Why are pushouts the right tool in these setups

Question

$\newcommand{\cat}[1]{\mathcal{#1}}$ $\newcommand{\cod}{\operatorname{cod}}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\colim}{colim}$

The question is about two pushout constructions G.M. Kelly is using in his paper A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on (§14.1) to construct colimts and adjoints in a comma category setting. Sadly, he neither gives a motivation for these constructions, nor does he prove they do what they are supposed to do. And I can't figure out why pushouts are the right tool in this context.

I don't know weather there is a common answer for both constructions, but they appear side by side in a similar setting, so I figured I might as well put them into the same question.

The first one is

Given a category $\cat{C}$ and an endofunctor $T\colon\cat{C}\to\cat{C}$, a diagram $D\colon I\to T\downarrow\cat{C}$ is given by two functors $X,Y\colon I\to\cat{C}$, and a natural transformation $\kappa\colon T\circ X\Rightarrow Y$. Then the colimit $\colim_I D$ is given by $(\colim X,f,x)$, where $f$ and $x$ are given by the pushout $$ \require{AMScd} \begin{CD} \colim T\circ X@>{\colim\kappa}>> \colim Y\\ @V{\tilde T}VV @VhVV \\ T\colim X@>{f}>> x \end{CD} $$ where $\tilde T$ is the canonical comparison map.

whereas the second is

Given a category $\cat{C}$, two endofunctors $T,T'\colon\cat{C}\to\cat{C}$ and a natural transformation $\alpha\colon T'\to T$, we obtain a functor $\alpha^*\colon T\downarrow\cat{C}\to T'\downarrow\cat{C}$, which sends $(x,f,y)$ to $(x,f\circ\alpha_x,y)$. This functor has a left adjoint $\alpha_*\colon T'\downarrow \cat{C}\to T\downarrow \cat{C}$ which sends $(x',f',y')$ to $(x',\bar f' ,\bar y' )$, where $\bar f'$ and $\bar y'$ are given by the pushout $$ \require{AMScd} \begin{CD} T'(x)@>{f'}>> y\\ @V{\alpha_x}VV @V\hat f' VV \\ T(x)@>{\bar f'}>> \bar y' \end{CD} $$

I'm not asking for a calculation why these constructions do what they suppose to do, but rather some kind of motivation/explanation, why the author considers using pushouts in the first place and why they are supposed to work in this setting.

[Edit] Ok, so the second diagram seems to use what the nlab calls a cobase change along every component of $\alpha$. The respective nlab article itself isn't very useful, though.

Answer 1

For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

Answer 2

The two questions are indeed related. I suspect Kelly did colimits first so he could use the Special Adjoint Functor Theorem to write down that formula for the left adjoint in your second box. A great reference to see more about this is Mac Lane's Categories for the Working Mathematician (see Theorem 2 on page 129 of the 2nd edition). It might also be helpful to check out the chapter on Kan Extensions where these formulas are written out fully, and to check out the chapter on monads. The nLab article on adjoint functor theorems is also not bad. By the way, in references I've read there is no difference between "cobase change" and "pushout" but then again I always work in a bicomplete setting.

Now let's think about your first box. This looks very much like the idea of an Internal Algebra Classifier that I learned about recently from Michael Batanin. These classifiers are used heavily in Batanin and Berger's paper Tame Polynomial Monads, where $T$ was a monad, e.g. the free monoid functor. The idea there is to replace free algebra extensions (i.e. certain pushouts in $T\downarrow C$) by Kan extensions that are easier to analyze. Something similar was done in the appendix of Schwede-Shipley Algebras and Modules in Monoidal Model Categories for the free monoid functor. They give tons of justification for why the pushout should be created as words taken from the upper right and lower left corners with relations in the upper left corner. This uses heavily the fact that $T$ is the free monoid functor, but the general method has been used by many authors since then for many different monads $T$.

The big picture idea is to break down pushouts in $T\downarrow C$ (i.e. after applying $T$) into pieces coming from $C$ and $T$. Often, the data comes packaged to you in terms of $C$ and $T$, and the ability to compute pushouts is something you must have before tackling what $T \downarrow C$ looks like. I'll bet Kelly was doing something similar to the idea of an internal algebra classifier with the pushout you wrote about, i.e. breaking down the pushout. Another good reference for this is Batanin's paper The Eckmann-Hilton argument and higher operads, though I would start with Schwede-Shipley to get the motivation and the general method of breaking down that pushout computation.

Answer 3

The introduction to Kelly's paper is quite well-written. He makes it clear that his goal is to tie together the disparate approaches which had appeared up to this point for the problem of computing colimits in the category $T$-alg where $T$ is an endofunctor (the nLab has a page on algebras over an endofunctor). As I mentioned in my previous answer, one approach is to use Freyd's adjoint functor theorem, but it's hard to extract a construction from this result in general. This is one easy way to see that when $T$ is a monad then $T$-alg is cocomplete (since the forgetful functor $U$ is so nice), and Kelly discusses this on page 2). Throughout this answer, I use $\mathcal{A}$ to denote the base category.

Suppose now that you want to make this story go through for an endofunctor which is not a monad. The endofunctor had better at least be pointed, i.e. come with a natural transformation $\eta:Id_{\mathcal{A}}$ to $T$, which is like the unit of a monad. Even better would be if the endofunctor also satisfied $\eta \circ T = T\circ \eta: T\to T^2$ which Kelly calls well-pointed (since this guarantees iterations of $T$ are well behaved, even if they are not associative). Kelly wants to compute colimits in $T$-alg by realizing $T$-alg as a subcategory of the comma category $T/\mathcal{A}$. This is a good idea, since colimits are well understood in comma categories and easy to compute from basic knowledge of $T$ and $\mathcal{A}$. Even better is if $T$-alg sits reflectively in $T/\mathcal{A}$ since this allows you to compute colimits in $T$-alg by applying the reflecting functor to the corresponding colimit in $T/\mathcal{A}$ (see nLab). He mentions in the introduction that he planned to use this reflectivity in a forthcoming paper, so that's why he works so hard to prove it.

It seems his goal in the section you cite is to prove that it's always enough to work in the well-pointed setting, since the pointed setting reduces to this case. To me, Proposition 14.1 is the most exciting punchline, since it reduces the question of computing general colimits in $T$-alg to a particular pushout (the colimit of which will be computed in $T/\mathcal{A}$). He introduces your first box (14.2) in order to demonstrate how to compute that colimit in $T/\mathcal{A}$. I know you said you didn't want a discussion of why that box was true, but I can't help point to this link where a ton of examples are worked out, including one about undercategories. As for your second box, this is going to be crucial when we try to reduce from pointed endofunctors to well-pointed endofunctors. In particular, Kelly wants to prove that free $T$-algebras exist (this is easy for monads but is a critical step in getting your hands on the category of $T$-algebras, as my previous answer mentioned). He uses that second box in the special case where $\alpha: 1_{\mathcal{A}} \to T$ is the pointing, and thereby gets a relationship between $T/\mathcal{A}$ and the arrow category $Arr({\mathcal{A})$. The goal is to figure out what the image of $T$-alg in $T/\mathcal{A}$ looks like, and on page 43 he uses it to characterize the image in terms of what happens in the arrow category. This is how he proves that any pointed endofunctor which is defined by one of these pushouts we care about is actually well-pointed, which is a great way to reduce the problem (i.e. to get the well-pointing for free).