I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes:

Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for every simplex $\sigma \in K$, $\varphi(\sigma) \cup \psi(\sigma)$ is a simplex of $L$.

Now I wonder whether this notion has been extended for more general contexts such as simplicial sets or $\Delta$-complexes.

Question. Has it been studied a notion of contiguity for simplicial maps between simplicial sets (or $\Delta$-complexes)? If that is the case could you provide me a reference.

Thanks in advance.


1 Answer 1


Maybe a partial answer to your question is provided in the paper(wanted to post as a comment but don't have enough reputation)

Grayson, Daniel R., Finite generation of K-groups of a curve over a finite field. (After Daniel Quillen), Algebraic (K)-theory, Proc. Conf., Oberwolfach 1980, Part I, Lect. Notes Math. 966, 69-90 (1982). ZBL0502.14004.

The exact references are Definition 1.2 and Definition 1.7. One should beware of a typo in the definition 1.2, where f,g are said to be adjacent if the set $\{f(x)|x \in \sigma, x \leq x'\} \cup \{g(x)|x \in \sigma, x \leq x'\}$. The correct definition should be

Definition: Two simplicial maps $f,g:X \rightarrow Y$ between ordered simplicial complexes $X,Y$ are said to be adjacent if for any simplex $\sigma$ and any $x' \in \sigma$, the set $$\{f(x)|x \in \sigma, x \leq x'\} \cup \{g(x)|x \in \sigma, x' \leq x\}$$ is a simplex in Y.

Then Corollary 1.3 says that adjacent simplicial maps are homotopic(meaning the induced maps between geometric realizations are homotopic).


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