Saturated directed spaces and generalized clocks
Abstract
We construct a Hausdorff saturated directed space whose directed path space between two distinct points has nontrivial fundamental group although the corresponding trace space is a square. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from paths to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces whose compact subspaces are metrizable, these quotients are trivial q-fibrations. We construct a globular clock containing the directed circle and admitting regular maps from all q-cofibrant globular realizations, whereas a finite cellular example admits no regular map to the circle. Normalization also gives a proof of the quotient theorem for execution paths and establishes the homotopy lifting property for nondecreasing reparametrizations.
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BibTeX
@misc{gaucher2026saturated,
title = {{Saturated directed spaces and generalized clocks}},
author = {Philippe Gaucher},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35353},
eprint = {2609.35353},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05768411v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Saturated directed spaces and generalized clocks
AU - Philippe Gaucher
PY - 2026
DA - 2026/09/28
PB - HAL
DO - 10.48550/arxiv.2609.35353
UR - https://hal.science/hal-05768411v2
AB - We construct a Hausdorff saturated directed space whose directed path space between two distinct points has nontrivial fundamental group although the corresponding trace space is a square. A clock is a directed space with submetrizable underlying space; a timed space is a saturated directed space equipped with a regular directed map to a clock. Over every fixed clock, timed spaces form a locally presentable category, and their quotients from paths to traces are trivial Hurewicz fibrations with continuous sections. For Hausdorff saturated directed spaces whose compact subspaces are metrizable, these quotients are trivial q-fibrations. We construct a globular clock containing the directed circle and admitting regular maps from all q-cofibrant globular realizations, whereas a finite cellular example admits no regular map to the circle. Normalization also gives a proof of the quotient theorem for execution paths and establishes the homotopy lifting property for nondecreasing reparametrizations.
ER -