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I just had my first magical category theory experience. A couple of pages of calculations and diagrams suddenly collapsed into one simple category theoretic statement. I hope that means I’m doing something right :)
I was trying to answer the question I asked on MO,
2-groups are to crossed modules as strict 2-categories are to … ?
After some help from Toby forcing me to think more clearly about this, and some inspirational examples from Urs, I think I’ve come up with an answer. It is not yet fully baked, but it has been in the oven long enough I thought I’d ask for some feedback.
Here are some of my notes:
Here’s the definition:
Definition
A crossed category $(C,D,s,t,\eta)$ consists of two categories $C$, $D$, two functors $s,t:D\to C$, and a natural transformation $\eta:s\Rightarrow t$.
With only a slight abuse of notation, we can identify the components of $\eta$ with their respective objects thus identifying objects of $D$ with morphisms of $C$. This is justified since it amounts to writing a morphism $f\in C$ as
$\eta_f = f:s(f)\to t(f),$where we think of the functor $s$ as the source of $f$ and $t$ as the target of $f$.
Given a morphism $f\to g$ in $D$, it follows from naturality that
$t(f\to g)\circ f = g\circ s(f\to g).$
This marks an improvement over my boundary idea.
From here, I hope to demonstrate that when $C$ and $D$ are 1-object groupoids, then the crossed category $(\mathbf{B}G,\mathbf{B}H,s,t,\eta)$ is equivalent to a crossed module $(G,H,t,\alpha)$.
Comments, questions, pointers to references, suggestions for better names, etc are all more than welcome.
First question: what do the functors $s$ and $t$ do on objects? If they are the identity then you have a 2-category (I think). If not something more interesting.
I have been playing with a related thing where everything is looked at over a fixed category. I will chat about that some time as it is quite neat.
Usually there is an action in a crossed thingie? Is there one hiddne here e.g. by whiskering or something?
Eric, I am not sure what you are after.
The “inspirational” examples that I vaguely mentioned are different, they consist of an algebra $A$ and a group $G$ and a map from $A$ to $G$ and an action of $G$ on $A$ that make all this similar to a crossed module $H \to G$ of groups.
There is a notion of crossed module of groupoids that are 2-groups: this is called a crossed square. Maybe that’s the kind of thing you want.
@Tim:
First question: what do the functors $s$ and $t$ do on objects? If they are the identity then you have a 2-category (I think). If not something more interesting.
If $s(f) = s(g)$ and $t(f) = t(g)$ we’ve got a bigon and I also suspect this is equivalent to the standard definition of a strict 2-category, but I didn’t see any reason to enforce that, so I left it free. I’m not sure if we need to restrict $s(f\to g) = 1_{s(f)}$ and $t(f\to g) = 1_{t(f)}$ though. I prefer just to let things be free like this and I think it probably still deserves the right to be called a strict 2-category (or crossed category), but maybe based on just a slightly more wiggly shape than a bigon.
@Urs
The “inspirational” examples that I vaguely mentioned are different, they consist of an algebra $A$ and a group $G$ and a map from $A$ to $G$ and an action of $G$ on $A$ that make all this similar to a crossed module $H \to G$ of groups.
The thing I found inspiring was that they were crossed module-like constructions, but the higher level was not invertible. This made me have some hope we could take it further so that even the lower level was not invertible, but we still have something crossed module-like.
When things are not invertible, all we need is two functors $s$ and $t$. When things are invertible, I’m hoping to show that $t$ and $\alpha$ of the crossed modules may be expressed in terms of $s$ and $t$ of the crossed category. I think that to map $(s,t)$ to $(t,\alpha)$ only requires the lower level to be invertible, which is consistent with your example.
Eric, I am not sure what you are after.
I’m ultimately after understanding higher transport on higher diamonds, which are “directed” and are definitely not groupoids. So I’m trying to “extrude” the stuff you did on functors and forms to directed spaces, which are not modeled on groupoids. But I’m not ready to talk about this yet…
@Tim again
Usually there is an action in a crossed thingie? Is there one hiddne here e.g. by whiskering or something?
I think it is hidden in there when, as I mention above, $C$ is a groupoid, but I’m not sure yet. I think the action can be expressed in terms of $s$ and $t$ when morphisms in $C$ are invertible. Still a work in progress though.
This made me have some hope we could take it further so that even the lower level was not invertible, but we still have something crossed module-like.
I find this a contradiction in terms. The “crossed” in “crossed-something” refers to the fact that we have an action of the groupoid of 1-morphisms on everything. That’s the crucial fact that distinguishes a crossed complex from a chain complex. The word “crossed” comes from “crossed product of groups”, one acting on the other.
In your definition nothing is acting. In fact, your definition looks a bit too empty to me to be anything.
I’m ultimately after understanding higher transport on higher diamonds, which are “directed” and are definitely not groupoids.
So then you want higher functors with values in higher categories. Dropping the invertibility from crossed complexes leads you to strict omega-categories. Maybe strict 2-categories to start with. This is what Tim already suggested above. And in fact, what your suggested definition is closest to being is the definition of a double category, one of the models for strict 2-categories.
Thanks Urs. The last line of my first comment was
Comments, questions, pointers to references, suggestions for better names, etc are all more than welcome.
I wasn’t sure if “crossed category” was the best choice and from your comment, I suppose it is not. Really, if this is just another way of looking at some strict 2-category, e.g. maybe a double category, I’m happy enough without introducing a new name. Like I said, this came about when a couple pages of calculations and diagrams suddenly collapsed to this one statement about the collection $(C,D,s,t,\eta)$. It is very simple and it felt like I was “following the dao” :)
When I get a chance and when I have the energy (I’ve been bedridden with acute bronchitis) the next thing I want to do is try to define an action $\alpha: G\to Aut(H)$ from $s$ and $t$ for cases where the lower level is invertible.
By the way, what completes this analogy: “A 2-group is to a crossed module as a 2-groupoid is to a … “?
Edit: Removed some distracting material.
By the way, what completes this analogy: “A 2-group is to a crossed module as a 2-groupoid is to a … “?
…a crossed module over a groupoid. an ordinary crossed module is over a one-object groupoid i.e. a group, so is a special case of the latter.
By the way, what completes this analogy: “A 2-group is to a crossed module as a 2-groupoid is to a … “?
.. crossed complex concentrated in degree 0, 1, and 2.
This is a diagram
$G_2 \stackrel{\delta_1}{\to} G_1 \stackrel{\to}{\to} G_0$equipped with certain action morphisms, such that $G_1 \stackrel{\to}{\to} G_0$ is a groupoid and such that $G_2 \to G_1$ behaves like a crossed module over each point of $G_0$. Details are at crossed complex.
Notice that crossed complexes in general are an equivalent way to talk about strict omega-groupoids. As Ronnie Brown likes to emphasize, these may be modeled by cubical $\infty$-groupoids. You can see this in the references in the entry on crossed complexes.
Ack. Bronchitis + asthma + moving flats = no fun.
What is worse is that my new flat is “closer” so I can walk to work = no more 30 minute relaxing train rides where I can read some maths (to keep me sane). So you can expect my journey to dementia to be accelerated.
Anyway, Toby has convinced me that my idea about boundaries isn’t completely correct, but the neat calculations tell me there is something interesting (if not new) to be said about it. Given a 2-morphism $\alpha:f\Rightarrow g$, the boundary was supposed to be an endomorphisms such that
$\partial\alpha\circ f = g.$Although not quite correct, I think the new improved version is closer to being correct. Instead of a single endomorphism, I needed two endomorphisms:
$t(\alpha)\circ f = g\circ s(\alpha).$I spent a few pages drawing diagrams expressing the kinds of relations I expected these to satisfy and noticed that everything “clicked” when I considered two categories $C$ and $D$ with 2 functors $s,t:D\to C$ and a natural transformation $\eta:s\to t$ between them.
The functor $e:C\to D$ is pretty neat too and helped me understand some things about 2-groups and crossed modules, e.g. why the identity for horizontal composition is the way it is.
Now I’m thinking what I have is a pair $(\mathcal{C},\eta:s\Rightarrow t)$, where $\mathcal{C}$ is a double category and I’m hoping to try to relate this to strict 2-categories.
Is this a reasonable thing to try to do?
Well, your original pair of categories with two functors is a category internal to (non-reflexive) graphs together with the natural transformation, the last of which doesn’t immediately look like something familiar to me. But with a double category together with the natural transformation, it looks almost like an edge-symmetric double category - but not quite. An edge symmetric double category is one where there you are given an isomorphism from the objects of the arrow category to the arrows of the object category, and this respects (some of?) the structure maps. This looks like a special case of what you have in your penultimate paragraph.
There are comparison functors between the 1-categories of 2-categories and double categories, one of which takes a double category and defines a 2-cat with the same objects, formal pairs $(h,v)$ of a horizontal and vertical arrow that ’match up’ at the target of the former and the source of the latter, and formal ’diagonal’ composites of squares as 2-cells (this construction is all due to Ehresmann). This construction is a functor and is adjoint to one of (or poss. both) the inclusions of 2Cat into DCat - or even the double category of 2-commuting squares. Can’t remember just now the details I’m afraid.
Now if you have edge-symmetric double groupoids, as studied somewhat by Ronnie Brown et al. as arising from homotopy double groupoids (vertical and horiz. arrows are just paths, squares are literally (homotopy classes of) squares $I^2 \to X$), there are some more interesting comparison functors, because then formal composites of vertical and horiz. arrows can be made into real composites using the isomorphism. But it would be interesting to how this could be relaxed, and what sort of structure one gets by replacing the vertical=horizontal isomorphism with your natural transformation…
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