Subsection 3.2 (0PN1) · ScalingStacks

3.2 Commutative cubes

Let $\mathcal{I}$ be a finite set. Denote by $|\mathcal{I}|$ the cardinality of $\mathcal{I}$ and by $r(\mathcal{I})$ the set of all pairs $(\mathcal{L},a)$ where $\mathcal{L}$ is a subset of $\mathcal{I}$ and $a$ an element of $\mathcal{I}$ that does not belong to $\mathcal{L}.$ To simplify notation we will often

(a) denote a one-element set $\{a\}$ by $a,$

(b) denote a finite set $\{a,b,\dots,d\}$ by $ab\dots d,$

(c) denote the disjoint union $\mathcal{L}_{1}\sqcup\mathcal{L}_{2}$ of two sets $\mathcal{L}_{1},\mathcal{L}_{2}$ by $\mathcal{L}_{1}\mathcal{L}_{2},$ in particular, we denote by $\mathcal{L}a$ the disjoint union of a set $\mathcal{L}$ and a one-element set $\{a\},$ similarly, $\mathcal{L}ab$ means $\mathcal{L}\sqcup\{a\}\sqcup\{b\},$ etc.

0PKY

Definition 1 Let $\mathcal{I}$ be a finite set and $\mathcal{B}$ a category. A commutative $\mathcal{I}$ -cube $V$ over $\mathcal{B}$ is a collection of objects $V(\mathcal{L})\in Ob(\mathcal{B})$ for each subset $\mathcal{L}$ of $\mathcal{I},$ morphisms

\xi^{V}_{a}(\mathcal{L}):V(\mathcal{L})\longrightarrow V(\mathcal{L}a)

(27)

for each $(\mathcal{L},a)\in r(\mathcal{L}),$ such that for each triple $(\mathcal{L},a,b),$ where $\mathcal{L}$ is a subset of $\mathcal{I}$ and $a,b,a\not=b$ are two elements of $\mathcal{I}$ that do not lie in $\mathcal{L},$ there is an equality of morphisms

\xi^{V}_{b}(\mathcal{L}a)\xi^{V}_{a}(\mathcal{L})=\xi^{V}_{a}(\mathcal{L}b)\xi^{V}_{b}(\mathcal{L}),

(28)

i.e., the following diagram is commutative

\begin{CD}V(\mathcal{L})@>{\xi^{V}_{a}(\mathcal{L})}>{}>V(\mathcal{L}a)\\ @V{}V{\xi^{V}_{b}(\mathcal{L})}V@V{}V{\xi^{V}_{b}(\mathcal{L}a)}V\\ V(\mathcal{L}b)@>{\xi_{a}^{V}(\mathcal{L}b)}>{}>V(\mathcal{L}ab)\end{CD}

We will call a commutative $\mathcal{I}$ -cube an $\mathcal{I}$ -cube or, sometimes, a cube when it is clear what $\mathcal{I}$ is. Maps $\xi_{a}^{V}$ are called the structure maps of $V.$

Example If $\mathcal{I}$ is the empty set, an $\mathcal{I}$ -cube is an object in $\mathcal{B}.$ If $\mathcal{I}$ consists of one element, an $\mathcal{I}$ -cube is a morphism in $\mathcal{B}.$ If $\mathcal{I}$ consists of two elements, $\mathcal{I}=\{a,b\},$ an $\mathcal{I}$ -cube is a commutative square of objects and morphisms in $\mathcal{B}$ :

\begin{CD}V(\emptyset)@>{}>{}>V(a)\\ @V{}V{}V@V{}V{}V\\ V(b)@>{}>{}>V(ab)\end{CD}

In general, an $\mathcal{I}$ -cube can be visualized in the following manner. Let $n$ be the cardinality of $\mathcal{I}.$ We take an $n$ -dimensional cube in standard position in the Euclidean $n$ -dimensional space, i.e. each vertex has coordinates $(a_{1},\dots,a_{n})$ where $a_{i}\in\{0,1\}$ and orient each edge in the direction of the vertex with the bigger sum of the coordinates. Then the edges of any 2-dimensional facet of this cube are oriented as shown on the diagram below.

Choose a bijection between elements of $\mathcal{I}$ and coordinates of $\mathbb{R}^{n}.$ This bijection defines a bijection between vertices of the $n$ -cube and subsets of $\mathcal{I},$ with the $(0,\dots,0)$ vertex associated to the empty set. Oriented edges of the $n$ -cube correspond to pairs $(\mathcal{L},a)\in r(\mathcal{I}).$

Given an $\mathcal{I}$ -cube $V,$ put object $V(\mathcal{L})$ into the vertex associated to the set $\mathcal{L}$ and assign morphism $\xi^{V}_{a}(\mathcal{L})$ to the arrow going from the vertex associated to $\mathcal{L}$ to the vertex associated to $\mathcal{L}a.$ Equation (28) is equivalent to the commutativity of diagrams in all 2-dimensional faces of the $n$ -cube.

Given two $\mathcal{I}$ -cubes $V,W$ over a category $\mathcal{B},$ an $\mathcal{I}$ -cube map $\psi:V\longrightarrow W$ is a collection of maps

\psi(\mathcal{L}):V(\mathcal{L})\longrightarrow W(\mathcal{L}),\hskip 21.68121pt\mbox{ for all }\mathcal{L}\subset\mathcal{I}

that make diagrams

\begin{CD}V(\mathcal{L})@>{\psi(\mathcal{L})}>{}>W(\mathcal{L})\\ @V{}V{\xi^{V}_{a}(\mathcal{L})}V@V{}V{\xi^{W}_{a}(\mathcal{L})}V\\ V(\mathcal{L}a)@>{\psi(\mathcal{L}a)}>{}>W(\mathcal{L}a)\end{CD}

(29)

commutative for all $(\mathcal{L},a)\in r(\mathcal{I}).$ The map $\psi$ is called an isomorphism if $\psi(\mathcal{L})$ is an isomorphism for all $\mathcal{L}\subset\mathcal{I}.$ The map $\psi$ of $\mathcal{I}$ -cubes over an abelian category $\mathcal{B}$ is called injective/surjective if $\psi(\mathcal{L})$ is injective/surjective for all $\mathcal{L}\subset\mathcal{I}.$

The class of $\mathcal{I}$ -cubes over an abelian category $\mathcal{B}$ and maps between $\mathcal{I}$ -cubes constitute an abelian category in the obvious way. In particular, direct sums of $\mathcal{I}$ -cubes are defined.

For a finite set $\mathcal{I}$ and $a\in\mathcal{I},$ let $\mathcal{J}$ be the complement, $\mathcal{I}=\mathcal{J}\sqcup\{a\}.$ Given an $\mathcal{I}$ -cube $V,$ let $V_{a}(\ast 0),V_{a}(\ast 1)$ be $\mathcal{J}$ -cubes defined as follows:

V_{a}(\ast 0)(\mathcal{L})=V(\mathcal{L}),\hskip 14.45377ptV_{a}(\ast 1)(\mathcal{L})=V(\mathcal{L}a),\hskip 14.45377pt\mbox{ for }\mathcal{L}\subset\mathcal{J}

(30)

and the structure maps of $V_{a}(\ast 0),V_{a}(\ast 1)$ are determined by the structure maps $\xi_{b}^{V},b\in\mathcal{I}_{1}$ of $V$ in the obvious fashion. Sometimes we will write $V(\ast 0)$ for $V_{a}(\ast 0),$ etc. The structure map $\xi_{a}^{V}$ of $V$ defines an $\mathcal{I}_{1}$ -cube map $\xi_{a}^{V}:V_{a}(\ast 0)\longrightarrow V_{a}(\ast 1).$ This provides a one-to-one correspondence between $\mathcal{I}$ -cubes and maps of $\mathcal{I}_{1}$ -cubes.

We say that a map $\psi:V\to W$ of $\mathcal{I}$ -cubes over the category $R$ -mod of graded $R$ -modules and graded maps is graded of degree $i$ if the map $\psi(\mathcal{L}):V(\mathcal{L})\to W(\mathcal{L})$ has degree $i$ for all $\mathcal{L}\subset\mathcal{I}.$

For a cube $V$ over $R{\mbox{-mod}_{0}}$ denote by $V\{i\}$ the cube $V$ with the grading shifted by $i$ :

V\{i\}(\mathcal{L})=V(\mathcal{L})\{i\}\mbox{ for all }\mathcal{L}\subset\mathcal{I}

(31)

and the structure maps being appropriate shifts of the structure maps of $V.$ A degree $i$ map $\psi:V\to W$ of $\mathcal{I}$ -cubes over $R{\mbox{-mod}_{0}}$ induces a grading-preserving map $V\to W\{i\},$ also denoted $\psi.$

Mikhail Khovanov

Original source: arXiv:math/9908171v2