Not signed in

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

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 9th 2012
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexIn Cartan and Eilenberg there are several instances of squares which "anticommute", that is, we have $h \circ f = - k \circ g$ instead of $h \circ f = k \circ g$. I was wondering if we could make this into an instance of a square commuting "up to a specified 2-morphism" and it turned out the answer was yes. Let $\mathbf{C}$ be a (small) category. We attach to every parallel pair of 1-morphisms $f, g : X \to Y$ the set of all natural transformations $\alpha : id_\mathbf{C} \Rightarrow id_\mathbf{C}$ such that $g = \alpha_Y \circ f$. The vertical composition is obvious, and if we have another parallel pair $h, k : Y \to Z$ and a 2-morphism $\beta : h \Rightarrow k$, the horizontal composition of $\alpha$ and $\beta$ is just $\beta \circ \alpha$, since $k \circ g = (\beta_Z \circ h) \circ (\alpha_Y \circ f) = (\beta_Z \circ \alpha_Z) \circ (h \circ f)$, by naturality of $\alpha$. This yields a (strict) 2-category structure on $\mathbf{C}$. Note that we have to remember which natural transformation is needed to make the triangle commute in order to have a well-defined horizontal composition. In the specific case of $\mathbf{C} = R\text{-Mod}$, the set (class?) of natural transformations $id_\mathbf{C} \Rightarrow id_\mathbf{C}$ include the scalar action of $R$, so in particular the anticommutative squares of Cartan and Eilenberg can be regarded as a square commuting up to a 2-morphism. My question now: Is there a name for this construction?

   In Cartan and Eilenberg there are several instances of squares which “anticommute”, that is, we have instead of . I was wondering if we could make this into an instance of a square commuting “up to a specified 2-morphism” and it turned out the answer was yes.

   Let be a (small) category. We attach to every parallel pair of 1-morphisms the set of all natural transformations such that . The vertical composition is obvious, and if we have another parallel pair and a 2-morphism , the horizontal composition of and is just , since , by naturality of . This yields a (strict) 2-category structure on . Note that we have to remember which natural transformation is needed to make the triangle commute in order to have a well-defined horizontal composition.

   In the specific case of , the set (class?) of natural transformations include the scalar action of , so in particular the anticommutative squares of Cartan and Eilenberg can be regarded as a square commuting up to a 2-morphism.

   My question now: Is there a name for this construction?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 10th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThat's cute! I've never seen it before, but here's one way to describe it in more abstract language. Whenever a monoidal or categorical object $G$ acts on an object $X$, there is a "category of elements". For instance, if $G$ is a group acting on a set $X$, the category of elements is the [[action groupoid]] $X\sslash G$, whose objects are the elements of $X$ and whose morphisms $x\to y$ are elements of $G$ such that $g(x)=y$. If $G$ is a category and $X$ a presheaf of sets (or categories) on $G$, then you have the usual [[category of elements]], a.k.a. [[Grothendieck construction]]. I think your construction can be obtained as follows. Given a category $C$, it is of course acted on by the (strict) monoidal category $Aut(C)$. Now the full subcategory of $Aut(C)$ containing only the object $Id_C$ inherits a monoidal structure. This smaller monoidal category could naturally be called $B Z(C)$, since it is the [[delooping]] of the [center](http://nlab.mathforge.org/nlab/show/center#OfCategories) of $C$. Of course, $B Z(C)$ inherits an action on $C$, and I think your construction should be the category of elements of this action. This is, of course, a (non-full, but wide and locally full) sub-2-category of the category of elements of the action of all of $Aut(C)$ on $C$. In the latter category, I think the objects would still be those of $C$, but a morphism from $x$ to $y$ would consist of an automorphism $F$ of $C$ together with a morphism $x\to F(y)$ in $C$, and the 2-cells would be natural transformations $F\to G$ making the evident triangle commute. Obviously if we only allow $F=Id_C$ we recover your description above. I'm slightly surprised that I've never encountered the category of elements of the action of $Aut(C)$ on $C$ before; it seems like such a naturally "universal" construction. Unless I have encountered it and not realized it. Does this look familiar to anyone? I suppose when $C$ is an ($n$-)groupoid, we get the total space of the universal bundle with fiber $C$ (whose base is $B Aut(C)$).

   That’s cute! I’ve never seen it before, but here’s one way to describe it in more abstract language. Whenever a monoidal or categorical object acts on an object , there is a “category of elements”. For instance, if is a group acting on a set , the category of elements is the action groupoid , whose objects are the elements of and whose morphisms are elements of such that . If is a category and a presheaf of sets (or categories) on , then you have the usual category of elements, a.k.a. Grothendieck construction.

   I think your construction can be obtained as follows. Given a category , it is of course acted on by the (strict) monoidal category . Now the full subcategory of containing only the object inherits a monoidal structure. This smaller monoidal category could naturally be called , since it is the delooping of the center of . Of course, inherits an action on , and I think your construction should be the category of elements of this action.

   This is, of course, a (non-full, but wide and locally full) sub-2-category of the category of elements of the action of all of on . In the latter category, I think the objects would still be those of , but a morphism from to would consist of an automorphism of together with a morphism in , and the 2-cells would be natural transformations making the evident triangle commute. Obviously if we only allow we recover your description above.

   I’m slightly surprised that I’ve never encountered the category of elements of the action of on before; it seems like such a naturally “universal” construction. Unless I have encountered it and not realized it. Does this look familiar to anyone? I suppose when is an (-)groupoid, we get the total space of the universal bundle with fiber (whose base is ).

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 10th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexErr, where by $Aut(C)$ I meant $End(C)$, since your natural transformations aren't necessarily invertible.

   Err, where by I meant , since your natural transformations aren’t necessarily invertible.