On Mahavier products

Charatonik, Włodzimierz J.; Roe, Robert Paul

doi:10.1016/j.topol.2014.02.008

Cited by 3 publications

(1 citation statement)

References 2 publications

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“…Generalised inverse limits of compacta were introduced by Ingram and Mahavier in 2006 in [1] and have since received much attention (e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]). Recall that an inverse limit of a sequence…”

Section: Introductionmentioning

confidence: 99%

Mahavier completeness and classifying diagrams

Maehara

Weiss

2017

Topology and its Applications

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Generalised inverse limits of compacta were introduced by Ingram and Mahavier in 2006. The main difference between ordinary inverse limits and their generalised cousins is that the former concerns diagrams of singlevalued functions while the latter permits multivalued functions. However, generalised inverse limits are not merely limits in the Kleisli category of a hyperspace monad, a fact that independently motivated each of the authors of this article to come up with the same formalism which restores the link with category theory through the concept of Mahavier limit of an order diagram in an order extension of a category B. Mahavier limits of diagrams in B coincide with ordinary limits in B, and so Mahavier limits are an extension of ordinary limits along the functor that views an ordinary diagram as a diagram in the extension. Within that context it is natural to consider Mahavier completeness, namely when all small diagrams admit Mahavier limits, as well as classifying diagrams, namely the existence of a right adjoint to the mentioned functor on diagrams. In this work we show that these two conditions are equivalent, and we study some of the properties of classifying diagrams and of the adjunction.

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Section: Introductionmentioning

confidence: 99%

Mahavier completeness and classifying diagrams

Maehara

Weiss

2017

Topology and its Applications

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Towards the Complete Classification of the Two Lines Inverse Limit

Rogina

2023

J Dyn Control Syst

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For positive numbers $$r<1$$ r < 1 and $$\rho >1$$ ρ > 1 , let $$L_{r,\rho }$$ L r , ρ be the union of two line segments in $$[0,1] \times {[}0,1{]}$$ [ 0 , 1 ] × [ 0 , 1 ] , one from (0, 0) to (1, r) and the other one from (0, 0) to $$(\frac{1}{\rho },1)$$ ( 1 ρ , 1 ) . In Banič et al. (2022), it was proven for all such r and $$\rho$$ ρ that, if r and $$\rho$$ ρ never connect, then the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is homeomorphic to the Lelek fan. In this paper, we show that for all such r and $$\rho$$ ρ , if r and $$\rho$$ ρ do connect, the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is a fan F with top v with some additional properties. More precisely, it is the union of a countable family $$\mathcal {C}$$ C of Cantor fans such that for each $$C_1,C_2\in \mathcal {C}$$ C 1 , C 2 ∈ C , if $$C_1\ne C_2$$ C 1 ≠ C 2 , then $$C_1\cap C_2 = \{v\}$$ C 1 ∩ C 2 = { v } and the set of limit points of the set of end-points of F, forms in each arc from v to an end-point, a harmonic sequence. This solves the open problem from Banič et al. (2022).

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Shift maps and their variants on inverse limits with set-valued functions

Kawamura

Kennedy

2018

Topology and its Applications

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On Mahavier products

Cited by 3 publications

References 2 publications

Mahavier completeness and classifying diagrams

Mahavier completeness and classifying diagrams

Towards the Complete Classification of the Two Lines Inverse Limit

Shift maps and their variants on inverse limits with set-valued functions

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