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1. Invertible topological field theories

   https://doi.org/10.1112/topo.12335  

   Schommer‐Pries, Christopher (June 2024, Journal of Topology)

   A ‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal ‐category of ‐bordisms (embedded into and equipped with a tangential ‐structure) that lands in the Picard subcategory of the target symmetric monoidal ‐category. We classify these field theories in terms of the cohomology of the ‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the ‐category of bordisms with as an ‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math.202(2009), no. 2, 195–239) in the case , and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the ‐uple case. We also obtain results for the ‐category of ‐bordisms embedding into a fixed ambient manifold , generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN2011(2011), no. 3, 572–608) in the case . We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of ‐vector spaces (for ), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math.25(2013), no. 5, 1067–1106. arXiv:0912.4706). 

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2. Stability in the homology of Deligne–Mumford compactifications

   https://doi.org/10.1112/S0010437X21007582  

   Tosteson, Philip (December 2021, Compositio Mathematica)

   Abstract Using the theory of $${\mathbf {FS}} {^\mathrm {op}}$$ modules, we study the asymptotic behavior of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ , the Deligne–Mumford compactification of the moduli space of curves, for $$n\gg 0$$ . An $${\mathbf {FS}} {^\mathrm {op}}$$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of $${\overline {\mathcal {M}}_{g,n}}$$ the structure of an $${\mathbf {FS}} {^\mathrm {op}}$$ module and bound its degree of generation. As a consequence, we prove that the generating function $$\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$$ is rational, and its denominator has roots in the set $$\{1, 1/2, \ldots, 1/p(g,i)\},$$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ into irreducible $$\mathbf {S}_n$$ representations. 

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3. Formalizing Colimits in 𝒞at

   https://doi.org/10.4230/lipics.itp.2025.20  

   Carneiro, Mario; Riehl, Emily (September 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Forster, Yannick; Keller, Chantal (Ed.)

   {"Abstract":["Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite ∞-category theory has not been formalized. To support future work on formalizing ∞-category theory in Lean’s mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that 𝒞at has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete."]} 

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4. A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

   https://doi.org/10.32408/compositionality-6-2  

   Aduddell, Rebekah; Fairbanks, James; Kumar, Amit; Ocal, Pablo S; Patterson, Evan; Shapiro, Brandon T (May 2024, Compositionality)

   Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology. 

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5. Bi-incomplete Tambara functors

   Blumberg, Andrew J.; Hill, Michael A. (January 2021, Equivariant Topology and Derived Algebra)

   Balchin, S.; Barnes, D.; Kędziorek, M.; Szymik, M. (Ed.)

   For an equivariant commutative ring spectrum R, \pi_0 R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N_\infty ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and \pi_0 R has the structure of an incomplete Tambara functor. However, if R is an N_\infty ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures. 

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