PreprintarXiv licencearXiv
Cartesian closedness of lax comma categories
Abstract
For a finitely complete category X, we prove that the lax comma category Cat//X is cartesian closed if and only if its free coproduct completion Fam(X) is cartesian closed. The proof uses a change-of-base theorem: postcomposition with a functor induces a left adjoint between lax comma categories precisely when that functor is a left multiadjoint. We describe the exponentials and derive examples from known results on family categories.
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BibTeX
@misc{clementino2026cartesian,
title = {{Cartesian closedness of lax comma categories}},
author = {Maria Manuel Clementino and Fernando Lucatelli Nunes},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01184},
eprint = {2611.01184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01184},
}
RIS
TY - UNPB
TI - Cartesian closedness of lax comma categories
AU - Maria Manuel Clementino
AU - Fernando Lucatelli Nunes
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01184
UR - https://arxiv.org/abs/2611.01184
AB - For a finitely complete category X, we prove that the lax comma category Cat//X is cartesian closed if and only if its free coproduct completion Fam(X) is cartesian closed. The proof uses a change-of-base theorem: postcomposition with a functor induces a left adjoint between lax comma categories precisely when that functor is a left multiadjoint. We describe the exponentials and derive examples from known results on family categories.
ER -