Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projective Naturality in Heegaard Floer Homology   Let $\text{Man}_{*}$ denote the category of closed, connected, oriented and
based $3$-manifolds, with basepoint preserving diffeomorphisms between them.
Juh\'asz, Thurston and Zemke showed that the Heegaard Floer invariants are
natural with respect to diffeomorphisms, in the sense that there are functors
$HF^{\circ}: \text{Man}_{*} \rightarrow \mathbb{F}_{2}[U]\text{-}\text{Mod}$
whose values agree with the invariants defined by Ozsv\'ath and Szab\'o. The
invariant associated to a based $3$-manifold comes from a transitive system in
$\mathbb{F}_{2}[U]\text{-}\text{Mod}$ associated to a graph of embedded
Heegaard diagrams representing the $3$-manifold. We show that the Heegaard
Floer invariants yield functors $HF^{\circ}: \text{Man}_{*} \rightarrow
\text{Trans}(P(\mathbb{Z}[U]\text{-}\text{Mod}))$ to the category of transitive
systems in a projectivized category of $\mathbb{Z}[U]$-modules. In doing so, we
will see that the transitive system of modules associated to a $3$-manifold
actually comes from an underlying transitive system in the projectivized
homotopy category of chain complexes over $\mathbb{Z}[U]\text{-}\text{Mod}$. We
discuss an application to involutive Heegaard Floer homology, and potential
generalizations of our results.

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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat