Monads in topology

Manes, Ernest G.

doi:10.1016/j.topol.2009.12.013

Cited by 7 publications

(3 citation statements)

References 24 publications

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“…Then, (T, µ, η) is a monad and the inclusion transformation i : T −→ S is a monad morphism. In this case, T is called a submonad of (S, µ, η) [33,34]. Given a set B, the contravariant functor…”

Section: The Problemmentioning

confidence: 99%

“…The multiplication of the filter monad helps us to express iterative limits in topology; the Eilenberg-Moore algebras of the monad of (not necessarily proper) filters are precisely the continuous lattices and hence the injective T 0 spaces; the Eilenberg-Moore algebras of the monad of ultrafilters are precisely the compact and Hausdorff spaces; and etc. There exists a large number of works that are related to applications of the filter monad in topology and order theory, for example, the monographs [10,14,32] and the articles [1,6,8,31,34,38].…”

Section: Introductionmentioning

confidence: 99%

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The saturated prefilter monad

Lai¹,

Zhang²,

Zhang

2020

Preprint

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This paper addresses the problem whether the functor of (quantale-valued) saturated prefilters is a monad on the category of sets. It is shown that when the quantale is the unit interval equipped with a continuous t-norm &, the following conditions are equivalent:(1) The functor of saturated prefilters is a monad.(2) The functor of ⊤-filters is a monad.(3) The functor of bounded saturated prefilters is a monad.(4) The implication operator of & is continuous at each point off the diagonal.

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Section: The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The saturated prefilter monad

Lai¹,

Zhang²,

Zhang

2020

Preprint

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“…[20,Example 2.11]) and its submonads, the filter monad and the ultrafilter monad in particular, are among the fundamental constructions of sets and play important roles in category theory, algebra, topology, and other disciplines, see e.g. [13,18,20,21]. Since the Goguen category Set(L) is not a topos unless L is a singleton set, its behavior is quite different to the category of sets when "powerobjects" are concerned [27].…”

Section: Introductionmentioning

confidence: 99%