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