Dimitri Chikhladze’s research while affiliated with Macquarie University and other places

... As an application of this technique, we construct a free Talgebra functor and the underlying T-monoid functor, which are analogues of the free monoidal category functor and the underlying multicategory functor going between the categories of monoidal categories and multicategories. We then generalize the results of[5]. The second theme of the paper is the lax formal theory of monads within a tricategory. ...

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Lax formal theory of monads, monoidal approach to bicategorical structures and generalized operads

... We use this method to show how several 2-monads on the 2-category Cat of small categories and functors can be extended to pseudomonads on the bicategory Prof of small categories and profunctors (also known as bimodules or distributors) [8,42,60]. This result has applications in the theory of variable binding [22,24,55,61], concurrency [13], species of structures [23], models of the differential λ-calculus [21], and operads and multicategories [15][16][17]25,27]. ...

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Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures

... Using the aforementioned adjunction, we can extend the monad L to a monad on (L, V )-Cat, denoted by L as well. Moreover, one can prove (see [5]) that there is an equivalence ...

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Injective Hulls of Quantale-Enriched Multicategories

... This is a rather philosophical consideration which corresponds to many proper mathematical approaches or methods for establishing "quantum category theory" (e.g.Rennela, Staton 2018;Moskaliuk, Moskaliuk 2013;Chikhladze 2011;Davis 2006; Holdsworth 1977, etc.). ...

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What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

... where ∆ (2) (1) = 1 (1) ⊗ 1 (2) ⊗ 1 (3) and ∆(1) = 1 [1] ⊗ 1 [2] . (1) ε t (xε t (y)) = ε t (xy), ε s (ε s (x)y) = ε s (xy). ...

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Reconstruction of weak bialgebra maps and its applications

... Let (C, ⊗, I, a, l, r) be a monoidal category, (F, δ, ε) be a comonad on C, and (F, F 2 , F 0 ) : C → C be a monoidal functor. Then recall from [18] or [19] that F is called a monoidal comonad (or a bicomonad) on C if δ and ε are both monoidal natural transformations, i.e. the following compatibility conditions hold for any X, Y ∈ C: ...

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On quasi-monoidal comonads and their corepresentations

... Given a coalgebra D, the (co)opposite coalgebra is denoted by D o . The following definition is adapted from the definition of the underlying quiver of a quantum category explicitly defined by Chikhladze [10] inspired by Day-Street [12], to our context of representable quantum sets. ...

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Introduction to Quantum Combinatorics

... In fact, they restriced their analysis to Kan complexes, as this condition implies the admissibility of these objects for the corresponding Galois structure. Later Chikhladze introduced relative factorization systems, and showed that the induced relative factorization system for Kan fibrations is locally stable, so that the Galois structures induces a relative monotone-light factorization ( [15]). ...

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Fundamental groupoids for simplicial objects in Mal'tsev categories

... Our aim is to extend the other three theorems, finding a common setting that includes both the ordinary and the additive context. Note that an enriched version of Barr's Embedding Theorem has already been considered in [10], but the notion of regularity appearing there is more restrictive than ours: see Remark 5.2. ...

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Enriched Regular Theories