If *K* and *L* are geometric digital simplicial complexes, then a digital simplicial map f:K\to L is determined by taking the vertices \{{v}_{i}\} of *K* to vertices \{f({v}_{i})\} of *L* such that if [{v}_{{i}_{0}},\dots ,{v}_{{i}_{m}}] is a ({\kappa}_{1},m)-simplex of *K*, then f({v}_{{i}_{0}}),\dots ,f({v}_{{i}_{m}}) are all vertices (not necessarily unique) of some ({\kappa}_{2},m)-simplex in *L*. Given such a function {K}^{0}\to {L}^{0}, the rest of f:K\to L is determined by linear interpolation on each ({\kappa}_{1},m)-simplex (if x\in K can be represented by x={\sum}_{j=1}^{m}{t}_{j}{v}_{{i}_{j}} in barycentric coordinates of the ({\kappa}_{1},m)-simplex spanned by the {v}_{{i}_{j}}, then f(x)={\sum}_{j=1}^{m}{t}_{j}f({v}_{{i}_{j}})) (see Figures 4, 5).

**Corollary 2.1** *Every digital simplicial map is digitally continuous*.

*Proof* Let *K* and *L* be digital simplicial complexes having {\kappa}_{1} and {\kappa}_{2} adjacency, respectively, and let f:K\to L be a digital simplicial map. Since *K* is a simplicial complex, then the point of simplices of *K* have {\kappa}_{1} adjacency. So, *f* maps the points of the simplices of *K* to the points of the simplices of *L*. Therefore the image of {\kappa}_{1}-adjacent points under *f* is {\kappa}_{2}-adjacent points. Consequently, *f* is ({\kappa}_{1},{\kappa}_{2})-continuous, *i.e.*, digitally continuous. □

**Example 2.2** A simple but interesting and important example is the inclusion of a (\kappa ,m)-simplex into a digital simplicial complex. If *X* is a digital simplicial complex and {v}_{{i}_{0}},\dots ,{v}_{{i}_{m}} is a collection of vertices of *X* that spans a (\kappa ,m)-simplex of *X*, then K=[{v}_{{i}_{0}},\dots ,{v}_{{i}_{m}}] is itself a digital simplicial complex. We then have a simplicial map K\to X that takes each {v}_{{i}_{j}} to the corresponding vertex in *X* and hence takes *K* identically to itself inside *X*.

**Example 2.3** Let [{v}_{0},{v}_{1},{v}_{2}] be a (8,2)-simplex, one of whose (8,1)-faces is [{v}_{0},{v}_{1}]. Consider the simplicial map f:[{v}_{0},{v}_{1},{v}_{2}]\to [{v}_{0},{v}_{1}], determined by f({v}_{0})={v}_{0}, f({v}_{1})={v}_{1}, f({v}_{2})={v}_{1}, that collapses the (8,2)-simplex down to the (8,1)-simplex (see Figure 5). The great benefit of the theory of digital simplicial sets is a way to generalize these kinds of maps in order to preserve information so that we can still see the image of the (\kappa ,2)-simplex hiding in the (\kappa ,1)-simplex as a degenerate (\kappa ,m)-simplex.

### 2.1 Face maps in digital images

Given a (\kappa ,n)-simplex, we would like a handy way of referring to its (n-1)-dimensional faces. This is handled by the face maps. On the standard (\kappa ,n)-simplex, we have n+1 face maps {d}_{0},\dots ,{d}_{n}, defined so that {d}_{j}[0,\dots ,n]=[0,\dots ,\stackrel{\u02c6}{j},\dots ,n], where, as usual, the ˆ denotes a term that is being omitted. Thus applying {d}_{j} to [0,\dots ,n] yields the (n-1)-face missing the vertex *j*. It is important to note that each {d}_{j} simply assigns one of its faces to the (\kappa ,n)-simplex (see Figures 6, 7).

### 2.2 Delta sets and delta maps in digital images

Delta sets (sometimes called {\mathrm{\Delta}}_{\kappa}-sets) constitute an intermediary between digital simplicial complexes and digital simplicial sets.

**Definition 2.4** A delta set consists of a sequence of sets {X}_{0},{X}_{1},\dots which have {\kappa}_{0},{\kappa}_{1},\dots adjacency respectively, and the maps {d}_{i}:{X}_{n+1}\to {X}_{n} for each *i* and n\ge 0, 0\le i\le n+1, such that {d}_{i}{d}_{j}={d}_{j-1}{d}_{i} whenever i<j.

Of course this is just an abstraction, and generalization, of the definition of an ordered digital simplicial complex, in which {X}_{n} are the sets of (\kappa ,n)-simplices and the {d}_{i} are the digital face maps.

### 2.3 Degenerate maps in digital images

Degeneracy maps in digital images are, in some sense, the conceptual converse of face maps. Recall that the face map {d}_{j} takes a (\kappa ,n)-simplex and gives us back its *j* th (n-1)-face. On the other hand, the *j* th digital degeneracy map {s}_{j} takes a (\kappa ,n)-simplex and gives us back the *j* th degenerate (\kappa ,(n+1))-simplex in digital images living inside it.

As usual, we illustrate with the standard (\kappa ,n)-simplex, which will be a model for what happens in all digital simplicial sets. Given the standard (\kappa ,n)-simplex as {|{\mathrm{\Delta}}^{2}|}_{\kappa}=[0,\dots ,n], there are (n+1)-digital degeneracy maps {s}_{0},\dots ,{s}_{n}, defined by {s}_{j}[0,\dots ,n]=[0,\dots ,j,j,\dots ,n]. In other words, {s}_{j}[0,\dots ,n] gives us the unique degenerate (\kappa ,(n+1))-simplex in {|{\mathrm{\Delta}}^{n}|}_{\kappa} with only the *j* th vertex repeated.

Again, the geometric concept is that {s}_{j}{|{\mathrm{\Delta}}^{n}|}_{\kappa} can be thought of as the process of collapsing {\mathrm{\Delta}}^{n+1} down into {\mathrm{\Delta}}^{n} by the digital simplicial map {\pi}_{j} defined by {\pi}_{j}(i)=i for i<j, {\pi}_{j}(j)={\pi}_{j}(j+1)=j and {\pi}_{j}(j)=i-1 for i>j+1.

This idea extends naturally to digital simplicial complexes, to digital delta sets, and also to (\kappa ,n)-1 simplices that are already degenerate. If we have a (possibly degenerate) (\kappa ,n)-simplex [{v}_{{i}_{0}},\dots ,{v}_{{i}_{n}}] with {i}_{k}\le {i}_{k+1} for each *k*, 0\le k<n, then we set {s}_{j}[{v}_{{i}_{0}},\dots ,{v}_{{i}_{n}}]=[{v}_{{i}_{0}},\dots ,{v}_{{i}_{j}},{v}_{{i}_{j}},\dots ,{v}_{{i}_{n}}], *i.e.*, {v}_{{i}_{j}} always repeats. Thus we may say that this is a digital degenerate simplex in [{v}_{{i}_{0}},\dots ,{v}_{{i}_{n}}].

Also, as for the {d}_{i}, there are certain natural relations that the degeneracy maps possess. In particular, if i\le j, then {s}_{i}{s}_{j}[0,\dots ,n]=[0,\dots ,i,i,\dots ,j,j,\dots ,n]={s}_{j+1}{s}_{i}[0,\dots ,n]. Note that we have {s}_{j+1} in the last formula, not {s}_{j}, since the application of {s}_{i} pushes *j* one slot to the right.

Furthermore, there are relations amongst the face and degeneracy operators. These are a little bit advantageous to write down since there are three possibilities:

These situations are all clear. For example, applying [0,\dots ,n] to either side of the first formula yields [0,\dots ,\stackrel{\u02c6}{\u0131},\dots ,j,j,\dots ,n]. Note also that the middle formula takes care of both i=j and i=j+1.