Homology on Pointed Sets 2

We wish to address four of the Eilenberg-Steenrod axioms for this category.  First note that for dimension we have H(x_0)=x_0 where x_0 denotes the trivial pointed set.

If (X,x_0) and (Y,y_0) are pointed sets, then we define the intuitive product pointed set as the pointed set (X\times Y,(x_0,y_0)).  If X and Y are also differential sets with differentials d_1 and d_2, then we define the product differential component-wise: d_1\times d_2 (x,y)=(d_1(x),d_2(y)).  This is clearly a differential on X\times Y, which we in turn call the product differential set.

If we then define the relation \sim on X\times Y where (x,y)\sim (x',y') iff (d_1(x),d_2(y))=(d_1(x'),d_2(y')), then we have [(x,y)]_{d_1\times d_2}=[x]_{d_1}\times [y]_{d_2}.  Hence we have

\begin{array}{lcl}H(X\times Y)&=&[(x,y)]-(d(X\times Y)-(x_0,y_0))\\&=&[x]\times[y]-\left((d_1(X),d_2(Y))-(x_0,y_0)\right)\\&=&\left([x]-(d_1(X)-\{x_0\}),[y]-(d_2(Y)-\{y_0\})\right)\\&=&H(X)\times H(Y).\end{array}

We can similarly define a formal sum of pointed spaces \sqcup_i(X_i,x_i) whose corresponding homology is just defined as


So we have two forms of additivity.  We also have excision, for if (X,A) is a pair of sets with A a sub-differential set of X and U\subseteq A, then


We lastly address homotopy and omit the long exact sequence as we have not developed a dimensional concept (although recall we constructed a short exact sequence under an assumption on A).  Let f:(X,A)\to (Y,B) be a map between paired sets.  We say this is a paired set morphism if f(A)\subseteq B.  If we furthermore have that these are paired pointed sets (X,A,x_0) and (Y,B,y_0) with A and B sub-pointed sets, a paired set morphism f between them is a paired pointed morphism if we also have that f(x_0)=y_0.  If both are differential sets as well, we can define a paired differential morphism if we further have f\circ d_1=d_2\circ f.  Let f be a paired differential morphism between such sets.  Then we would like for its restriction f:H(X,A)\to H(Y,B) to be well-defined.  Hopefully some combination of requirements like normality and being a paired differential morphism will work, but I have not yet convinced myself.  If it worked, we would in turn just define two maps f,g:(X,A)\to (Y,B) to be homotopic mod A provided they agreed on H(X,A).


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