I recall the definitions I am using.

A $\Delta$-complex is a topological space $X$ together with a partition of $X$ into cells and, for every $n$-cell $A$ of the decomposition, a continuous map $\Phi_A : \Delta ^n \to X$, called characteristic map of $A$, such that:

1) its restriction of $\Phi _A $ at $\text{int}(\Delta ^n )$ is a bijection onto $A$;

2) for every $(n-1)$-closed face of $\Delta ^n$, the restriction of $\Phi _A $ at that face is the characteristic map of some $(n-1)$-cell of the decomposition;

3) a subset of $X$ is closed iff $X\cap \overline A$ is closed in $\overline A $ for every cell $A$ of the decomposition.

A simplicial complex is a $\Delta$-complex such that every characteristic map is injective and is uniquely determined by the image of the set of vertices.

The class of simplicial complexes can be turned into a category, by defining simplicial maps. What is the analogous definition of morphism for $\Delta$-complexes?

Thank you.


It is natural to assume that any reasonable definition of a $\Delta$-map should be based on the idea that we assign to every simplex in $X$ a simplex in $Y$ such that this assignment is compatible with the face operator. To see what that means, I'll introduce some combinatorics:

Let us think of a $\Delta$-complex as a sequence $X_0,X_1,\dots$ of sets, and for each monotone injection $f:[k]\to[n],$ where $[k]=\{0,1,\dots,k\},$ there is a function $B_f:X_n\to X_k$ such that $B_f\circ B_g = B_{gf}$ and $B_{\text{id}_{[n]}}=\text{id}_{X_n}.$ An element $\sigma\in X_n$ is called an $n$-simplex.
To see how this relates to the definition you are using, consider the map $\delta_i:[n-1]\to[n]$ missing the element $i\in[n]$. Then $d_i=B_{\delta_i}:S_n\to S_{n-1}$ is the $i$-th face map, assigning to each $n$-simplex $\sigma$ its $i$-th face, which is the face opposite of the $i$-th vertex in $\Delta^n$ or, if you want, the restriction of $\Phi_\sigma$ to that subsimplex. The topology of the space we are covering by these simplices is actually determined by the combinatorics, i.e. by the maps $B_f$.

Now one possible candidate for a $\Delta$-map $F:X\to Y$ would be a sequence of maps $F_n:X_n\to Y_n$ such that $$B_f(F_n(\sigma))=F_k(B_f(\sigma))$$ for any monotone injection $f:[k]\to[n]$ and $n$-simplex $\sigma$.

Another more flexible candidate for a $\Delta$-map $F:X\to Y$ consists of the following data: For any $\sigma\in X_n$, there is a monotone surjection $s_\sigma:[n]\to[n_\sigma]$, an $n_\sigma$-simplex $F(\sigma)$, and for each monotone injection $f:[k]\to[n]$, we have $$ s_\sigma f = g s_{B_f(\sigma)} $$ for some monotone injection $g$, and also $$ B_g(F(\sigma)) = F(B_f(\sigma)) $$

The first way of defining a $\Delta$-map is then just a special case of the second way by requiring $s_\sigma = \text{id}_{[n]}$

We call the space covered by the characteristic maps of the simplices, as described in your post, the realization of $X$, denoted $|X|$. One can show that a $\Delta$-map $F:X\to Y$ induces a continuous map $|F|:|X|\to|Y|$.

Here's a (very simple) example of a $\Delta$-map of the second type:
Let $X_n=\emptyset$ for $n>2$, and $$X_2=\{\sigma\}, \quad X_1=\{d_0σ,d_1σ,d_2σ\}, \quad X_0=\{v_0,v_1,v_2\}$$ This is just a triangle with vertices $v_i$ such that $d_iσ$ is the edge opposite of $v_i$. Let $Y$ be the $\Delta$-complex $$ Y_1=\{e\}, \quad Y_0=\{w_0,w_1\} $$ an edge $e$ from $w_0$ to $w_1$. A $\Delta$-map $F:X\to Y$ which has $F(\sigma)=e$ has $$s_\sigma: \quad 0\mapsto 0, \quad 1\mapsto 1, \quad 2\mapsto 1$$ Now if $\delta_0:[1]\to[2]$ sends $0\mapsto 1, 1\mapsto 2$, that means $s_{d_0σ}$ must be the constant map $[1]\to[0]$, and $g[0]\to[1]$ sends $0\mapsto 1$. We then have $$ F(d_0σ) = d_0(F(σ)) = d_0(e) = w_1 $$ We can determine the rest of the data comprising $F$ in a similar manner. Geometrically this is the map which collapses the edge $d_0σ$ to the vertex $w_1$.

  • $\begingroup$ Thank you. Is really necessary to preserve the order of vertices? What about this definition: a $\Delta$-map $f:X\to Y$ between $\Delta$-complexes (I am using definitions in my first post) is a map such that: 1) for every cell $A$ of $X$, $f(\overline A) = \overline B$ for some cell $B$ of $Y$, and 2) if $f(\overline A ) = \overline B$, and $\dim A =n , \dim B=k$, then there exists a linear map $\tilde f : \Delta ^n \to \Delta ^k $ such that $ \Phi_B \circ \tilde f = f \circ \Phi_A$. My goal is to find a "good" definition of $\Delta$-map for defining a simplicial homology functor $\endgroup$ – Ervin Nov 23 '15 at 8:10

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