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In 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?
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 $G$ 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 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 $B Aut(C)$).
Err, where by $Aut(C)$ I meant $End(C)$, since your natural transformations aren’t necessarily invertible.
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