added (some of) what we covered in the mini-course today

2_28_more_tom_dieck.tex

@@ -150,9 +150,9 @@ \subsection*{The Burnside category.}
 $\sO_G\to B_G$, and restricting to spans of the form $Y\gets X\stackrel\id\to X$ defines a contravariant functor
 $\sO_G\op\to B_G$. Composing with a Mackey functor $F$ defines a pair of functors $\sO_G\op\to\Ab$ and
 $\sO_G\to\Ab$. We'll denote the image of an $f\colon X\to Y$ under these functors as $f^*$ and $f_*$, and given a
-diagram
+pullback diagram
 \[\xymatrix{
-	X'\ar[r]^{f'}\ar[d]^{g'} & X\ar[d]^g\\
+	X'\ar[r]^{f'}\ar[d]^{g'}\pushoutcorner & X\ar[d]^g\\
 	Y'\ar[r]^f & Y,
 }\]
 Mackey functors satisfy the \term{Chevalley condition} $g^*f_* = f_*'(g')^*$.

3_2_mackey_functors.tex

@@ -40,11 +40,10 @@
 	\[\pi_n^H(X)\cong [S^n\wedge G/H_+, X]\cong [S^n, X\wedge G/H_+],\]
 	so it's clear how a map $G/H\to G/K$ induces both a right-way and a wrong-way map.
 	\item Let $\underline\Z$ denote the \term{constant Mackey functor} in $\Z$, which assigns $\Z$ to every object.
-	You have to figure out what the transfers and restrictions are, but in the end the restrictions are the
-	identity and the transfers are multiplication by the Euler characteristic.
+	The restriction maps are all the identity, and the transfer $G/H\to G/K$ is multiplication by $\abs{H/K}$.
 	\item The \term{Burnside Mackey functor} $A(G)$ is defined by letting $A(G)(G/H)$ be the Grothendieck group of
-	the monoid on $H$-sets under coproduct. The transfer and restriction maps come from induction and restriction,
-	respectively, between $H\Set$ and $K\Set$.
+	the symmetric monoidal category of finite $H$-sets under coproduct. The transfer and restriction maps come from
+	induction and restriction, respectively, between $H\Set$ and $K\Set$.
 	\item The \term{representation Mackey functor} $R(G)$ is defined by letting $R(G)(G/H)$ be the Grothendieck
 	group of finite-dimensional $H$-representations, with transfer and restriction given by
 	$\Z[K]\otimes_{\Z[H]}\bl$.

4_13_composition_and_bispans.tex

@@ -219,7 +219,10 @@
 	\item Just as we described Mackey functors as a pair of a covariant and a contravariant functor satisfying a
 	push-pull axiom, there's a similar (more complicated) definition for Tambara functors, with three functors
 	satisfying some interoperability conditions. We won't need this for the time being, so we're not going to write
-	it out. \qedhere
+	it out.
+	\item If you forget the norm map, a Tambara functor defines a Green functor. A Green functor may extend to a
+	Tambara functor, but such an extension need not exist or be unique~\cite{MazurThesis}.
+	\qedhere
 \end{comp}
 \begin{exm}
 Let $R$ be a ring with a $G$-action. Then, $\Map_G(\bl,R)$ is a Tambara functor. In this case, given a map $f\colon

references.bib

@@ -1229,3 +1229,10 @@ @article{Bredon
 	year = "1967"
 }
 
+@phdthesis{MazurThesis,
+	author = "Kristen Mazur",
+	title = "On the Structure of {M}ackey Functors and {T}ambara Functors",
+	school = "University of Virginia",
+	year = "2013",
+	note = "\url{https://sites.lafayette.edu/mazurk/files/2013/07/Mazur-Thesis-4292013.pdf}"
+}