In his paper "Categories and cohomology theories" Graeme Segal considers the category of finite length chain complexes of finite dimensional vector spaces: Let $n = (n\_i)\_{i \in \mathbb{Z}}$ be a sequence of positive integers almost all zero. Then he claims that the space $K\_n$ of chain complexes $E$ with $E^i = \mathbb{R}^{n\_i}$ is a real algebraic variety. I am no algebraic geometer, but I suspect this to be similar to flag varieties somehow. So my first question is:



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> What is the topology on $K\_n$?
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As morphisms in the above mentioned category he takes *chain homotopy equivalences* between these complexes and claims that these also form a topological space. So:



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> What is the topology on Mor($E,F$) for two chain complexes $E$ and $F$ as described above?
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Chain complexes of vector bundles
The space $K\_n$ sits inside the space of sequences of linear maps 
$$L\_n = \Pi\_i Hom(E^i,E^{i+1}).$$
This is just a space of sequences of matrices, so it is a real vector space of dimension $\sum\_i (n\_i \cdot n\_{i+1})$. We give it the usual euclidean topology for real vector spaces.


The subspace $K\_n$ consists of those sequences of linear maps $\{f\_i\}$ which form a chain complex - i.e., $f\_{i+1} \circ f\_i = 0$. This condition is polynomial in the entries in the matrices, so $K\_n$ is a real algebraic affine subvariety inside $L\_n$. We give $K\_n$ the subspace topology.


The topology on the space of morphism can be described similarly by embedding the morphism set $Mor(E,F)$ into the space of sequences of linear maps $\Pi\_i Hom(E^i,F^i)$, which is again a real vector space. The condition of being a morphism of chain complexes is real algebraic so the morphism space is again a real algebraic affine variety.


