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Hi,
I am attempting to show that Fong’s Causal theories are internal categories in an endofunctor category.
In section 4.2 of his paper, we see an interpretation of causal theories on SET. In particular, he talks about projections and diagonal maps. From this, I am trying to construct a bimonad, though I really only need a monad/comonad that is exact. Here it is, on SET:
$(F, \mu, \nu, \eta, \phi)$The endofunctor is
$F : SET \rightarrow SET$and is diagonal, so
$F:x \rightarrow (x,x)$$X \in Obj(SET)$$X=(a,b,c)$$F:(a,b,c) \rightarrow ((a,a),(b,b),(c,c))$Now for the natural transformations:
$\eta : 1_{SET} \rightarrow F$Example (actually, I am not sure how this works so please suggest a proper solution),
$\eta : a \rightarrow (a)$ $\mu : F^2 \rightarrow F$Example,
$((a,a),(a,a)) \rightarrow (a,a)$ $\nu : F \rightarrow 1_{SET}$Example (again, I am not sure how this one works),
$\nu : (a) \rightarrow a$ $\phi : F \rightarrow F^2$Example,
$\phi : (a,a) \rightarrow ((a,a),(a,a))$In order to show that this monad is an internal category in an endofunctor category, we take a look at the definition of internal categories in a monoidal category. What we want to do is take an endofunctor category as a monoidal category where the monoidal product is just functor composition. The monads and comonads form the internal monoids and comonoids as per the definition in the link. My understanding is that if you have a monad that is also a comonad and the underlying functor is exact, then you should have an internal category in an endofunctor category.
So far, I have shown that the interpretatation in SET for Fong’s causal theories is both a monad and comonad. What I can’t show is that the underlying functor is exact. Can someone show that the underlying functor is exact?
There is a lot of reasoning in here that I can’t verify, so please post if you know if this is all correct or needs correction.
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