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This https://arxiv.org/abs/2501.17769 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Colimits of internal categories
We show that for an extensive $1$-category $\mathcal{E}$ with pullbacks and pullback stable coequalisers in which the forgetful functor $\mathcal{U}: \mathbf{Cat}(\mathcal{E})_1 \to \mathbf{Gph}(\mathcal{E})$ has left adjoint, the $2$-category $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations has finite $2$-colimits. In addition, $\mathbf{Cat}(\mathcal{E})$ is extensive, has pullbacks and codescent coequalisers are stable under pullback along discrete Cond…

A model for the assembly map of bordism-invariant functors
Jordan Levin, Guglielmo Nocera, Victor Saunier
https://arxiv.org/abs/2506.05238 https://

A model for the assembly map of bordism-invariant functors
We study oplax colimits of stable categories, of hermitian categories and of Poincaré categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincaré categories which is also a Verdier projection, whose kernel we explicitly describe. As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincaré categories proved by Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle …