The Stacks project

12.17 Additive monoidal categories

Some material about the interaction between a monoidal structure and an additive structure on a category.

Definition 12.17.1. An additive monoidal category is an additive category $\mathcal{A}$ endowed with a monoidal structure $\otimes , \phi $ (Categories, Definition 4.43.1) such that $\otimes $ is an additive functor in each variable.

Lemma 12.17.2. Let $\mathcal{A}$ be an additive monoidal category. If $Y_ i$, $i = 1, 2$ are left duals of $X_ i$, $i = 1, 2$, then $Y_1 \oplus Y_2$ is a left dual of $X_1 \oplus X_2$.

Proof. Follows from uniqueness of adjoints and Categories, Remark 4.43.7. $\square$

Lemma 12.17.3. In a Karoubian additive monoidal category every summand of an object which has a left dual has a left dual.

Proof. We will use Categories, Lemma 4.43.6 without further mention. Let $X$ be an object which has a left dual $Y$. We have

\[ \mathop{\mathrm{Hom}}\nolimits (X, X) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{1}, X \otimes Y) = \mathop{\mathrm{Hom}}\nolimits (Y, Y) \]

If $a : X \to X$ corresponds to $b : Y \to Y$ then $b$ is the unique endomorphism of $Y$ such that precomposing by $a$ on

\[ \mathop{\mathrm{Hom}}\nolimits (Z' \otimes X, Z) = \mathop{\mathrm{Hom}}\nolimits (Z', Z \otimes Y) \]

is the same as postcomposing by $1 \otimes b$. Hence the bijection $\mathop{\mathrm{Hom}}\nolimits (X, X) \to \mathop{\mathrm{Hom}}\nolimits (Y, Y)$, $a \mapsto b$ is an isomorphism of the opposite of the algebra $\mathop{\mathrm{Hom}}\nolimits (X, X)$ with the algebra $\mathop{\mathrm{Hom}}\nolimits (Y, Y)$. In particular, if $X = X_1 \oplus X_2$, then the corresponding projectors $e_1, e_2$ are mapped to idempotents in $\mathop{\mathrm{Hom}}\nolimits (Y, Y)$. If $Y = Y_1 \oplus Y_2$ is the corresponding direct sum decomposition of $Y$ (Section 12.4) then we see that under the bijection $\mathop{\mathrm{Hom}}\nolimits (Z' \otimes X, Z) = \mathop{\mathrm{Hom}}\nolimits (Z', Z \otimes Y)$ we have $\mathop{\mathrm{Hom}}\nolimits (Z' \otimes X_ i, Z) = \mathop{\mathrm{Hom}}\nolimits (Z', Z \otimes Y_ i)$ functorially as subgroups for $i = 1, 2$. It follows that $Y_ i$ is the left dual of $X_ i$ by the discussion in Categories, Remark 4.43.7. $\square$

Example 12.17.4. Let $F$ be a field. Let $\mathcal{C}$ be the category of graded $F$-vector spaces. Given graded vector spaces $V$ and $W$ we let $V \otimes W$ denote the graded $F$-vector space whose degree $n$ part is

\[ (V \otimes W)^ n = \bigoplus \nolimits _{n = p + q} V^ p \otimes _ F W^ q \]

Given a third graded vector space $U$ as associativity constraint $\phi : U \otimes (V \otimes W) \to (U \otimes V) \otimes W$ we use the “usual” isomorphisms

\[ U^ p \otimes _ F (V^ q \otimes _ F W^ r) \to (U^ p \otimes _ F V^ q) \otimes _ F W^ r \]

of vectors spaces. As unit we use the graded $F$-vector space $\mathbf{1}$ which has $F$ in degree $0$ and is zero in other degrees. There are two commutativity constraints on $\mathcal{C}$ which turn $\mathcal{C}$ into a symmetric monoidal category: one involves the intervention of signs and the other does not. We will usually use the one that does. To be explicit, if $V$ and $W$ are graded $F$-vector spaces we will use the isomorphism $\psi : V \otimes W \to W \otimes V$ which in degree $n$ uses

\[ V^ p \otimes _ F W^ q \to W^ q \otimes _ F V^ p,\quad v \otimes w \mapsto (-1)^{pq} w \otimes v \]

We omit the verification that this works.

Lemma 12.17.5. Let $F$ be a field. Let $\mathcal{C}$ be the category of graded $F$-vector spaces viewed as a monoidal category as in Example 12.17.4. If $V$ in $\mathcal{C}$ has a left dual $W$, then $\sum _ n \dim _ F V^ n < \infty $ and the map $\epsilon $ defines nondegenerate pairings $W^{-n} \times V^ n \to F$.

Proof. As unit we take By Categories, Definition 4.43.5 we have maps

\[ \eta : \mathbf{1} \to V \otimes W\quad \epsilon : W \otimes V \to \mathbf{1} \]

Since $\mathbf{1} = F$ placed in degree $0$, we may think of $\epsilon $ as a sequence of pairings $W^{-n} \times V^ n \to F$ as in the statement of the lemma. Choose bases $\{ e_{n, i}\} _{i \in I_ n}$ for $V^ n$ for all $n$. Write

\[ \eta (1) = \sum e_{n, i} \otimes w_{-n, i} \]

for some elements $w_{-n, i} \in W^{-n}$ almost all of which are zero! The condition that $(\epsilon \otimes 1) \circ (1 \otimes \eta )$ is the identity on $W$ means that

\[ \sum \nolimits _{n, i} \epsilon (w, e_{n, i})w_{-n, i} = w \]

Thus we see that $W$ is generated as a graded vector space by the finitely many nonzero vectors $w_{-n, i}$. The condition that $(1 \otimes \epsilon ) \circ (\eta \otimes 1)$ is the identity of $V$ means that

\[ \sum \nolimits _{n, i} e_{n, i}\ \epsilon (w_{-n, i}, v) = v \]

In particular, setting $v = e_{n, i}$ we conclude that $\epsilon (w_{-n, i}, e_{n, i'}) = \delta _{ii'}$. Thus we find that the statement of the lemma holds and that $\{ w_{-n, i}\} _{i \in I_ n}$ is the dual basis for $W^{-n}$ to the chosen basis for $V^ n$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FN9. Beware of the difference between the letter 'O' and the digit '0'.