. (8.3.11 of An introduction to homological algebra by C. Weibel) in terms of endofunctors of $\Delta$ ?

We say $f$ and $g$ are (strictly) simplicial homotopic if if for each $p\geq 0$, there exists morphisms $$ h_i=h^p_i:X_p\to Y_{p+1} \text{ for } i=0,\ldots,p $$ such that the following conditions hold.

  1. $$\partial_0 h_0=f_p, \partial_{p+1}h_p=g_p;$$

  2. $$ \partial_ih_j=\begin{cases}h_{j-1}\partial_i & i<j\\ \partial_ih_{i-1} &i=j\\ h_j\partial_{i-1} & i>j+1; \end{cases}$$

  3. $$ s_ih_j=\begin{cases}h_{j+1}\partial_i & i\leq j\\ h_js_{i-1} & i>j. \end{cases} $$

One way to try to do it is as follows.

Let $[\times 2]:\Delta\to\Delta$ denote the endomorphism "doubling" each linear order: $$[0\le 1\le \dotsb \le n] \longmapsto [0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'] $$ $$ f:[0\le 1\le \dotsb \le n]\to [0\le 1\le \dotsb \le m] \,\longmapsto\, f':[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'] \to [0 \le 0' \le 1 \le 1'\le \dotsb \le m \le m']$$ $$ f'(i)=f(i), f'(i')=f(i)' \forall 0\leq i\leq n.$$ If I understand correctly the answer by Dmitri Pavlov, a standard argument gives that $Y_\bullet[\times 2]$ is a path object: $$Y_\bullet\xrightarrow{w_\bullet} Y_\bullet[\times 2]\xrightarrow{p_\bullet} Y_\bullet\times Y_\bullet$$ where each $w_n:Y_n \to Y_{2n+1}$ corresponds to the map of preorders $$[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n']\to [0 \le 1 \le \dotsb \le n] , i,i'\mapsto i $$ "repeating each coordinate twice", and the projections $p^\text{even}_\bullet,p^\text{odd}_\bullet:Y_\bullet[\times 2]\to Y_\bullet$ are maps "leaving only even/odd coordinates", i.e. are given by $p_n^\text{even}: Y_{2n+1}\to Y_n $ corresponding to $$[0 \le 1 \le \dotsb \le n]\to [0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'], i\mapsto i$$ and $p_n^\text{odd}: Y_{2n+1}\to Y_n $ corresponding to $$[0 \le 1 \le \dotsb \le n]\to[0 \le 0' \le 1 \le 1'\le \dotsb \le n \le n'], i\mapsto i'$$

Question. Is it true (under some “niceness” assumptions) that in a simplicial category for $X_\bullet$ connected, two maps $f_\bullet,g_\bullet:X_\bullet\to Y_\bullet$ are (strictly) homotopic iff both maps $ X_\bullet\xrightarrow {f_\bullet} Y_\bullet$ and $ X_\bullet\xrightarrow {g_\bullet} Y_\bullet$ factor via $Y_\bullet[\times 2]$ as $$ X_\bullet\xrightarrow {\tilde h_\bullet} Y_\bullet\circ[\times 2] \xrightarrow{p^\text{even}_\bullet} Y_\bullet$$ $$ X_\bullet\xrightarrow {\tilde h_\bullet} Y_\bullet\circ[\times 2] \xrightarrow{p^\text{odd}_\bullet} Y_\bullet$$ where $p^\text{even}_\bullet:X_\bullet\circ[\times 2]\to X_\bullet$ and $p^\text{odd}_\bullet:X_\bullet\circ[\times 2]\to X_\bullet$ are maps induced by “forgetting” the odd/even half of the “doubled” linear order $0\le 1 \le \dotsb \le n \le 0' \le 1'\le \dotsb\le n'$, as above.

This appears to be true if one of the maps is trivial (i.e. is a map to a point), see Defining homotopy via endofunctors of a simplicial category.

  • $\begingroup$ An important keyword here is décalage. $\endgroup$
    – Tim Campion
    Aug 14, 2022 at 14:21
  • $\begingroup$ Could you explain what is the current status of the answer for you? Did you change the question after it? $\endgroup$ Aug 16, 2022 at 16:35
  • $\begingroup$ I replaced one obviously incorrect description of the ''doubling'' endofunctor by another incorrect description. I think the status of the answer is that it is an answer, $\endgroup$
    – user420620
    Aug 16, 2022 at 16:49
  • $\begingroup$ but I haven't yet understood it. An explicit reference to an explicit definition and treatment of homotopy in terms of this "doubling" functor would still be nice, but it may well follow from the references given. $\endgroup$
    – user420620
    Aug 16, 2022 at 17:00

1 Answer 1


Once we correct the definition of $[×2]$ to make it a functor (currently it does not preserve identities) by setting $f'=f⊔f$, it is true that the construction described in the main post gives rise to an equivalent notion of homotopy, assuming $Y$ is Kan.

Indeed, $[⨯2]$ is a right Quillen functor from simplicial sets to simplicial sets, with its left adjoint being the unique left Quillen functor that sends $Δ^n$ to $Δ^n\ast Δ^n$, the join of $Δ^n$ with itself. We also have canonical inclusions given by the adjoints of projection maps in the main post: $X→X\ast X$ can be included as the left or right factor. The induced natural transformation $X⊔X→X\ast X$ exhibits $X\ast X$ as a (generalized) cylinder object for $X$, where instead of a genuine codiagonal map $X\ast X→X$ we only have a zigzag $X\ast X→Y←X$, where $X→Y$ is a weak equivalence. But this is sufficient to answer the original question in the affirmative.

Equivalently, $Y[⨯2]$ together with the two natural maps $Y[⨯2]→Y$ is a (generalized) path object. A bit of elementary model category theory (see the cited articles) then implies that the notion of homotopy induced by this new cylinder (or path) object coincides with the old notion, assuming $Y$ is Kan.

  • $\begingroup$ Thanks! Do your considerations apply to any category $sA$ of simplicial objects of a category $A$ such that $sA$ is a model category (with Kan meaning fibrant) ? $\endgroup$
    – user420620
    Aug 15, 2022 at 10:12
  • $\begingroup$ @user420620: It applies to any simplicial model category without changes. $\endgroup$ Aug 15, 2022 at 16:07
  • $\begingroup$ I corrected the definition and use of the "doubling" endofunctor $[\times 2]$. I am not sure it fits your description $f\mapsto f\sqcup f$, one rather needs to use "even and odd coordinates" to be able to simplicially construct $Y_\bullet \to Y_\bullet[\times 2]$. $\endgroup$
    – user420620
    Aug 16, 2022 at 12:04
  • 1
    $\begingroup$ @user420620: 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. $\endgroup$ Aug 16, 2022 at 16:13
  • 2
    $\begingroup$ @user420620: 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. $\endgroup$ Aug 16, 2022 at 18:03

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