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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJan 29th 2010
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   Author: TobyBartels
   Format: MarkdownI redesigned [[cycle category]], as had been requested there for some time. I'm not sure if the discussion decided whether the first definition was even correct; that discussion is now towards the bottom of the page. I also incorporated material from the erstwhile separate [[category of cycles]].

   I redesigned cycle category, as had been requested there for some time. I'm not sure if the discussion decided whether the first definition was even correct; that discussion is now towards the bottom of the page. I also incorporated material from the erstwhile separate category of cycles.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 29th 2010
   • (edited Jan 29th 2010)
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   Author: zskoda
   Format: MarkdownI am still personally feeling against the name cycle category. One of Mike's counterarguments was that parallel term the category of simplices is used in another meaning (comma category Delta over X for fixed simplicial set X). But e.g. Goerss-Jardine's book calls "simplex category" of X what you call the category of simplices of X, and many others call (what nlab calls) simplex category, "category of combinatorial simplices" just for ephasis.

   I am still personally feeling against the name cycle category. One of Mike's counterarguments was that parallel term the category of simplices is used in another meaning (comma category Delta over X for fixed simplicial set X). But e.g. Goerss-Jardine's book calls "simplex category" of X what you call the category of simplices of X, and many others call (what nlab calls) simplex category, "category of combinatorial simplices" just for ephasis.

3. • CommentRowNumber3.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 12th 2017
   • (edited Jul 12th 2017)
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   Author: Peter Heinig
   Format: MarkdownItex[technical note: chose this thread since the [[cycle category]] appears to be an example of a category isomorphic to its dual] Is there already an "official' nLab term for _category isomorphic to its opposite category_? Do you agree that if not, we should create one (and an article for it)? Do you agree that, at the very minimum, one should add to [[cycle category]] that it is isomorphic to its dual? Do you know whether anyone anywhere has systematically distinguished (0) categories which are equivalent but not isomorphic to their duals from (1) categories which are isomorphic to their duals? (Or is the variant (0) moot for some reason?) Motivation is that currently I have a hard technical motivation to understand this better, and to have a large collection of examples available. (One has been mentioned above, another not-entirely-unimportant one is what is sometimes called the walking quiver, i.e., the four-morphism category with two parallel morphisms. I mean, that the latter is isomorphic to its dual is obvious, but _that_ it is can be seen to have some larger significance, which to get into here would lead too far afield)

   [technical note: chose this thread since the cycle category appears to be an example of a category isomorphic to its dual]

   Is there already an “official’ nLab term for category isomorphic to its opposite category?

   Do you agree that if not, we should create one (and an article for it)?

   Do you agree that, at the very minimum, one should add to cycle category that it is isomorphic to its dual?

   Do you know whether anyone anywhere has systematically distinguished

   (0) categories which are equivalent but not isomorphic to their duals

   from

   (1) categories which are isomorphic to their duals?

   (Or is the variant (0) moot for some reason?)

   Motivation is that currently I have a hard technical motivation to understand this better, and to have a large collection of examples available. (One has been mentioned above, another not-entirely-unimportant one is what is sometimes called the walking quiver, i.e., the four-morphism category with two parallel morphisms. I mean, that the latter is isomorphic to its dual is obvious, but that it is can be seen to have some larger significance, which to get into here would lead too far afield)

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 12th 2017
   • (edited Jul 12th 2017)
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   Author: Todd_Trimble
   Format: MarkdownItexSome off-the-cuff reactions: What would be the compelling point of imposing an isomorphism $F: C^{op} \to C$ rather than just having an equivalence? Of course such isomorphisms crop up; notably an equivalence is an isomorphism if the categories involved are skeletal. But I'm not sure that's really enough to make a special point of talking about the isomorphism version. Yes, I agree that it is good to point out that the cycle category is self-dual (which is how I would informally refer to the equivalence version of the notion), and we could even say isomorphic here since the cycle category as defined is skeletal. We don't seem to have an entry "self-dual category" although there are stronger variations on this theme we talk about. One interesting situation is where the equivalence comes from a [[dualizing object]], where we obtain a self-dual category in [this sense](https://mathoverflow.net/questions/92355/what-is-a-self-dual-category). An example there is [[Pontryagin duality]]. Another is [[dagger-category]] (which is an "evil" concept). Both of these are "involutive" dualities (in an appropriate sense). Then there are more specialized concepts such as those discussed at [[self-dual object]] and [[star-autonomous category]]. I don't know of discussions where people have distinguished, "systematically" or otherwise, (0) from (1) at this level of generality. I lean more toward the possibility that it is (1) that is "moot" (again, at this level of generality -- less moot perhaps in e.g. the more specialized situation of dagger-categories). My lone opinion is that it might be okay to create an entry "self-dual category" defined as a category that admits an *equivalence* $F: C^{op} \to C$ -- but I'd want also a fairly convincing example of such in the wild that is not of "involutive" type (it's easy of course to create artificial examples).

   Some off-the-cuff reactions:

   What would be the compelling point of imposing an isomorphism $F: C^{op} \to C$ rather than just having an equivalence?

   Of course such isomorphisms crop up; notably an equivalence is an isomorphism if the categories involved are skeletal. But I’m not sure that’s really enough to make a special point of talking about the isomorphism version.

   Yes, I agree that it is good to point out that the cycle category is self-dual (which is how I would informally refer to the equivalence version of the notion), and we could even say isomorphic here since the cycle category as defined is skeletal.

   We don’t seem to have an entry “self-dual category” although there are stronger variations on this theme we talk about. One interesting situation is where the equivalence comes from a dualizing object, where we obtain a self-dual category in this sense. An example there is Pontryagin duality. Another is dagger-category (which is an “evil” concept). Both of these are “involutive” dualities (in an appropriate sense). Then there are more specialized concepts such as those discussed at self-dual object and star-autonomous category.

   I don’t know of discussions where people have distinguished, “systematically” or otherwise, (0) from (1) at this level of generality. I lean more toward the possibility that it is (1) that is “moot” (again, at this level of generality – less moot perhaps in e.g. the more specialized situation of dagger-categories).

   My lone opinion is that it might be okay to create an entry “self-dual category” defined as a category that admits an equivalence $F: C^{op} \to C$ – but I’d want also a fairly convincing example of such in the wild that is not of “involutive” type (it’s easy of course to create artificial examples).

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJul 12th 2017
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   Author: Mike Shulman
   Format: MarkdownItexI don't know whether this counts as "in the wild", but suppose $D$ is an arbitrary category and consider the category $C$ of endofunctors $F_0:D\to D$ that are part of an infinite string of adjunctions $$ \cdots \dashv F_{-2} \dashv F_{-1} \dashv F_0 \dashv F_1 \dashv F_2 \dashv \cdots $$ Since taking mates under adjunctions reverses the direction of transformations, the assignation $F_0 \mapsto F_1$ is a contravariant functor $C^{op}\to C$, which has as an inverse equivalence $F_0\mapsto F_{-1}$. But it is not involutive, and in general doesn't even have finite order.

   I don’t know whether this counts as “in the wild”, but suppose $D$ is an arbitrary category and consider the category $C$ of endofunctors $F_0:D\to D$ that are part of an infinite string of adjunctions

   $$\cdots \dashv F_{-2} \dashv F_{-1} \dashv F_0 \dashv F_1 \dashv F_2 \dashv \cdots$$

   Since taking mates under adjunctions reverses the direction of transformations, the assignation $F_0 \mapsto F_1$ is a contravariant functor $C^{op}\to C$, which has as an inverse equivalence $F_0\mapsto F_{-1}$. But it is not involutive, and in general doesn’t even have finite order.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 12th 2017
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   Author: Mike Shulman
   Format: MarkdownItexI would say that (1) is of interest for [[strict categories]], which include the cycle category. I would say it's also of interest if the auto-isomorphism is the identity on objects, since an isomorphism of this sort can be formulated "non-evilly" in the dependently typed language of categories without referring to equality of objects (it consists of functions $C(x,y) \to C(y,x)$ preserving composition) --- when we add involutivity to this we obtain exactly the notion of dagger-category. (I don't know of any identity-on-objects contravariant automorphisms that are not involutive, but I haven't really tried to find any.) So I support the idea of making a page [[self-dual category]] discussing both notions, these restricted situations under which (1) is "sensible", and the fact that most naturally occurring examples seem to be involutive and what that means, as well as some obvious properties that don't depend on involutivity such as "if a self-dual category is complete then it is cocomplete".

   I would say that (1) is of interest for strict categories, which include the cycle category. I would say it’s also of interest if the auto-isomorphism is the identity on objects, since an isomorphism of this sort can be formulated “non-evilly” in the dependently typed language of categories without referring to equality of objects (it consists of functions $C(x,y) \to C(y,x)$ preserving composition) — when we add involutivity to this we obtain exactly the notion of dagger-category. (I don’t know of any identity-on-objects contravariant automorphisms that are not involutive, but I haven’t really tried to find any.)

   So I support the idea of making a page self-dual category discussing both notions, these restricted situations under which (1) is “sensible”, and the fact that most naturally occurring examples seem to be involutive and what that means, as well as some obvious properties that don’t depend on involutivity such as “if a self-dual category is complete then it is cocomplete”.

7. • CommentRowNumber7.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 13th 2017
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   Author: Peter Heinig
   Format: MarkdownItexOne off-the-cuff reaction to: > I would say it’s also of interest if the auto-isomorphism is the identity on objects, since an isomorphism of this sort can be formulated “non-evilly” in the dependently typed language of categories without referring to equality of objects and to > when we add involutivity to this we obtain exactly the notion of dagger-category. This immediately reminds me of the fact that in (from what I know of) finite-digraph theory, digraphs $D$ isomorphic to the graph obtained by reversing each arc, are mostly studied with the _additional_ stronger requirement that there be at least one isomorphism $\varphi$ witnessing this which _is an involution and does not fix any vertex_ of the graph. Such $D$ are called _skew-symmetric_. I have never asked myself why this much stronger notion of "digraph isomorphic to its own reverse graph" is (apparently) the more important. I will look into that matter. In a sense, this is an example of the "an A is more than just a B, it is a B together with a specified C" pattern _within combinatorics_ (this pattern abounds in category theory, an example being "an _adjunction_ is more than just a _pair of adjoint functors_, it is a _pair of adjoint functors_ together with a specified _unit and counit_). Here we have "a _skew-symmetric digraph_ is more than just a _digraph isomorphic to its reversal_, it is a _digraph isomorphic to its reversal_ together with a specified _fixed-point-free involution_ to witness the isomorphism. Maybe we are onto something real here.

   One off-the-cuff reaction to:

   I would say it’s also of interest if the auto-isomorphism is the identity on objects, since an isomorphism of this sort can be formulated “non-evilly” in the dependently typed language of categories without referring to equality of objects

   and to

   when we add involutivity to this we obtain exactly the notion of dagger-category.

   This immediately reminds me of the fact that in (from what I know of) finite-digraph theory, digraphs $D$ isomorphic to the graph obtained by reversing each arc, are mostly studied with the additional stronger requirement that there be at least one isomorphism $\varphi$ witnessing this which is an involution and does not fix any vertex of the graph. Such $D$ are called skew-symmetric.

   I have never asked myself why this much stronger notion of “digraph isomorphic to its own reverse graph” is (apparently) the more important. I will look into that matter. In a sense, this is an example of the “an A is more than just a B, it is a B together with a specified C” pattern within combinatorics (this pattern abounds in category theory, an example being “an adjunction is more than just a pair of adjoint functors, it is a pair of adjoint functors together with a specified unit and counit). Here we have “a skew-symmetric digraph is more than just a digraph isomorphic to its reversal, it is a digraph isomorphic to its reversal together with a specified fixed-point-free involution to witness the isomorphism. Maybe we are onto something real here.