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%opening
\title{A2 --- Stacks Project Workshop}
\author{Simon Felten}

\begin{document}

\maketitle

Let $C$ be a $k$-linear symmetric monoidal category. Then a commutative algebra in $C$ is a pair $(R,\mu)$ of an object $R \in C$ and a morphism $\mu: R \otimes R \rightarrow R$ such that an associativity relation and a commutativity relation are satisfied. We assume that $(R,\mu)$ is unital, i.e., there is a morphism $\eta: 1 \rightarrow R$ such that a relation is satisfied. Such a morphism $\eta$ is unique. We denote the category of commutative algebras by $\mathrm{ComAlg}(C)$.

There is an obvious forgetful functor $\mathrm{ComAlg}(C) \rightarrow C$. Constructing a left adjoint requires:
\begin{enumerate}
\item For every $V \in C$, the direct sum $T(V) := \bigoplus_{i \geq 0} V^{\otimes i}$ exists.
\item The coequalizer $\mu, \mu \circ \beta: T(V) \otimes T(V) \rightarrow T(V)$ exists. Here $\mu$ is the multiplication, and $\beta$ is the symmetry.
\end{enumerate}
These conditions are satisfied if $C$ admits all filtered colimits and is abelian. Namely, coequalizers can be constructed as the cokernel of the difference of the morphisms. The left adjoint is the coequalizer, which we denote by $S_\beta(V)$. It is called the symmetric algebra of $V$. 

Under these conditions, there is a second construction of the symmetric algebra. Namely, every permutation $\sigma \in S_n$ gives rise to a symmetrization morphism $s = \sum_{\sigma \in S_n} \beta_\sigma(V): V^{\otimes n} \to V^{\otimes n}$. Because $C$ is abelian, we can form its image $S^n_\beta(V) := s(V^{\otimes n})$. Then the direct sum $\bigoplus_{i \geq 0} S^n_\beta(V)$ exists and carries a natural multiplication. Q: What is the natural multiplication?

Now assume $C$ an \emph{abelian} $k$-linear symmetric monoidal category such that the functor $C \rightarrow C, X \mapsto X \otimes V$ is right exact for every object $V \in C$---we don't assume that all filtered colimits exist. Given a commutative algebra $(R,\mu)$ in $C$, a module is a pair $(M,m)$ with an object $M \in C$ and a morphism $m: R \otimes M \to M$ satisfying two relations. Modules form a $k$-linear abelian (?) category $R-\mathrm{Mod}(C)$. We obtain a bifunctor 
$$\otimes_R: R-\mathrm{Mod}(C) \times R-\mathrm{Mod}(C) \rightarrow R-\mathrm{Mod}(C)$$
by taking the coequalizer of the two maps $M \otimes (R \otimes N) \rightarrow M \otimes N$ and 
$$M \otimes (R \otimes N) \cong (M \otimes R) \otimes N \cong (R \otimes M) \otimes N \rightarrow M \otimes N.$$ 
It turns $R-\mathrm{Mod}(C)$ into a $k$-linear abelian symmetric monoidal category. Q: If $C$ admits all filtered colimits, does $R-\mathrm{Mod}(C)$ admit all filtered colimits?
\end{document}
