Abstract
This paper deals with the notion of weak Lawvere–Tierney topology on a topos. Our motivation to study such a notion is based on the observation that the composition of two Lawvere–Tierney topologies on a topos is no longer idempotent, when seen as a closure operator. For a given topos \({\mathcal {E}}\), in this paper, we investigate some properties of this notion. Among other things, it is shown that the set of all weak Lawvere–Tierney topologies on \({\mathcal {E}}\) constitutes a complete residuated lattice provided that \({\mathcal {E}}\) is (co)complete. Furthermore, when the weak Lawvere–Tierney topology on \({\mathcal {E}}\) preserves binary meets, we give an explicit description of the (restricted) associated sheaf functor on \({\mathcal {E}}\).
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Communicated by Fariborz Azarpanah.
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Khanjanzadeh, Z., Madanshekaf, A. Weak Topologies on Toposes. Bull. Iran. Math. Soc. 47, 461–486 (2021). https://doi.org/10.1007/s41980-020-00393-7
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DOI: https://doi.org/10.1007/s41980-020-00393-7