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Branko Nikolić

Publications and source records attributed to Branko Nikolić.

7 recordsLinked to original sources
arXiv2022

Monoidal centres and groupoid-graded categories

We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by $\mathrm{Mod}$; the tensor product is cartesian product of categories. For a groupoid $\scr{G}$, we study the monoidal centre $\mathrm{ZPs}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of the monoidal bicategory $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$ of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory $\scr{M}$, and it is a braided monoidale in the monoidal centre $\mathrm{Z}\scr{M}$ of $\scr{M}$. Each fibration $π: \scr{H} \to \scr{G}$ between groupoids provides an example of a full monoidal centre of a monoidale in $\mathrm{Ps}(\scr{G},\mathrm{Mod}^{\mathrm{op}})$. For a group $G$, we explain how the $G$-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context. We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of $k$-linear categories on which $G$ acts.

math.CT↗
arXiv2021

Comonadic base change for enriched categories

For our concepts of change of base and comonadicity, we work in the general context of the tricategory $\mathrm{Caten}$ whose objects are bicategories $\mathscr{V}$ and whose morphisms are categories enriched on two sides. For example, for any monoidal comonad $G$ on a cocomplete closed monoidal category $\mathscr{C}$, the forgetful functor $U : \mathscr{C}^G\to \mathscr{C}$ is comonadic when regarded as a morphism in $\mathrm{Caten}$ between one-object bicategories. We show that the forgetful pseudofunctor $\mathscr{U}:\mathscr{V}^\mathscr{G}\rightarrow \mathscr{V}$ from the bicategory of Eilenberg-Moore coalgebras for a comonad $\mathscr{G}$ on $\mathscr{V}$ in $\mathrm{Caten}$ induces a change of base pseudofunctor $\widetilde{\mathscr{U}}:\mathscr{V}^\mathscr{G}\text{-}\mathrm{Mod}\rightarrow \mathscr{V}\text{-}\mathrm{Mod}$ which is comonadic in a bigger version of $\mathrm{Caten}$. We define Hopfness for such a comonad $\mathscr{G}$ and prove that having that property implies $\mathscr{U}$ creates left (Kan) extensions in the bicategory $\mathscr{V}^\mathscr{G}$. We provide conditions under which Hopfness carries over from $\mathscr{G}$ to the comonad $\widetilde{\mathscr{G}}=\widetilde{\mathscr{U}}\circ \widetilde{\mathscr{R}}$ generated by the adjunction $\widetilde{\mathscr{U}}\dashv \widetilde{\mathscr{R}}$. This has implications for characterizing the absolute colimit completion of $\mathscr{V}^\mathscr{G}$-categories.

math.CT↗
arXiv2021

Cauchy completeness for DG-categories

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DG-categories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit.

math.CT↗
arXiv2018

Strictification tensor product of 2-categories

Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category $\mathcal{C}\boxtimes\mathcal{D}$ satisfying $\textrm{Lax}(\mathcal{C},\textrm{Lax}(\mathcal{D},\mathcal{E})) \cong [\mathcal{C}\boxtimes \mathcal{D},\mathcal{E}]_\mathrm{lnt}$, hence generalising the case of the free distributive law $1\boxtimes 1$. We also discuss dual constructions.

math.CT↗
arXiv2018

Hopf rings for grading and differentials

In the category of abelian groups, Pareigis constructed a Hopf ring whose comodules are differential graded abelian groups. We show that this Hopf ring can be obtained by combining grading and differential Hopf rings using semidirect product in fairly general symmetric monoidal additive categories.

math.CT↗
arXiv2017

Cauchy completeness and causal spaces

Following Lawvere's description of metric spaces using enriched category theory, we introduce a change in the base of enrichment that allows description of some aspects of (relativistic) causal spaces. All such spaces are Cauchy complete, in the sense of enriched category theory. Furthermore, we give sufficient conditions on a base monoidal category for which enriched categories are Cauchy complete if and only if their underlying categories are (their idempotent arrows split).

math.CT↗