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\title{Notes on Picard groupoids}

\author{J.P. May}

\date{January 1, 2009}

\begin{document}

\maketitle

We consider connective spectra $X$.  If the only nonzero homotopy
group of $X$ is the zero$^{th}$, then $X$ is an Eilenberg--Mac\,Lane
spectrum $K(\pi_0(X),0)$, and $\pi_0$ establishes an equivalence between
the homotopy category of such spectra and the category of 
Abelian groups.  In this paper, for reasons that will shortly 
become apparent, we perversely call Abelian groups
``symmetric $1$-groups''.  Categorically, we think of a symmetry
isomorphism that has no choice but to be the identity map.   Instead
of considering spectra, we could consider the category of spaces $X$ with 
a unique non-zero homotopy group for any fixed $n\geq 2$.  That is, we are 
in the stable range for such spaces at dimension $2$.   We want to understand 
the analogous statements for connective spectra $X$ such that $\pi_i(X)=0$ for 
all $i>q$ for some fixed $q$.  The relevant stable range for spaces with non-zero 
homotopy groups between $n$ and $n+q$ is $n\geq q+2$. 

In these notes, I'll discuss the case $q=1$, which is used non-trivially
by Sasha Beilinson and Deepam Patel, to whom the basic result is due.

\begin{defn} A Picard groupoid, or weak symmetric $2$-group, is a small 
symmetric monoidal groupoid 
\[ (\sP,\otimes,I,c, a, \ell, r)\]
in which every object is invertible up to isomorphism under $\otimes$. 
Thus, for each object $A$, there must be an object $A^{-1}$ such that
$A\otimes A^{-1}\iso I\iso A^{-1}\otimes A$.  This is structure on $\sP$.
If we require specified isomorphisms (satisfying the triangle identities) 
as part of the data defining $\sP$, we call $A$ a coherent symmetric $2$-group.
\end{defn}

Using strong monoidal functors as $1$-morphisms and monoidal natural transformations
as $2$-morphisms, we obtain strict $2$-categories $WS2G$ and $CS2G$ of weak symmetric 
$2$-groups and coherent symmetric $2$-groups, with a forgetful pseudofunctor
$U\colon CS2G\rtarr WS2G$.  Working in the non-symmetric case, Baez and Lauda \cite{BD} 
have made an exhaustive study of these notions.  Their arguments carry over
verbatim upon restriction to the symmetric case, and we'll summarize their results here.  
They justify the term
``coherent'' with a coherence theorem \cite[3.6]{BL} that is essentially due to
LaPlaza \cite{LaP}.  They then prove the following results \cite[5.1, 5.3, 5.4]{BL}. 

\begin{thm} Any weak symmetric $2$-group $\sP$ can be given a structure of a coherent
symmetric $2$-group $Coh\sP$.  There results a pseudofunctor 
$Coh\colon WS2G\rtarr CS2G$ that acts as the identity on $1$-morphisms and $2$-morphisms,
and $U$ and $Coh$ extend to a $2$-equivalence between $WS2G$ and $CS2G$. 
\end{thm}

Thus the neglect of structure functor $U$ loses no essential information and these
variant notions of weak $2$-group can be used interchangeably. The technical reason 
that this works so simply is that the $1$ and $2$ morphisms in these $2$ categories 
are defined in exactly the same way. The justification for the definitions is that any 
$1$-morphism between coherent symmetric $2$-groups automatically preserves inverses 
up to canonical isomorphism \cite[6.1]{BL}. 

Moreover, they go on to define the notion of a coherent $2$-group in any category $\sV$ 
with finite limits \cite[7.1]{BL}.   The case $\sV = \sC\!at$ recovers the original 
definition of coherent $2$-groups \cite[7.2]{BL}.  Again, there is a strict $2$-category 
$CS2G\sV$ of such objects.  Working in this greater generality, they compare weak $2$-groups 
to strict $2$-groups.  Returning to our original symmetric $2$-groups in $\sC\!at$, we
define strict objects as follows.

\begin{defn}  An elementary Picard groupoid, or strict symmetric $2$-group, is a small 
permutative groupoid in which every object is strictly invertible.  Thus, for each object 
$A$, there must be an object $A^{-1}$ such that $A\otimes A^{-1} = I = A^{-1}\otimes A$.
\end{defn}

Since the term strict permutative groupoid is sometimes used for the strictly commutative 
case, which is uninteresting to us, we have suggested the alternative name ``elementary 
Picard groupoids''.  Ignoring the symmetry, these are just groups in $\sC\!at$ and so are 
often called $\sC\!at$-groups.  The symmetry is our focus of interest.  

\marginpar{Crossed modules!?\cite[p.462]{BL}}
\marginpar{RoleH3?\cite[471]{BL}}

Standard categorical yogi (e.g. \cite[4.2]{MayPer}, \cite[VI.3.2]{MQR}) shows that
any symmetric monoidal category is equivalent as a symmetric monoidal category to a 
permutative category, in which $\otimes$ is strictly associative and unital, with unit 
object $I$.  However, a skeleton of a permutative category need only be symmetric monoidal, 
not necessarily permutative.  Passage to the skeleton of a permutative category preserves 
the property of being a groupoid, and Baez and Lauda prove the following result \cite[8.3.3]{BL}.

\begin{prop} Every coherent symmetric $2$-group is equivalent in $CS2G$ to a skeletal
symmetric $2$-group whose unit coherence isomorphisms are the identity and in which
every object is strictly invertible.
\end{prop} 

However, the associativity is not strict, leading to the appearance of a cohomology
class coming from the associator in their classification theorem for coherent $2$-groupoids 
\cite[8.3.7, 8.3.8]{BL}.  On the other hand, Baez and Lauda \cite[8.3.9]{BL} also prove that 
one can arrange strictness, but at the price of not being skeletal.  

\begin{prop} Every coherent symmetric $2$-group is equivalent in $CS2G$ to a strict
symmetric $2$-group.
\end{prop}

We want to give a new way of classifying strict symmetric $2$-groupoids.  
Fix such a category $\sP$.  We abbreviate $\pi_0$ for the group of isomorphism 
classes of objects of $\sP$ under $\otimes$.  [Maybe must use the not necessarily
Abelian group of all objects of $\sP$: problem of choice of an object in its
isomorphism class in getting $c_I$ below; if we use all objects,
then may need to use all maps = isos between them?  Not equivalent].  
Because of the symmetry isomorphism
$c\colon A\otimes B \rtarr B\otimes A$, $\pi_0$ is an Abelian group. 
Let $\pi_1$ denote the group of automorphisms of the unit object $I$.  
As in any monoidal category (symmetric not being needed for this), 
$\pi_1$ is also an Abelian group. Using $I\otimes I \iso I$, we see as usual 
that the tensor product and 
composite of automorphisms of $I$ agree.  

Here is a construction, starting from Abelian groups $\pi_0$ and $\pi_1$,
that does give lots of examples. Think of $\pi_0$ as a discrete category 
(only identity maps), and write $A$
for both an object of $\pi_0$ and its identity map.  It still 
makes sense to think of $\pi_0$ as a group, but now the product and inverse are 
thought of as functors between discrete categories.  This makes $\pi_0$ a discrete 
monoidal category, and we could view it as a strict permutative groupoid.

Think of the Abelian group $\pi_1$ as a category with a single object $\ast$.  
Form the product category 
$\pi_1 \times \pi_0$.  Give it the evident 
product $\otimes$:
\[ (ast,A)\otimes (\ast,B) = (\ast, A\otimes B) \]
on objects, and
\[ (f,A) \otimes (g,B) = (fg, A\otimes B) \] 
on morphisms. The unit object is $(*,I)$.  This product is
strictly associative and unital, and it makes sense for any pair of Abelian
groups $(\pi_0,\pi_1)$. We have just perversely denoted the group structure 
on $\pi_0$ as $\otimes$.  

We can ask for a commutativity isomorphism.  It should assign to each pair
$(A,B)$ a morphism
\[ (c_I(A,B), A\otimes B) \]
where $c_I(A,B)\in \pi_1$.  Thus $c_I$ is a function 
\[ c_I\colon \pi_0\times \pi_0 \rtarr \pi_1. \]
Since $\pi_1$ is an Abelian group,
we automatically have the ``naturality condition'' $f c_I(A,B) = c_I(A,B) f$  for 
all $f\in \pi_1$.  We must impose the symmetry condition $$c_I(A,B)^{-1} = c_I(B,A)$$
and the coherence conditions $$c_I(I,A) = id = c_I(A,I)$$ for all objects $A$ and the
triple permutation condition $$c_I(A\otimes B,C) = c_I(A,C)c_I(B,C)$$ for all triples of
objects $A$, $B$, $C$.  In the presence of the first of these three conditions, if we 
change to additive notation for the product (and inverse) in both $\pi_0$ and $\pi_1$, 
we see that the second and third conditions amount to bilinearity, so that $c_I$ is really 
nothing but a homomorphism
\[  c_I\colon \pi_0\otimes \pi_0 \rtarr \pi_1 \]
of Abelian groups satisfying $c_I(A\otimes B) = - c_I(B\otimes A)$.  
Written in this alternative notation, the trivial choice $c_I(A,B)=\id$ for all pairs $A$, $B$
is given by $c_I(A\otimes B) = 0$.  This gives a strict symmetric $2$-group, but there are other 
choices with the same underlying monoidal category.  Any choice of a commutativity 
isomorphism $c_I$, meaning a function
$c_I\colon \pi_0\times \pi_0 \rtarr \pi_1$
satisfying the anticommutativity relation $c_I(A\otimes B) = - c_I(B\otimes A)$ gives an
example 
\[ \sP(\pi_0,\pi_1,c_I) \]
of a symmetric $2$-group.  Up to isomorphism, we would like to say that these are the only examples.

\begin{prop} Any strict symmetric $2$-group $\sP$ is isomorphic to one of the form 
$\sP(\pi_0,\pi_1,c_I)$.
\end{prop}
\begin{proof}
We have already described how to construct $\pi_0$ and $\pi_1$ from $\sP$.  We begin 
by defining inverse equivalences of categories
\[ F\colon \sP \rtarr \pi_1 \times \pi_0  \]
\[ F^{-1} \colon  \pi_1 \times \pi_0 \to \sP.  \]
Note that since we start with a skeletal groupoid, there are no maps $A \rtarr B$ for 
$A\neq B$.  For a map $g\colon A\to A$, define 
\[g_I = g\otimes A^{-1}\colon I = A\otimes A^{-1}\to A\otimes A^{-1} = I. \]
For a map $f\colon I\to I$, define
\[f_A = f\otimes A: A = I\otimes A \to I\otimes A = A. \]
We then define $F$ by $F(A) = (*,A)$ on objects  and $F(g) = (g_I,A)$ on
morphisms $g\colon A\rtarr A$. We define $F^{-1}$ by $F^{-1}(*,A) = A$ on objects
and  $F^{-1}(f,A) = f_A$ on morphisms $f\colon I\rtarr I$.  This already gives an
isomorphism of monoidal categories.  What about $c$?  We are given isomorphisms
\[ c(A,B): A\otimes B\rtarr B\otimes A\]
in $\sP$.   Applying $F$ to this morphism gives a morphism
\[ (c_I(A,B), A\otimes B)\colon A\otimes B\rtarr A\otimes B, \]
where $c_I(A,B)\in \pi_1$.  Coherence in $\sP$ says that these elements satisfy the algebraic
properties of a commutativity isomorphism that we prescribed above.  Conversely,
we can reconstruct the coherence isomorphism $c$ from a commutativity isomorphism
$c_I\colon \pi_0\times \pi_0\rtarr \pi_1$.
\end{proof} 

By the two results above, any Picard groupoid is equivalent to an elementary Picard 
groupoid $\sP(\pi_1,\pi_0,c_I)$.  Consider the classifying space functor $B$.  As a
space, and ignoring $c_I$, $B(\pi_1 \times \pi_0)$ is just the disjoint union
over the elements $A\in \pi_0$ of copies of the group $\pi_1$, so that 
$B(\pi_1 \times \pi_0)$ is the disjoint union over $A$ of copies of the 
classifying space $B\pi_1=K(\pi_1,1)$.  Therefore the homotopy groups of 
$B(\pi_1 \times \pi_0)$ are $\pi_0$ (an Abelian group by definition) 
and $\pi_1$.  We need to understand what $c_I$ corresponds to on the
classifying space level, and it is clear conceptually that it must 
correspond to the unique $k$-invariant of a two-stage spectrum or
space (in a stable range).

Form the associated spectrum $E\sP(\pi_1,\pi_0,c_I)$, as characterized axiomatically
in \cite{May}.  The unique axiom on such a functor $E$ from permutative categories $\sC$
to spectra is that there is a natural group completion map $\et\colon B\sC\rtarr E_0\sA$.
In our case, $B\sP(\pi_1,\pi_0,c_I)$ is grouplike ($\pi_0$ is a group), and this implies 
that $\et$ is a weak homotopy equivalence.  Therefore $\pi_0$ and $\pi_1$ are the only
non-vanishing homotopy groups of $E\sP(\pi_1,\pi_0,c_I)$.  

Conversely, consider spectra  X  with two non-vanishing homotopy groups 
$\pi_0$ and $\pi_1$.  Such an $X$ fits into an exact triangle (cofiber sequence)
\[ K(\pi_1,1) \rtarr X \rtarr K(\pi_0,0) \rtarr K(\pi_1,2)  \]
of spectra; call the last map $k$: it is the unique $k$-invariant of this 
two-stage Postnikov system of spectra.   Working with spaces in the stable range, we can think of 
this as a two-stage Postnikov system 
\[ K(\pi_1,n+1) \rtarr X \rtarr K(\pi_0,n) \rtarr K(\pi_1,n+2)\]   
of spaces for $n\geq 3$.

We know how to describe such spaces purely algebraically, for example
by work of Loday \cite{Lod} and, especially relevant, \cite{Con}; see also \cite{Bau, P}.  
For $n= 1$, this goes all the way back to very old work of J.H.C. Whitehead, but we 
are interested only in the simpler case of large $n$.  We need only compare definitions to see that 
one of the several known algebraic descriptions of such two-stage spaces is equivalent to the 
description given by a commutativity isomorphism $c_I$, as above.  This amounts to 
describing algebraically how to calculate  $H^{n+2}(K(\pi_0,n); \pi_1)$.  In fact,
the triple $(\pi_0,\pi_1, C_I)$ is a very special case of a stable crossed module 
\cite[3.1]{Con} or stable quadratic module \cite[IV.C.1]{Bau}, and one way to proceed 
is to check that every stable quadratic module is suitably equivalent to one of this form.  
This exercise is carried out, not very explicitly, in \cite[3.10]{Con}.     

An alternative, purely stable, verification that $E$ specifies an equivalence from the 
category of Picard groupoids to the homotopy category of spectra whose only non-vanishing
homotopy groups are $\pi_0$ and $\pi_1$ can be obtained by using the Segal machine
to construct $E$ and then reconstructing from such a two-stage spectrum a very 
special $\GA$-space that is equivalent to one that comes from a Picard groupoid.
This approach is worked out in Deepam Patel's thesis \cite{DP}. 

What happens for larger $q$ might make a good thesis topic.
We know the connection between symmetric monoidal categories and spectra.
But making $\pi_0$ an Abelian group necessarily kills all $\pi_i$ with $i>2$,
so it doesn't seem that an answer in terms of symmetric monoidal categories
with special properties makes sense.  Rather, we should think in terms of
higher categorical versions of grouplike symmetric monoidal categories,
starting with the appropriate formulation of the notion of a Picard $2$-groupoid.

Interesting!

\begin{thebibliography}{9}

\bibitem{BL}
J.C. Baez and A.D. Lauda. Higher-dimensional algebra V: $2$-groups.
Theory and Applications of Categories 12(2004), 423--491. 

\bibitem{Bau}
H.-J. Baues.  
Combinatorial homotopy and 4-dimensional complexes.
Walter de Gruyter. Berlin, 1991. 

\bibitem{Con}
D. Conduch\'e.
Modules crois�s g�n�ralis�s de longueur 2.
J. Pure and Applied Algebra 34(1984), 155-178.   

\bibitem{LaP}
M. L. Laplaza. 
Coherence for categories with group structure: an alternative approach. 
J. Algebra 84(1983), 305�323. 
  
\bibitem{Lod}
Jean-Louis Loday.
Spaces with finitely many nontrivial homotopy groups.
J. Pure Appl. Algebra 24 (1982), no. 2, 179--202. 

\bibitem{MayPer}
J.P. May.
$E_{\infty}$ spaces, group completions, and permutative categories.    
London Math. Soc.  Lecture Notes Series Vol. 11, 1974, 61--93.

\bibitem{MQR}
J.P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). 
$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra.  
Lecture Notes in Mathematics Vol. 577.  Springer-Verlag 1977.

\bibitem{May}
J.P. May.
The spectra associated to permutative categories.  
Topology 17(1978), 225--228.

%\bibitem{MT}
%F. Muro and A. Tonks.
%The $1$-type of a Waldhausen $K$-theory spectrum.
%Advances in Mathematics 216 (2007), no. 1, 178-211. 

\bibitem{P}
S. Paoli.
Internal categorical structures in homotopical algebra.
To appear in a book published by the IMA.
http://www.math.uchicago.edu/~may/IMA/Paoli.pdf

\bibitem{DP}
D. Patel.
PhD thesis.
The University of Chicago. 2008. 


\end{thebibliography}

\end{document} 
   


