June 26, 2008

Category Algebras

Posted by Urs Schreiber

Currently Masoud Khalkali over from Noncommutative Geometry Blog is giving an introductory lecture series on – right – noncommutative geometry, here at the Hausdorff institute.

He is following his notes

Masoud Khalkali
Very basic noncommutative geometry
arXiv:math/0408416 .

Today he started talking about noncommutative algebras arising as groupoid algebras (or “groupoid convolution algebras”, special cases of category algebras, see p. 58) motivating them by their ubiquitousness in noncommutative geometry:

The good news is that most of the noncommutative spaces which are currently in use in noncommutative geometry are constructed by this method.
p. 53

Here I want to use this opportunity to point out how category algebra works from the point of view of groupoidification along the lines of the discussion in An Exercise in Groupoidification: The Path Integral.

For $V$ a (finite dimensional) vector space, I write $FinSet_V$ for the category of sets over $V$. An object is a set $S$ with a map to $V$, $S\to V$ and a morphism is a commuting triangle $\array{ S &\to& S' \\ & \searrow \swarrow \\ & V } \,.$ The $V$-cardinality operation is a monoidal map $|\cdot| : (FinSet_V, \oplus) \to (V, +)$ sending a finite set labeled by $V$ to the sum of the labels of its elements.

Let $\mathbf{B} End(V)$ be the category whose single object is the vector space $V$ and whose space of morphisms is $End(V)$, with composition of morphisms being the composition of these endomorphisms.

Take $C$ to be any finite category whose category algebra we want to consider. Form the cartesian product category $C \times \mathbf{B} End(V)$ and consider finite sets over the collection of morphisms: $\array{ S \\ \downarrow \\ Mor(C \times \mathbf{B} End(V)) } \,.$ Notice that for each fixed morphisms of $C$ this is an element in $FinSet_{End(V)}$. Under the $End(V)$-cardinality it is therefore an $End(V)$-valued function on $Mor(C)$. If $V = \mathbb{C}$ (and assuming we are talking about complex vector spaces), then this is a complex function on $Mor(C)$.

Given two such sets over $Mor(C \times \mathbf{B}End(V))$ $\array{ S &&&& S' \\ \downarrow &&&& \downarrow \\ Mor(C \times \mathbf{B} End(V)) &&&& Mor(C \times \mathbf{B} End(V)) }$ representing two $End(V)$-valued functions on $Mor(C)$ we take the correspondence space given by the composition operation in $C \times \mathbf{B}End(V)$

$\array{ S && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) && S' \\ \downarrow & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \downarrow \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) } \,,$ pull back both sets $S$ and $S'$ to the total space and take fiberwise cartesian product there to get the new set $d_0^* S \otimes d_1^* S'$ $\array{ && d_0^* S \otimes d_1^* S' \\ && \downarrow \\ && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) && \\ & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) }$ and then push that down along the composition map $\circ$ $\array{ && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) \\ & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) \\ && \uparrow \\ && \int_\circ d_0^* S \otimes d_1^* S' } \,.$ Under $End(V)$-cardinality, the resulting finite set $\int_\circ d_0^* S \otimes d_1^* S'$ is the category algebra product of the two original $End(V)$-valued functions on $Mor(C)$.

We can also discuss the action of this category algebra on $V$-valued functions on $Obj(C)$ in a similar manner:

For $A$ a monoid write $\mathbf{B}A$ for the corresponding one-object category. A representation of $A$ on $V$ is a functor $\rho : \mathbf{B}A \to Vect$ such that $\rho : (\bullet \stackrel{a}{\to} \bullet) \mapsto (V \stackrel{\rho(a)}{\to} V) \,.$ We write $V//A$ for the corresponding action category, being the pullback $\array{ V//A &\to& Vect_* &\to& Set_* \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}A &\stackrel{\rho}{\to}& Vect &\to& Set }$ which has $V$ as its space of objects and one morphism of the form $v \stackrel{a}{\to} \rho(a)(v)$ for each $v \in V$ and each $a \in A$, with the obvious composition coming from the product in $A$.

For the discussion of category algebras, consider again the monoid $A = End(V)$ and hence the action category $V//End(V)$. Notice that every $End(V)$-valued set over $Mor(C \times \mathbf{B}End(V))$ pulls back along the canonical projection $p : V//A \to \mathbf{B}A$ to a finite set over $Mor(C \times V//End(V))$.

So consider a $V$-valued set over $Obj(C)$ in the form of a finite set $\array{ S \\ \downarrow \\ Obj(C \times V//End(V)) ) = Obj(C)\times V }$ and then the correspondence space of the category $C$ $\array{ && p^* S' \\ && \downarrow \\ S && Mor(C \times V//End(V)) \\ \downarrow & {}^s\swarrow && \searrow^t \\ Obj(C)\times V &&&& Obj(C)\times V } \,,$

where $p^* S'$ is the pullback along the above $p$ of an $End(V)$-valued set over $Mor(C)$ representing, by the previous discussion, an element in the category algebra of $C$ (with coefficients in $V$).

Then pull back $S$ along the source map to $Mor(C \times V//End(V))$, tensor there with $p^* S'$

$\array{ && s^*S \otimes p^* S' \\ && \downarrow \\ && Mor(C \times V//End(V)) \\ & {}^s\swarrow && \searrow^t \\ Obj(C)\times V &&&& Obj(C)\times V } \,,$

and then push down along the target map

$\array{ && Mor(C \times V//End(V)) && \int_t s^* S \otimes p^* S' \\ & {}^s\swarrow && \searrow^t & \downarrow \\ Obj(C)\times V &&&& Obj(C)\times V }$

to get a $V$-valued set over $Obj(C)$. Under $V$-cardinality, this is the image of the original $V$-valued function represented by $S$ under the action of the category algebra element represented by $S'$.

The reasoning needed for seeing this is precisely the one used for the disucssion of the path integral in this context.

Posted at June 26, 2008 8:50 PM UTC

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Re: Category Algebras

Urs wrote:
> The V-cardinality operation is a monoidal
> map
> ∣⋅∣:(FinSet V,⊕)→(V,+)
> sending a finite set labeled by V to the
> sum of the labels of its elements.

Do you want to put a weight of this sum of elements? For instance if you have a finite group acting on a vector space V you could take the S sitting over v in V to be all the G translates of v and the map would then be 1/|G| \sum gv, i.e. the projection of v on the invariant subspace.

Posted by: Maarten Bergvelt on July 1, 2008 5:36 PM | Permalink | Reply to this

Re: Category Algebras

Very good point.

Indeed, these weights will come in when we do push-forward of sets over groupoids (instead of just sets over sets, as I have considered here).

You can see these factors appear in definition 5 of John Baez’s HDA VII.

The way I like to think about them is this:

ultimately, as I was just telling Eric # ,these sets over a groupoid will be thought of as functors from the groupoid to $\mathrm{Set}$. Then we can consider the push-forward literally as the colimit of that set-valued functor and Tom Leinster’s formula applies to give us these weights automatically (or almost automatically).

Posted by: Urs Schreiber on July 1, 2008 5:55 PM | Permalink | Reply to this

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