# Definition of a subcomplex of a $\Delta$-complex

I am taking the following as the definition of a $\Delta$-complex.

(i) one starts with an indexing set $I_n$ for each $n \in \mathbb{Z}_{\ge 0}$.

(ii) for each $\alpha \in I_n$, one takes a copy $\sigma_\alpha^n$ of the standard $n$-simplex.

(iii) one forms the disjoint union of all of these simplices, for all $n\geq 0$.

(iv) now require that for each $(n-1)$-dimensional face of $\sigma_\alpha^n$, there is an associated $\sigma_\beta^{n-1}$ for some $\beta \in I_{n-1}$.

(v) now form the quotient space by identifying each $(n-1)$-dimensional face of each $\sigma_\alpha^n$ with $\sigma_\beta^{n-1}$ using the canonical homeomorphism. In particular, these homeomorphisms preserve the ordering of vertices.

The sources I've consulted don't make it clear what a subcomplex is. Can someone give me a rigorous definition?

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With that (slightly strange way of phrasing the) definition, a subcomplex should be a sequence $(J_n)_{n\geq0}$ such that
• for each $n\geq0$ we have $J_n\subseteq I_n$, and
• if $\alpha\in J_n$ and $\beta\in I_{n-1}$ are such that $\sigma_b^{n-1}$ is associated to one of the faces of $\sigma_\alpha^n$, then $\beta\in J_{n-1}$.
The definition is from lectures I took on algebraic topology that don't have typeset notes, so I cannot link you to an electronic version. However, I did reproduce above the lecturer's definition word-for-word. It appears not to be too different from the one in Hatcher's Algebraic Topology. What definition would you normally use for $\Delta$-complexes? Thank you for the recommendation; I'll try to find it in my library. – Sputnik May 18 '11 at 2:27
Incidentally, I've since learned that Eilenberg and Zilber first invented the notion of $\Delta$-complexes under the name "semi-simplicial complexes" in this paper. – Sputnik May 18 '11 at 15:24