We study unital <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads by their arity restrictions. Given <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we develop a model for unital k-restricted <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads, which are variants of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads which have only <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\le k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>≤</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-arity morphisms, as complete Segal presheaves on closed k-dendroidal trees, which are closed trees built from corollas with valences <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\le k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we prove that the restriction functors from unital <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads to unital k-restricted <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k, the left and right adjoints give a filtration and a co-filtration for any unital <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operad by k-restricted <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∞</mi> </math></EquationSource> </InlineEquation>-operads.

Unital k-restricted $\infty $-operads

期刊论文
发表日期：

2025年11月7日

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Amartya Shekhar Dubey,
Yu Leon Liu

摘要

We study unital $\infty $ $\infty$ -operads by their arity restrictions. Given $k \ge 1$ $k \geq 1$ , we develop a model for unital k-restricted $\infty $ $\infty$ -operads, which are variants of $\infty $ $\infty$ -operads which have only $(\le k)$ $(\leq k)$ -arity morphisms, as complete Segal presheaves on closed k-dendroidal trees, which are closed trees built from corollas with valences $\le k$ $\leq k$ . Furthermore, we prove that the restriction functors from unital $\infty $ $\infty$ -operads to unital k-restricted $\infty $ $\infty$ -operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k, the left and right adjoints give a filtration and a co-filtration for any unital $\infty $ $\infty$ -operad by k-restricted $\infty $ $\infty$ -operads.

Unital k-restricted \(\infty \)-operads

摘要