The n-Category Café

JS: We don’t care if the effect fails to determine the cause.

Here’s a couple of other things to think about:

One

Discussions of the 2-place relation “$x \mathop{determines} y$” are frequently confounded by several different senses of the word “determine”.

There are at least these two senses:

1. Formal, informational, or mathematical determination:
   • “two points determine a line”
2. Material, causal, or physical determination:
   • “push determines shove”

What exactly do we mean by “the” cause and “the” effect, anyway?

Do we mean a totality of events that we call “the cause” of a given effect?

Do we mean a totality of events that we call “the effect” of a given cause?

Two

In the propositions as types analogy, we have a suggestive relation between the function arrow “$\to$” and the implication arrow “$\Rightarrow$”.

But if we say that function arrows are arrows of time, do we really want to say that implication arrows are arrows of time, and in the same sense, too?

Perhaps the two halves of the post are not so much in tune. If we think of which of algebra and coalgebra more closely resembles the arrow of time, it’s the latter.

Maps $X \to F X$, for some endofunctor $F$, are like one step time evolution on a state space, as observed here, and discussed in the paper mention in the comment below that one. But then that’s due to the resemblance between functor applied to object and function applied to element.

A comonoid in $C$ is a monoid in $C^{op}$ is a monad in $B C^{op}$ is a monad in $(B C)^{co}$ is a comonad in $B C$.

That looks right to me.

If $Rel$ is the “time-symmetric” version of $Set$, then presumably Prof is the “time-symmetric” version of $Cat$?

Of course, the obvious kind of coalgebra in $Cat$ is basically that mentioned here:

• $X\to P(X)$, the presheaf category: endo-profunctor of $X$

We can get some more by composing this $P$ with some other endofunctors. For instance, if $T$ is the “free symmetric strict monoidal category” monad, then we have

• $X \to P(T(X))$: generalized structure type

If it’s worth anything, cofibrant replacement is a comonad, and a 2-comonad in appropriate contexts. For instance, there’s a 2-comonad $Q$ on the 2-category $MonCat_{str}$ of monoidal categories and strict monoidal functors whose co-Kleisli 2-category is the 2-category $MonCat$ of monoidal categories and weak monoidal functors. A $Q$-coalgebra (qua pointed endofunctor) is a “flexible” or “cofibrant” monoidal category, i.e. an $A$ for which every monoidal functor $A\to B$ is equivalent to a strict one.

Of course that’s not “honestly 2-categorical” since morally, $MonCat_{str}$ is only of technical interest (for instance, as a way to produce and study $MonCat$). But the lax and colax versions of this are still interesting even in the fully weak setting. That is, there’s a 2-comonad $Q_l$ on $MonCat$ whose co-Kleisli 2-category is the 2-category $MonCat_l$ of monoidal categories and lax monoidal functors. I haven’t thought much about the coalgebras for $Q_l$ in this case, but Steve Lack and I used the coalgebras for an analogous 2-comonad (on a 2-category of weights) to characterize the limits that lift to 2-categories of algebras and lax morphisms for a 2-monad.

Just thinking out loud (figuratively speaking). (-:

There is an obvious notion of time-symmetric category, or ‘undirected’ category, different from the dagger category concept. An undirected category is defined to have a collection of objects together with a collection of morphisms over every unordered pair of objects. Composition and identities can be taken to be defined in the same way as for ordinary categories.

Has anyone ever worked this out in detail? What about functors between undirected categories and adjointness?

The first problem arises with Rel: we get an undirected category only once we restrict to symmetric endorelations over every object by throwing out non-symmetric ones; the reason being that composition with a non-symmetric endorelation is not well-defined.