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<h1 id="chaincomplexes">ChainComplexes</h1>
<h2 id="overview">Overview</h2>
<p>A package for working with <a href="https://en.wikipedia.org/wiki/Chain_complex">chain complexes</a> over the finite field with two elements and their morphisms.</p>
<p>The homology groups of chain complexes can be calculated and whether a morphism of chain complexes is a <a href="https://en.wikipedia.org/wiki/Quasi-isomorphism">quasi-isomorphism</a>.</p>
<p>The type <code>ChainComplex{T}</code> contains a single field <code>differentials</code>, a vector of matrices of type <code>T</code>. The ith element of the <code>differentials</code> field represents the ith differential in the chain complex.</p>
<p>The type <code>AlignedChainComplex{T}</code> can be constructed from a <code>ChainComplex{T}</code> type. It finds a filtered basis for each underlying vector space of the chain complex such that the differentials of this chain complex can be written in block form:</p>
<p><img src="./images/blockform.svg" /></p>
<p>and thus the ith <a href="https://ncatlab.org/nlab/show/Betti+number">Betti number</a> can be found by the number of zero columns of the ith differential minus the number of non-zero rows of the (i-1)th differential.</p>
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