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Stream: learning: questions

Topic: lax bicolimits

fosco (Sep 21 2020 at 17:03):

I am trying to work out the definition of lax bicolimit of a functor between bicategories, in the laxest possible sense.

First, let's fix notation:

1. A functor between bicategories $\cal A,B$ is a lax functor $F$, with structure maps $Ff\circ Fg \Rightarrow F(fg)$ and $1_{Fc}\Rightarrow F(1_c)$
2. Given two lax functors $F,G$ the category $\text{Bicat}({\cal I},{\cal A})(F,G)$ has objects the oplax natural transformations $\theta : F \Rightarrow G$ and morphisms the modifications between said oplax natural transformations. This means that there are squares

$\begin{array}{ccc} Fi &\to& Gi \\\downarrow & \Rightarrow &\downarrow \\Fj &\to& Gj\end{array}$

filled by 2-cells $\theta_f$ in $\cal A$, and subject to suitable coherence conditions (for example, the request that $\theta_{1_i} = 1_{\theta_i}$).

Now, a lax bicolimit for a lax functor $T : {\cal I} \to {\cal A}$ consists of an object $\text{lbcolim } T$ of $\cal A$, such that there is an equivalence of categories

${\cal A}(\text{lbcolim } T, A) \cong \text{Bicat}({\cal I}, {\cal A})(T, \Delta A)$

where $\Delta$ is the constant-at-$A$ functor.

Tracking the identity map of $\text{lbcolim } T$ one obtains an oplax transformation $t : T \Rightarrow \text{lbcolim } T$ with components $t_i : Ti \to \text{lbcolim } T$ 1-cells of $\cal A$ equipped with 2-cells $t_f : t_j\circ Tf \Rightarrow t_i$, one for every $f : i\to j$ in $\cal I$, filling the diagram

main.png

Now for the universal property of such guy: given another lax cocone $z : T \Rightarrow E$, with codomain another constant functor at $E\in\cal A$, there exists a unique $\bar z : \text{lbcolim } T \to E$ such that the diagram

main.png

"commutes"; this commutativity meaning that the whiskering of $\bar z$ with the transformation $t$, equals the transformation $z$: $\bar z * t_f = z_f$.

Now, "how much" this diagram is forced to commute, given all this? Tracking again the identity, and the identity of the identity, it seems to me that the equality $\bar z\circ t_i = z_i$ holds on the nose.

Is this correct? Or the definition can be made even more lax?

Also. This definition is very heavy. I am trying to learn this stuff since forever, and I still can't find to avoid forgetting pieces of coherence that has to be imposed. Is there a trick I can use to avoid forgetting something, or it's just plain patience and computational skills (in this latter case, I feel doomed)?