shifting from standard simplicial homology to persistent homology, there is something that I don't understand.
In simplicial homology one builds a chain complex of the form
$$\dots \rightarrow C_n(K) \overset{\partial_n}\rightarrow C_{n-1}(K) \overset{\partial_{n-1}}\rightarrow \dots$$
Where $C_n(K)$ denotes the vector space on the field $\mathbb{Z}_2$ spanned by $n$– dimensional simplices. (i.e. $C_n(K) = \{\sum_{i=1}^p\alpha_i\sigma_i\}$ where each $\sigma_i$ is an $n$-simplex.
Then the $n$-th dimensional homology is defined as $H_n(K) = ker\partial_n(K)/Im\partial_{n+1}(K)$.
Instead, when we shift to persistent homology, what we have is a chain complex for each sub-complex one constructs by varying the filration parameter $\varepsilon$,
$$\emptyset = K^{(0)} \subseteq K^{(1)} \subseteq K^{(2)} \subseteq \dots \subseteq K^{(m)} = K$$
and the filtration is connected by inclusion homomorphism
$$H_n(K^{(0)}) \overset{i_n(0,1)}\longrightarrow H_n(K^{(1)}) \longrightarrow \dots \longrightarrow H_n(K^{(m-1)}) \overset{i_n(m-1,m)}\longrightarrow H_n(K^{(m)}) = H_n(K) $$
for every dimension $n$. At this point, for $i \le j$ the $n$-th persistent homology group is defined as
$$H_n^{(i,j)} := ker\partial_n(K^{(i)})/((Im \partial_{n+1} (K^{(j)}) \cap ker\partial_n (K^{(i)})) $$
I'm just thinking about this modified version of the homology and why do we include the second term in the denominator. If anyone could briedly comment on this and giving some question, it would be highly appreciated.
Many thanks,
James
Best Answer
A short answer would be the following, but first, some background! If $H$ is a normal subgroup of a group $G$, then we are able to define a quotient group $G/H$. Since everything in sight is an abelian group, we could instead work with abelian groups: If $H$ is a subgroup of an abelian group $G$, then we are able to define a quotient group $G/H$. Or, since you have chosen field ($\mathbb{Z}/2\mathbb{Z}$) coefficients, we could instead work with vector spaces. If W is a vector subspace of a vector space $V$, then we can define the quotient space $V/W$. Note in all of these cases, the "denominator'' is a suboject of the "numerator".
In $$H_n^{(i,j)}:=\mathrm{ker}\partial_n(K^{(i)})/(\mathrm{Im}\partial_{n+1}(K^{(j)}) \cap \mathrm{ker}\partial_n(K^{(i)})),$$ if you don't take the intersection in the "denominator", then the "denominator" may not be a subobject of the "numerator"! To be more explicit, $\mathrm{Im}\partial_{n+1}(K^{(j)})$ need not be a subobject of $\mathrm{ker}\partial_n(K^{(i)})$ when $j>i$. That is why one needs to take the intersection. But once you intersect $\mathrm{Im}\partial_{n+1}(K^{(j)})$ with $\mathrm{ker}\partial_n(K^{(i)})$, then certainly the resulting intersection is a subobject of the "numerator" $\mathrm{ker}\partial_n(K^{(i)})$, and hence it is possible to define the quotient.