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We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.

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Learning disentangled representations of realworld data is a challenging open problem. Most previous methods have focused on either supervised approaches which use attribute labels or unsupervised approaches that manipulate the factorization in the latent space of models such as the variational autoencoder (VAE) by training with task-specific losses. In this work, we propose polarized-VAE, an approach that disentangles select attributes in the latent space based on proximity measures reflecting the similarity between data points with respect to these attributes. We apply our method to disentangle the semantics and syntax of sentences and carry out transfer experiments. Polarized-VAE outperforms the VAE baseline and is competitive with state-of-the-art approaches, while being more a general framework that is applicable to other attribute disentanglement tasks.

Using the description of the category of quasi-coherent sheaves on a root stack, we compute the G-theory of root stacks via localisation methods. We apply our results to the study of equivariant K-theory of algebraic varieties under certain conditions. Proof. This follows by combining Corollaries 3.10 and 3.15. So we have:G(X L, r ) ∼ = K(EP(X, L, r)), and we reduced the problem to describing the K-theory of the (abelian) category of coherent extendable pairs EP(X, L, r).We are going to use several splittings of the category of coherent extendable pairs to simplify the latter K-theory. The first step is this Lemma 3.29. If X is a locally noetherian scheme, then in notation of section 3.3 one has: