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    • CommentRowNumber1.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 9th 2012

    In Cartan and Eilenberg there are several instances of squares which “anticommute”, that is, we have hf=kgh \circ f = - k \circ g instead of hf=kgh \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 C\mathbf{C} be a (small) category. We attach to every parallel pair of 1-morphisms f,g:XYf, g : X \to Y the set of all natural transformations α:id Cid C\alpha : id_\mathbf{C} \Rightarrow id_\mathbf{C} such that g=α Yfg = \alpha_Y \circ f. The vertical composition is obvious, and if we have another parallel pair h,k:YZh, k : Y \to Z and a 2-morphism β:hk\beta : h \Rightarrow k, the horizontal composition of α\alpha and β\beta is just βα\beta \circ \alpha, since kg=(β Zh)(α Yf)=(β Zα Z)(hf)k \circ g = (\beta_Z \circ h) \circ (\alpha_Y \circ f) = (\beta_Z \circ \alpha_Z) \circ (h \circ f), by naturality of α\alpha. This yields a (strict) 2-category structure on C\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 C=R-Mod\mathbf{C} = R\text{-Mod}, the set (class?) of natural transformations id Cid Cid_\mathbf{C} \Rightarrow id_\mathbf{C} include the scalar action of RR, 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?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2012

    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 GG acts on an object XX, there is a “category of elements”. For instance, if GG is a group acting on a set XX, the category of elements is the action groupoid XGX\sslash G, whose objects are the elements of XX and whose morphisms xyx\to y are elements of GG such that g(x)=yg(x)=y. If GG is a category and XX a presheaf of sets (or categories) on GG, 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 CC, it is of course acted on by the (strict) monoidal category Aut(C)Aut(C). Now the full subcategory of Aut(C)Aut(C) containing only the object Id CId_C inherits a monoidal structure. This smaller monoidal category could naturally be called BZ(C)B Z(C), since it is the delooping of the center of CC. Of course, BZ(C)B Z(C) inherits an action on CC, 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)Aut(C) on CC. In the latter category, I think the objects would still be those of CC, but a morphism from xx to yy would consist of an automorphism FF of CC together with a morphism xF(y)x\to F(y) in CC, and the 2-cells would be natural transformations FGF\to G making the evident triangle commute. Obviously if we only allow F=Id CF=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)Aut(C) on CC 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 CC is an (nn-)groupoid, we get the total space of the universal bundle with fiber CC (whose base is BAut(C)B Aut(C)).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 10th 2012

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

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