ScalingStacks

4.1.1. Definition

Let 𝒰{\mathcal{U}} be the differential strict monoidal category generated by an object ee and a map Ο„:e2β†’e2\tau:e^{2}\to e^{2} subject to the relations

(4.1.1) d⁑(Ο„)=1,Ο„2=0​ and ​eβ€‹Ο„βˆ˜Ο„β€‹e∘e​τ=τ​e∘eβ€‹Ο„βˆ˜Ο„β€‹e.d(\tau)=1,\ \tau^{2}=0\text{ and }e\tau\circ\tau e\circ e\tau=\tau e\circ e\tau\circ\tau e.

There are isomorphisms of differential monoidal categories opp:π’°β†’βˆΌπ’°opp{\operatorname{opp}\nolimits}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\operatorname{opp}\nolimits} and rev:π’°β†’βˆΌπ’°rev\mathrm{rev}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\mathrm{rev}} given on generators by e↦ee\mapsto e and τ↦τ\tau\mapsto\tau.

The following result is clear.

0P5F

Proposition 4.1.1. The objects of the category 𝒰{\mathcal{U}} are the ene^{n}, nβ‰₯0n\geq 0. We have Hom⁑(en,em)=0\operatorname{Hom}\nolimits(e^{n},e^{m})=0 if nβ‰ mn\neq m and there is an isomorphism of differential algebras

Hnβ†’βˆΌEnd⁑(en),Ti↦eiβˆ’1​τ​enβˆ’iβˆ’1.H_{n}\xrightarrow{\sim}\operatorname{End}\nolimits(e^{n}),\ T_{i}\mapsto e^{i-1}\tau e^{n-i-1}.

There is a commutative diagram

HmβŠ—Hn\textstyle{H_{m}\otimes H_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}TiβŠ—Tj↦Ti​Tm+j\scriptstyle{T_{i}\otimes T_{j}\mapsto T_{i}T_{m+j}}can\scriptstyle{{\mathrm{can}}}∼\scriptstyle{\sim}Hm+n\textstyle{H_{m+n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}∼\scriptstyle{\sim}can\scriptstyle{{\mathrm{can}}}End⁑(Em)βŠ—End⁑(En)\textstyle{\operatorname{End}\nolimits(E^{m})\otimes\operatorname{End}\nolimits(E^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}βŠ—\scriptstyle{\otimes}End⁑(Em+n)\textstyle{\operatorname{End}\nolimits(E^{m+n})}

The isomorphism opp:π’°β†’βˆΌπ’°opp{\operatorname{opp}\nolimits}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\operatorname{opp}\nolimits} gives rise to the isomorphism of differential algebras

opp:Hnβ†’βˆΌHnopp,Ti↦Ti.{\operatorname{opp}\nolimits}:H_{n}\xrightarrow{\sim}H_{n}^{\operatorname{opp}\nolimits},\ T_{i}\mapsto T_{i}.

The isomorphism rev:π’°β†’βˆΌπ’°rev\mathrm{rev}:{\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\mathrm{rev}} gives rise to the isomorphism of differential algebras

ΞΉn:Hnβ†’βˆΌHn,Ti↦Tnβˆ’i.\iota_{n}:H_{n}\xrightarrow{\sim}H_{n},\ T_{i}\mapsto T_{n-i}.

The functor βˆ’βŠ—En-\otimes E^{n} induces an injective morphism of differential algebras Hr=End⁑(Er)β†’Hr+n=End⁑(Er+n),Ti↦TiH_{r}=\operatorname{End}\nolimits(E^{r})\to H_{r+n}=\operatorname{End}\nolimits(E^{r+n}),\ T_{i}\mapsto T_{i} and we will identify HrH_{r} with a subalgebra of Hr+nH_{r+n} via this morphism.

The functor EnβŠ—βˆ’E^{n}\otimes- induces a morphism of differential algebras

fn:Hr=End⁑(Er)β†’Hn+r=End⁑(En+r),Ti↦Tn+i.f_{n}:H_{r}=\operatorname{End}\nolimits(E^{r})\to H_{n+r}=\operatorname{End}\nolimits(E^{n+r}),\ T_{i}\mapsto T_{n+i}.

Note that HnH_{n} commutes with fn​(Hr)f_{n}(H_{r}) and that fn=ΞΉn+r∘ιrf_{n}=\iota_{n+r}\circ\iota_{r}.

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2