Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeOct 20th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI added to [[star-autonomous category]] a mention of "$\ast$-autonomous functors".

   I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 20th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI have a question about the definitions of [[star-autonomous category]]. The proof given there shows that the two definitions are equivalent as *properties* of a symmetric monoidal category. (Actually, it doesn't quite finish the proof by showing that $d_A$ is an isomorphism for $\bot\coloneqq I^*$ in the second definition, but I can believe that that works.) However, are they equivalent as *structure*? Or, [[stuff, structure, property|technically, I suppose]], as *stuff*? The first definition equips a symmetric monoidal category with the stuff of an object $\bot$, the (property-like) structure of being closed, and the property of the double-dualization map $d_A$ being an isomorphism. The second definition equips it with the stuff of a functor $(-)^*$, the structure of a natural isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$, and... no properties. This immediately makes me suspicious: nearly always non-property-like structure has to be equipped with coherence properties to be "correct". And I can easily imagine coherence properties that the second definition could satisfy, like the equality of two isomorphisms $\mathcal{C}(A\otimes (B\otimes C),D^*) \cong \mathcal{C}(A,(B\otimes (C\otimes D))^*)$ --- and it seems likely that when the data is constructed from the first definition then it would satisfy these properties. Are these properties automatically *always* satisfied by the second definition? Or (as seems more likely to me) does passing back and forth from the second definition to the first and back again "coherentify" the isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$ into a possibly-different one that does satisfy the coherence axiom, analogous to replacing an equivalence by an adjoint equivalence?

   I have a question about the definitions of star-autonomous category. The proof given there shows that the two definitions are equivalent as properties of a symmetric monoidal category. (Actually, it doesn’t quite finish the proof by showing that $d_A$ is an isomorphism for $\bot\coloneqq I^*$ in the second definition, but I can believe that that works.) However, are they equivalent as structure? Or, technically, I suppose, as stuff?

   The first definition equips a symmetric monoidal category with the stuff of an object $\bot$, the (property-like) structure of being closed, and the property of the double-dualization map $d_A$ being an isomorphism. The second definition equips it with the stuff of a functor $(-)^*$, the structure of a natural isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$, and… no properties. This immediately makes me suspicious: nearly always non-property-like structure has to be equipped with coherence properties to be “correct”. And I can easily imagine coherence properties that the second definition could satisfy, like the equality of two isomorphisms $\mathcal{C}(A\otimes (B\otimes C),D^*) \cong \mathcal{C}(A,(B\otimes (C\otimes D))^*)$ — and it seems likely that when the data is constructed from the first definition then it would satisfy these properties. Are these properties automatically always satisfied by the second definition? Or (as seems more likely to me) does passing back and forth from the second definition to the first and back again “coherentify” the isomorphism $\mathcal{C}(A\otimes B,C^*) \cong \mathcal{C}(A,(B\otimes C)^*)$ into a possibly-different one that does satisfy the coherence axiom, analogous to replacing an equivalence by an adjoint equivalence?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 21st 2017
   • (edited Oct 21st 2017)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexMike, I suspect I was the one who introduced the second definition (I haven't gone back through the history to check). I'm not sure that I have much time to investigate this, but if the me of yesteryear wanted to argue back, I might have said this: are you sure this is non-property-like structure? You probably recognize that the second definition is trying to define closed structure (where to make that clearer it should probably be written instead as $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$), and closed structure is property-like, right? So at the moment I'm a little puzzled where the problem might lie.

   Mike, I suspect I was the one who introduced the second definition (I haven’t gone back through the history to check). I’m not sure that I have much time to investigate this, but if the me of yesteryear wanted to argue back, I might have said this: are you sure this is non-property-like structure?

   You probably recognize that the second definition is trying to define closed structure (where to make that clearer it should probably be written instead as $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$), and closed structure is property-like, right? So at the moment I’m a little puzzled where the problem might lie.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 21st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIt's property-like to assert the existence of some new object together with a universal property for it. But it's not generally property-like to assert that some object that's *already* present in the doctrine has some *new* universal property. Put differently, that isomorphism could equivalently be phrased as saying (1) $\mathcal{C}$ is closed, with internal-homs $[B,C]$, and (2) we have a specified natural isomorphism $[B,C] \cong (B\otimes C^*)^*$. Part (1) is property-like, but part (2) is not.

   It’s property-like to assert the existence of some new object together with a universal property for it. But it’s not generally property-like to assert that some object that’s already present in the doctrine has some new universal property. Put differently, that isomorphism could equivalently be phrased as saying (1) $\mathcal{C}$ is closed, with internal-homs $[B,C]$, and (2) we have a specified natural isomorphism $[B,C] \cong (B\otimes C^*)^*$. Part (1) is property-like, but part (2) is not.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeOct 21st 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexOn the other hand, if the isomorphism $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ were asserting a universal property for the operation $(-)^*$, then together the two of them would be property-like. But it isn't quite doing that; it's saying that something constructed in a more complicated way using $(-)^*$ and $\otimes$ has a universal property.

   On the other hand, if the isomorphism $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ were asserting a universal property for the operation $(-)^*$, then together the two of them would be property-like. But it isn’t quite doing that; it’s saying that something constructed in a more complicated way using $(-)^*$ and $\otimes$ has a universal property.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 22nd 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOkay, thanks. At this point I don't know how to amend definition 2 in a way that doesn't make it tautologously the same as definition 1. Rats. We can scrap it if you don't see a fix-up either.

   Okay, thanks. At this point I don’t know how to amend definition 2 in a way that doesn’t make it tautologously the same as definition 1. Rats. We can scrap it if you don’t see a fix-up either.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 22nd 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexHere's a specific example to consider: the category $FDVect$ of finite-dimensional vector spaces over a field $k$, which is compact closed and hence star-autonomous. Then I believe the "obvious" definition-2 structure $\tau : \mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ can be "twisted" by multiplying by any nonzero element $t\in k$ to obtain another such structure $\hat{\tau}$. An isomorphism between this structure and the original one would consist of natural isomorphisms $\rho_A : A^\ast \cong A^\ast$, which I think would have to be multiplication by some nonzero $r\in k$, such that the square $$ \array{ \mathcal{C}(A \otimes B, C^\ast) & \xrightarrow{\tau} & \mathcal{C}(A, (B \otimes C)^\ast)\\ ^{\rho_{C}}\downarrow && \downarrow^{\rho_{B\otimes C}} \\ \mathcal{C}(A \otimes B, C^\ast) & \xrightarrow{\hat{\tau}} & \mathcal{C}(A, (B \otimes C)^\ast) }$$ commutes. But this says $t r = r$, hence $t=1$. So the twisted version is not equivalent to the untwisted one. But it seems clear that even the twisted one will give rise to the standard form of definition-1, since the only non-property-like data there is the dualizing object $k$, and that when we come backwards to definition 2 we will get only the untwisted version.

   Here’s a specific example to consider: the category $FDVect$ of finite-dimensional vector spaces over a field $k$, which is compact closed and hence star-autonomous. Then I believe the “obvious” definition-2 structure $\tau : \mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ can be “twisted” by multiplying by any nonzero element $t\in k$ to obtain another such structure $\hat{\tau}$. An isomorphism between this structure and the original one would consist of natural isomorphisms $\rho_A : A^\ast \cong A^\ast$, which I think would have to be multiplication by some nonzero $r\in k$, such that the square

   $$\array{ \mathcal{C}(A \otimes B, C^\ast) & \xrightarrow{\tau} & \mathcal{C}(A, (B \otimes C)^\ast)\\ ^{\rho_{C}}\downarrow && \downarrow^{\rho_{B\otimes C}} \\ \mathcal{C}(A \otimes B, C^\ast) & \xrightarrow{\hat{\tau}} & \mathcal{C}(A, (B \otimes C)^\ast) }$$

   commutes. But this says $t r = r$, hence $t=1$. So the twisted version is not equivalent to the untwisted one. But it seems clear that even the twisted one will give rise to the standard form of definition-1, since the only non-property-like data there is the dualizing object $k$, and that when we come backwards to definition 2 we will get only the untwisted version.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeOct 22nd 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexMy proposal would be to add to definition 2 the coherence condition that the composites $$ \mathcal{C}(A\otimes (B\otimes C), D^\ast) \cong \mathcal{C}(A, ((B\otimes C) \otimes D)^\ast) \cong \mathcal{C}(A, (B\otimes (C \otimes D))^\ast) $$ and $$ \mathcal{C}(A\otimes (B\otimes C), D^\ast) \cong \mathcal{C}((A\otimes B) \otimes C, D^\ast) \cong \mathcal{C} (A\otimes B, (C\otimes D)^\ast) \cong \mathcal{C}(A, (B\otimes (C \otimes D))^\ast) $$ are equal. In the twisted example this would entail $t^2 = t^3$, hence $t=1$. In general it would require some work to show that this is sufficient, but I think it's likely to be true. At first I thought Definition 2 doesn't appear in the literature (although I don't have a copy of Barr's original monograph yet), but I just noticed that it is the definition given in section 5 of [Day-Street](https://arxiv.org/abs/math/0301209). There they say > For the reader interested in checking that our $\ast$-autonomous monoidal categories > agree with Michael Barr's $\ast$-autonomous categories, we recommend Definition 2.3 of > [Ba2] as the appropriate one for comparison. Reference [Ba2] is Barr's [non-symmetric star-autonomous categories](http://www.math.mcgill.ca/barr/papers/asymm.pdf), and his definition 2.3 ("Definition C") is similar to our Definition 2 but includes only the special case of the isomorphism where $A$ is the unit object. Is that enough to eliminate the ambiguity? (Although Barr's "Definition D" therein also makes me worried about coherence, since it also consists of an isomorphism satisfying no conditions.) Also, sections 8 and 10 of [Mellies](https://www.irif.fr/~mellies/tensorial-logic/9-dialogue-categories-and-frobenius-monoids.pdf) add to the abstract Day-Street definition an associativity pentagon that I think reduces in the concrete case of profunctors to my proposed coherence axiom above. This is some additional justification for it, although it still remains to be proven that it's strongly equivalent to Definition 1.

   My proposal would be to add to definition 2 the coherence condition that the composites

   $$\mathcal{C}(A\otimes (B\otimes C), D^\ast) \cong \mathcal{C}(A, ((B\otimes C) \otimes D)^\ast) \cong \mathcal{C}(A, (B\otimes (C \otimes D))^\ast)$$

   and

   $$\mathcal{C}(A\otimes (B\otimes C), D^\ast) \cong \mathcal{C}((A\otimes B) \otimes C, D^\ast) \cong \mathcal{C} (A\otimes B, (C\otimes D)^\ast) \cong \mathcal{C}(A, (B\otimes (C \otimes D))^\ast)$$

   are equal. In the twisted example this would entail $t^2 = t^3$, hence $t=1$. In general it would require some work to show that this is sufficient, but I think it’s likely to be true.

   At first I thought Definition 2 doesn’t appear in the literature (although I don’t have a copy of Barr’s original monograph yet), but I just noticed that it is the definition given in section 5 of Day-Street. There they say

   For the reader interested in checking that our $\ast$-autonomous monoidal categories agree with Michael Barr’s $\ast$-autonomous categories, we recommend Definition 2.3 of [Ba2] as the appropriate one for comparison.

   Reference [Ba2] is Barr’s non-symmetric star-autonomous categories, and his definition 2.3 (“Definition C”) is similar to our Definition 2 but includes only the special case of the isomorphism where $A$ is the unit object. Is that enough to eliminate the ambiguity? (Although Barr’s “Definition D” therein also makes me worried about coherence, since it also consists of an isomorphism satisfying no conditions.)

   Also, sections 8 and 10 of Mellies add to the abstract Day-Street definition an associativity pentagon that I think reduces in the concrete case of profunctors to my proposed coherence axiom above. This is some additional justification for it, although it still remains to be proven that it’s strongly equivalent to Definition 1.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 22nd 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRight, so the message of #7 is that the possible structures in that case are in bijection with automorphisms of the unit object $I$, and I think that holds generally. By setting $B = I$ in $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$, we have by Yoneda an assigned isomorphism $C \cong C^{\ast\ast}$. Then, putting $C = I^\ast$, we get the universal property of $(-)^\ast$ via $\mathcal{C}(A \otimes B, I^\ast) \cong \mathcal{C}(A, B^\ast)$, whereupon your comment #5 kicks in. So it all comes down to which natural isomorphism $C \cong C^{\ast\ast}$ we are dealing with. The set of such isomorphisms is a torsor over the group of automorphisms of the identity functor $C \mapsto C$; by an enriched Yoneda argument this boils down to the group of automorphisms of $I$. Your suggestion for the coherence condition in #8 nails down which automorphism of $I$ we are dealing with to the identity map, similar to the way in which the pentagon condition for monoidal categories forces the associator for the tensor product on vector spaces to be the standard one. In fact, your condition is a pentagon! And based on your description I'm pretty sure Barr's specialization in his definition C would nail it down as well, even though I haven't looked at it yet. There may be other simple ways of writing down an appropriate coherence condition.

   Right, so the message of #7 is that the possible structures in that case are in bijection with automorphisms of the unit object $I$, and I think that holds generally. By setting $B = I$ in $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$, we have by Yoneda an assigned isomorphism $C \cong C^{\ast\ast}$. Then, putting $C = I^\ast$, we get the universal property of $(-)^\ast$ via $\mathcal{C}(A \otimes B, I^\ast) \cong \mathcal{C}(A, B^\ast)$, whereupon your comment #5 kicks in. So it all comes down to which natural isomorphism $C \cong C^{\ast\ast}$ we are dealing with. The set of such isomorphisms is a torsor over the group of automorphisms of the identity functor $C \mapsto C$; by an enriched Yoneda argument this boils down to the group of automorphisms of $I$.

   Your suggestion for the coherence condition in #8 nails down which automorphism of $I$ we are dealing with to the identity map, similar to the way in which the pentagon condition for monoidal categories forces the associator for the tensor product on vector spaces to be the standard one. In fact, your condition is a pentagon! And based on your description I’m pretty sure Barr’s specialization in his definition C would nail it down as well, even though I haven’t looked at it yet. There may be other simple ways of writing down an appropriate coherence condition.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeOct 22nd 2017
    • (edited Oct 22nd 2017)
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAnother condition one could ask for is a "unit" condition that $\mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ becomes the identity when $B=I$. (I actually prefer $\mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ to $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ because the former applies verbatim in the non-symmetric case, whereas the latter has to be modified to $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes {}^\ast C)^\ast)$, where ${}^\ast(-)$ is the inverse of $(-)^\ast$.)

    Another condition one could ask for is a “unit” condition that $\mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ becomes the identity when $B=I$. (I actually prefer $\mathcal{C}(A \otimes B, C^\ast) \cong \mathcal{C}(A, (B \otimes C)^\ast)$ to $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ because the former applies verbatim in the non-symmetric case, whereas the latter has to be modified to $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes {}^\ast C)^\ast)$, where ${}^\ast(-)$ is the inverse of $(-)^\ast$.)

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeOct 22nd 2017
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexBy "automorphisms of the identity functor" do you mean $\mathcal{C}$-enriched automorphisms? Otherwise I don't see how the unit object enters. Is it obvious that $(-)^{\ast\ast}$ is $\mathcal{C}$-enriched and that only enriched automorphisms work?

    By “automorphisms of the identity functor” do you mean $\mathcal{C}$-enriched automorphisms? Otherwise I don’t see how the unit object enters. Is it obvious that $(-)^{\ast\ast}$ is $\mathcal{C}$-enriched and that only enriched automorphisms work?

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 22nd 2017
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexRe #11: yes, I mean enriched. Admittedly I didn't check carefully yet that $(-)^\ast$ is enriched, but let's see: a contravariant strength would take the form $(B \otimes A)^\ast \otimes B \to A^\ast$ which we can derive from the identity $(B \otimes A)^\ast \to (B \otimes A)^\ast$, so I think it would work out.

    Re #11: yes, I mean enriched. Admittedly I didn’t check carefully yet that $(-)^\ast$ is enriched, but let’s see: a contravariant strength would take the form $(B \otimes A)^\ast \otimes B \to A^\ast$ which we can derive from the identity $(B \otimes A)^\ast \to (B \otimes A)^\ast$, so I think it would work out.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeOct 22nd 2017
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexOh, silly me: of course $(-)^{\ast\ast}$ is $\mathcal{C}$-enriched, since it's isomorphic to the identity. But I still don't quite understand #9: I see that the isomorphism $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ determines an isomorphism from $(-)^{\ast\ast}$ to the identity, and also a universal property of $(-)^\ast$, as special cases, but how do these special cases in turn determine it?

    Oh, silly me: of course $(-)^{\ast\ast}$ is $\mathcal{C}$-enriched, since it’s isomorphic to the identity.

    But I still don’t quite understand #9: I see that the isomorphism $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ determines an isomorphism from $(-)^{\ast\ast}$ to the identity, and also a universal property of $(-)^\ast$, as special cases, but how do these special cases in turn determine it?

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeOct 25th 2017
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexOk, finally got my copy of Barr's original monograph "$\ast$-autonomous categories" from ILL. Firstly I was not quite right in #8 that our Definition 2 is that of Day-Street, since the former requires a priori only that $(-)^\ast$ is fully faithful, whereas Day and Street require it to be an equivalence (as does Definition C of Barr's non-symmetric paper). But that doesn't affect its coherence or lack thereof. Our Definition 2 actually does appear in Barr's monograph in paragraph (4.3). It's not his *definition* of $\ast$-autonomous category, but he proves that this data "suffices to determine" a $\ast$-autonomous category, which isn't quite the same as claiming that it's an equivalent *definition*; so that could be compatible with what we're seeing. However, his actual definition doesn't seem quite right to me either. He defines a $\ast$-autonomous category to be a closed symmetric monoidal category $\mathcal{C}$ together with a $\mathcal{C}$-enriched functor (which he calls a "closed functor" for some reason) $(-)^\ast : \mathcal{C}^{op}\to \mathcal{C}$ and a $\mathcal{C}$-enriched natural isomorphism $A \cong A^{\ast\ast}$. It seems like this would have the same problem of being a torsor over the center of $\mathcal{C}$.

    Ok, finally got my copy of Barr’s original monograph “$\ast$-autonomous categories” from ILL.

    Firstly I was not quite right in #8 that our Definition 2 is that of Day-Street, since the former requires a priori only that $(-)^\ast$ is fully faithful, whereas Day and Street require it to be an equivalence (as does Definition C of Barr’s non-symmetric paper). But that doesn’t affect its coherence or lack thereof.

    Our Definition 2 actually does appear in Barr’s monograph in paragraph (4.3). It’s not his definition of $\ast$-autonomous category, but he proves that this data “suffices to determine” a $\ast$-autonomous category, which isn’t quite the same as claiming that it’s an equivalent definition; so that could be compatible with what we’re seeing.

    However, his actual definition doesn’t seem quite right to me either. He defines a $\ast$-autonomous category to be a closed symmetric monoidal category $\mathcal{C}$ together with a $\mathcal{C}$-enriched functor (which he calls a “closed functor” for some reason) $(-)^\ast : \mathcal{C}^{op}\to \mathcal{C}$ and a $\mathcal{C}$-enriched natural isomorphism $A \cong A^{\ast\ast}$. It seems like this would have the same problem of being a torsor over the center of $\mathcal{C}$.

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAdded Hyland-Schalk reference <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/45">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/45">v45</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added Hyland-Schalk reference

    diff, v45, current

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 29th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThought I'd add Mike's explanation of "autonomous". <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/47">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/47">v47</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Thought I’d add Mike’s explanation of “autonomous”.

    diff, v47, current

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2019
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThe links aren't appearing for me at [[star-autonomous category]] in this sentence >Regarding the use of "autonomous", this was once used as a bare adjective to describe a [[closed monoidal category]], or sometimes a [[compact closed monoidal category]],... Anyone else?

    The links aren’t appearing for me at star-autonomous category in this sentence

    Regarding the use of “autonomous”, this was once used as a bare adjective to describe a closed monoidal category, or sometimes a compact closed monoidal category,…

    Anyone else?

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeJan 8th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexSame here. Perhaps related to the square brackets around that sentence?

    Same here. Perhaps related to the square brackets around that sentence?

19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2019
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThat must be it. Have shifted to round ones.

    That must be it. Have shifted to round ones.

20. • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeJan 8th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexStill a bug, though, even if we can work around it easily. (-:

    Still a bug, though, even if we can work around it easily. (-:

21. • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2019
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAdded a list of relations to other structures, including the fact that a traced $\ast$-autonomous category must be compact closed (Hajgató and Hasegawa). <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/52">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/52">v52</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added a list of relations to other structures, including the fact that a traced $\ast$-autonomous category must be compact closed (Hajgató and Hasegawa).

    diff, v52, current

22. • CommentRowNumber22.
    • CommentAuthornLab edit announcer
    • CommentTimeFeb 12th 2020
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexChange the link from Vect to FinVect category in examples Oleg Nizhnik <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/53">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/53">v53</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Change the link from Vect to FinVect category in examples

    Oleg Nizhnik

    diff, v53, current

23. • CommentRowNumber23.
    • CommentAuthorJ-B Vienney
    • CommentTimeJul 28th 2022
    • PermaLink
    Author: J-B Vienney
    Format: MarkdownItexAdded an example of a subcategory of $Chu(Vect,K)$ which is a *-autononomous category and the reference $On *-autonomous categories of topological vector spaces$ by Michael Barr which develops the idea. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/54">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/54">v54</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added an example of a subcategory of $Chu(Vect,K)$ which is a *-autononomous category and the reference $On *-autonomous categories of topological vector spaces$ by Michael Barr which develops the idea.

    diff, v54, current

24. • CommentRowNumber24.
    • CommentAuthorJ-B Vienney
    • CommentTimeAug 1st 2022
    • PermaLink
    Author: J-B Vienney
    Format: MarkdownItexSuggest a new notion of weak $*$-autonomous category: « We can weaken the requirement that the canonical morphism $d_A: A \to (A \multimap \bot) \multimap \bot ,$ is an isomorphism to the one that it is a monomorphism. » $Vec_{\mathbb{K}}$ is a weak $*$-autonomous category. It allows to speak about duality in more general situations. I would be glad to have returns on this. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/55">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/55">v55</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Suggest a new notion of weak $*$-autonomous category: « We can weaken the requirement that the canonical morphism $d_A: A \to (A \multimap \bot) \multimap \bot ,$ is an isomorphism to the one that it is a monomorphism. »

    $Vec_{\mathbb{K}}$ is a weak $*$-autonomous category. It allows to speak about duality in more general situations.

    I would be glad to have returns on this.

    diff, v55, current

25. • CommentRowNumber25.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 15th 2022
    • PermaLink
    Author: J-B Vienney
    Format: MarkdownItexAdded definition of the internal hom from $\parr$. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/57">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/57">v57</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added definition of the internal hom from $\parr$.

    diff, v57, current

26. • CommentRowNumber26.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 2nd 2023
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexOne of the examples is > Suppose $\langle C,\otimes, I,\multimap\rangle$ is a closed symmetric monoidal category equipped with a "pre-dualizing object" $\bot$, in the sense that the contravariant self-adjunction $(-\multimap\bot) \dashv (-\multimap\bot)$ is [[idempotent adjunction|idempotent]], i.e. the double-dualization map $A \to (A \multimap \bot) \multimap \bot$ is an isomorphism whenever $A$ is of the form $B\multimap\bot$. (Note that idempotence is automatic if $C$ is a poset.) Then the category of [[fixed point of an adjunction|fixed points]] of this adjunction, i.e. the full subcategory of objects of the form $B\multimap\bot$, is $*$-autonomous. For it is closed under $\multimap$, as $(A\multimap (B\multimap \bot)) \cong (A\otimes B\multimap \bot)$, and reflective with reflector $(-\multimap\bot)\multimap\bot$, and it contains $\bot$ since $\bot\cong (I\multimap \bot)$. Hence it is closed symmetric monoidal with tensor product $((A\otimes B)\multimap\bot)\multimap\bot$, and all its double-dualization maps are isomorphisms by assumption. A historically important example is Girard's [[phase semantics]] of [[linear logic]]. Note that this category is a full subcategory of $Chu(C,\bot)$ closed under duality --- indeed, it is the intersection of the two embeddings of $C$ and $C^{op}$ therein --- but its tensor product is *not* the restriction of the tensor product of $Chu(C,\bot)$. Is there a citeable reference where this example is presented? Who should be credited for this observation?

    One of the examples is

    Suppose $\langle C,\otimes, I,\multimap\rangle$ is a closed symmetric monoidal category equipped with a “pre-dualizing object” $\bot$, in the sense that the contravariant self-adjunction $(-\multimap\bot) \dashv (-\multimap\bot)$ is idempotent, i.e. the double-dualization map $A \to (A \multimap \bot) \multimap \bot$ is an isomorphism whenever $A$ is of the form $B\multimap\bot$. (Note that idempotence is automatic if $C$ is a poset.) Then the category of fixed points of this adjunction, i.e. the full subcategory of objects of the form $B\multimap\bot$, is $*$-autonomous. For it is closed under $\multimap$, as $(A\multimap (B\multimap \bot)) \cong (A\otimes B\multimap \bot)$, and reflective with reflector $(-\multimap\bot)\multimap\bot$, and it contains $\bot$ since $\bot\cong (I\multimap \bot)$. Hence it is closed symmetric monoidal with tensor product $((A\otimes B)\multimap\bot)\multimap\bot$, and all its double-dualization maps are isomorphisms by assumption. A historically important example is Girard’s phase semantics of linear logic. Note that this category is a full subcategory of $Chu(C,\bot)$ closed under duality — indeed, it is the intersection of the two embeddings of $C$ and $C^{op}$ therein — but its tensor product is not the restriction of the tensor product of $Chu(C,\bot)$.

    Is there a citeable reference where this example is presented? Who should be credited for this observation?

27. • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2023
    • (edited Feb 2nd 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI don't know off the top of my head. But that paragraph was added in [revision 39](https://ncatlab.org/nlab/revision/diff/star-autonomous+category/39) by Mike Shulman.

    I don’t know off the top of my head. But that paragraph was added in revision 39 by Mike Shulman.

28. • CommentRowNumber28.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 5th 2023
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexOkay, so is this observation due to Mike Shulman? Should I cite this nLab entry as the original reference on “predualizing objects”?

    Okay, so is this observation due to Mike Shulman? Should I cite this nLab entry as the original reference on “predualizing objects”?

29. • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2023
    • (edited Mar 5th 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexIncidentally, Mike has a corresponding MathOverflow question/comment at [MO:343167](https://mathoverflow.net/q/343167).

    Incidentally, Mike has a corresponding MathOverflow question/comment at MO:343167.

30. • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeMar 7th 2023
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI think the generalization from posets (as in Girard's phase semantics) to more general idempotent adjunctions was original. I haven't written it or seen it anywhere else, so yes, I guess you could cite the nLab page.

    I think the generalization from posets (as in Girard’s phase semantics) to more general idempotent adjunctions was original. I haven’t written it or seen it anywhere else, so yes, I guess you could cite the nLab page.

31. • CommentRowNumber31.
    • CommentAuthorJohn Baez
    • CommentTimeMay 4th 2023
    • PermaLink
    Author: John Baez
    Format: MarkdownItexAdded a reference to Dold and Puppe's work which implies that a $\ast$-autonomous category with a natural isomorphism $(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$ is a [[compact closed category]]. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/61">v61</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added a reference to Dold and Puppe’s work which implies that a $\ast$-autonomous category with a natural isomorphism $(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$ is a compact closed category.

    diff, v61, current

32. • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2023
    • (edited May 5th 2023)
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded the publication data: * [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics **154** (1984) 85–103 &lbrack;[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/doldpup2.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/63">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/63">v63</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    added the publication data:

    • Albrecht Dold, Dieter Puppe, Duality, Trace and Transfer, Proceedings of the Steklov Institute of Mathematics 154 (1984) 85–103 [mathnet:tm2435, pdf, pdf]

    diff, v63, current

33. • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2024
    • PermaLink
    Author: Urs
    Format: MarkdownItexAdded pointer to: * [[Jürgen Fuchs]], [[Gregor Schaumann]], [[Christoph Schweigert]], [[Simon Wood]], *Grothendieck-Verdier duality in categories of bimodules and weak module functors* &lbrack;[arXiv:2306.17668](https://arxiv.org/abs/2306.17668)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/69">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/69">v69</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added pointer to:

    • Jürgen Fuchs, Gregor Schaumann, Christoph Schweigert, Simon Wood, Grothendieck-Verdier duality in categories of bimodules and weak module functors [arXiv:2306.17668]

    diff, v69, current

34. • CommentRowNumber34.
    • CommentAuthorvarkor
    • CommentTimeApr 20th 2024
    • PermaLink
    Author: varkor
    Format: MarkdownItexAdded the example of $V \times V^{op}$. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/70">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/70">v70</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    Added the example of $V \times V^{op}$.

    diff, v70, current

35. • CommentRowNumber35.
    • CommentAuthorMike Shulman
    • CommentTimeApr 24th 2024
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItex$V\times V^{op}$ is a special case of the Chu construction. <a href="https://ncatlab.org/nlab/revision/diff/star-autonomous+category/71">diff</a>, <a href="https://ncatlab.org/nlab/revision/star-autonomous+category/71">v71</a>, <a href="https://ncatlab.org/nlab/show/star-autonomous+category">current</a>

    $V\times V^{op}$ is a special case of the Chu construction.

    diff, v71, current