Initial Subcategories via Free Presentations of the Constant Functor

Abstract

Let 𝒞 be a small category. We characterize the full initial subcategories of 𝒞 as precisely those whose representables give a free presentation of the constant functor Δ_𝒞𝕂:𝒞→Vect. Such a presentation yields a formula for the limit of any functor 𝒞 → Vect as the kernel of a morphism between products of evaluations. When the linearized endomorphism ring of every object is local, we show that a minimal free presentation, whenever one exists, determines the unique minimal full initial subcategory, up to equivalence. Bass's characterization of perfect rings gives a sufficient condition for the existence of such presentations. In particular, every finite category with no nonidentity endomorphisms, such as a finite poset, contains a unique minimal initial full subcategory. In the poset case, our results recover the minimal initial subposet constructed in [BDL27] and [DL26]. When 𝒞 is the free category on a finite quiver, we give an explicit construction of its minimal initial full subcategories, show that every such subcategory arises from this construction, and provide algorithms with time-complexity bounds.

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BibTeX RIS

BibTeX

@misc{brustle2026initial,
  title = {{Initial Subcategories via Free Presentations of the Constant Functor}},
  author = {Thomas Brüstle and Justin Desrochers and Samuel Leblanc},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41068},
  eprint = {2610.41068},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41068},
}

RIS

TY  - UNPB
TI  - Initial Subcategories via Free Presentations of the Constant Functor
AU  - Thomas Brüstle
AU  - Justin Desrochers
AU  - Samuel Leblanc
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.41068
UR  - https://arxiv.org/abs/2610.41068
AB  - Let 𝒞 be a small category. We characterize the full initial subcategories of 𝒞 as precisely those whose representables give a free presentation of the constant functor Δ_𝒞𝕂:𝒞→Vect. Such a presentation yields a formula for the limit of any functor 𝒞 → Vect as the kernel of a morphism between products of evaluations. When the linearized endomorphism ring of every object is local, we show that a minimal free presentation, whenever one exists, determines the unique minimal full initial subcategory, up to equivalence. Bass's characterization of perfect rings gives a sufficient condition for the existence of such presentations. In particular, every finite category with no nonidentity endomorphisms, such as a finite poset, contains a unique minimal initial full subcategory. In the poset case, our results recover the minimal initial subposet constructed in [BDL27] and [DL26]. When 𝒞 is the free category on a finite quiver, we give an explicit construction of its minimal initial full subcategories, show that every such subcategory arises from this construction, and provide algorithms with time-complexity bounds.
ER  -

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