Definitions and coherence in "rigid" monoidal categories

Question

In "Catégories Tannakiennes" by Savedra Rivano (under A. Grothendieck supervision) at pag.78 he define a rigid category $\mathscr{C}$ as a monoidal symmetrical closed such that the natural morphisms $[X, X'] \otimes [Y, Y'] \to [X \otimes Y, X' \otimes Y' ]$ and $\theta: X \mapsto [[X, I], I] $ are isomorphisms.

The first is adjoint to $[X, X'] \otimes [Y, Y'] \otimes X \otimes Y \cong [X, X'] \otimes X \otimes [Y, Y'] \otimes Y \xrightarrow{t \otimes t} X' \otimes Y' $

the latter is adjoint to $X \otimes [X, I] \cong [X, I] \otimes X \xrightarrow{t} I$.

where $t_{A, B}: [A, B] \otimes B \to A$, $u_{X, Y}: X \to [Y, X \otimes Y]$ the co-unity and unity of the adjunction $(A \otimes B, C) \cong (A, [B, C])$.

Let us write $[X, I] := \widehat{X}$.

Follow the natural isomorphisms:

1) $[X, Y] \cong \hat{X} \otimes Y $

2) $([X, Y])^\wedge \cong [\widehat{Y}, \widehat{X}] $

Now, in mathematical literature are studied many kinds of specialized monoidal categories: compact closed, tortile, autonomous, rigid (but a different definition) ecc. ecc. and many work about coherence questions about.

I ask: this example of "rigid category" (of Savedra Rivano) is a particular case of some well studied type of monoidal categories?

A more simple question: let $\tau_X: I\xrightarrow{j} [X, X]\cong \widehat{X}\otimes X$.

I ask: (in a rigid category) is the morphisms $I\xrightarrow{\tau_X}\widehat{X}\otimes X \xrightarrow{s} X\otimes \widehat{X} \xrightarrow{\theta \otimes 1} \widehat{\widehat{X}}\otimes \widehat{X} $ equal to $\tau_{\widehat{X}}: I\to \widehat{\widehat{X}}\otimes \widehat{X}$ ?

Answer 1

Rigid monoidal categories in this sense are equivalent to compact closed categories. Their coherence was studied by Kelly and LaPlaza:

• G.M. Kelly and M. LaPlaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra, vol. 19, pp. 193-213, 1980.

In particular, the answer to your second (simpler) question is 'yes'.

Answer 2

The answer to both your questions is yes. The second one may be more elementary, but since I'm used to working with a definition of duality different that the one of Saavedra Rivano, I prefer to answer both questions in order.

Answer to the first question: I copy from the introduction of my degree thesis (http://arxiv.org/pdf/1110.5293v1.pdf) "In section 3 we define the important concept of dual pairing. In [1] and in [3] the authors use two (a priori) different definitions of duality, so we expose both and establish relations between them. We prove that they are equivalent, but in the process we also obtain a formulation of the concept of rigid category (of [3]) equivalent to the usual one but with fewer axioms and valid also in the non-symmetric case"

[1] A. Joyal and R. Street. An Introduction to Tannaka Duality and Quantum Groups in Category Theory, Proceedings, Como 1990, Lecture Notes in Math. 1488, pages 413-492.

[3] P. Deligne and J.S. Milne. Tannakian Categories. Hodge Cocycles Motives and Shimura Varieties, pages 101-228, 1982.

Note: the definition of [3] is the same as the one of Saavedra Rivano.

Note: the full text of the thesis is in spanish.

After having written my thesis, I found out that Deligne also does this (better of course) in Pierre Deligne, Catégories tannakiennes. This is in The Grothendieck Festschrift, Volume 2, 111--195. Birkhauser, 1990. This is done in section 2, Rappels et compléments: catégories tensorielles (text is in french). This is stated explicitly in proposition 2.3 and in 2.5.

To summarize, a rigid category as in Saavedra-Rivano or as in Deligne-Milne is equivalent to an autonomous category (as in Joyal-Street §9) that is symmetric. This is by definition a symmetric monoidal category where every object $X$ has a (right and left) dual $X^*$, i.e. there are arrows $\eta: I \rightarrow X^* \otimes X$, $\varepsilon: X \otimes X^* \rightarrow I$ satisfying two triangular identities), you can also find this in the wikipedia article http://en.wikipedia.org/wiki/Rigid_category (there it says that there are at least two equivalent definitions but only states this one, maybe someone reading this is willing to correct it?).

Answer to the second question: now we know that a rigid category as in Saavedra-Rivano satisfies that every object has a dual $X^*$ as above. Since the category is symmetric, right and left duals coincide and ${X^*}^* = X$. Therefore the first composition of your second question (without the last arrow if we take ${X^*}^* = X$ as definition of ${X^*}^*$) is precisely the way one defines the first arrow ($\eta: I \rightarrow {X^*}^* \times X^*$) that expresses the duality between ${X^*}^*$ and $X^*$ (remember ${X^*}^*=X$).

In other words, the symmetry morphism $s$ is the one that yields the equality ${X^*}^* = X$, by taking the $\eta$ corresponding to the duality ${X^*}^* \dashv X^*$ ($\tau_{\widehat{X}}$ in your notation) as the first composition of your second question by definition. I state this well known fact explicitly in proposition 3.9 of my degree thesis.