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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeMay 27th 2014
   • PermaLink
   Author: Fosco
   Format: MarkdownItexI'm trying to expand a little bit on the page about [$\kappa$-ary factorization systems](http://ncatlab.org/nlab/show/k-ary+factorization+system). In particular I want to understand the "convergence condition" given by * each morphism $f\colon X \to Y$ is both the [[inverse limit]] $\underset{i \to \infty}\lim \im_i f$ in the [[slice category]] $C/Y$ and the [[direct limit]] $\underset{i \to -\infty}\colim \coim_i f$ in the [[coslice category]] $X/C$, and * for each $f\colon X \to Y$, $\id_Y$ is $\underset{i \to -\infty}\colim \im_i f$ and $\id_X$ is $\underset{i \to \infty}\lim \coim_i f$. I can understand that this is an abstraction of what happens in the finite case, where every $k$-ary factorization system (for $2\le k \,<\, \omega$) can be thought "completed" by $$ \big(\mathbb{F}_0=(Iso, All) \subset \! \big)\; \mathbb{F}_1 \subset\dots\subset \mathbb{F}_{k-1}\;\big(\!\subset (All, Iso) = \mathbb{F}_k\big) $$ Nevertheless it seems to me that the whole situation deserves a more subtle analysis. Let me switch to a less peculiar notation to denote the factorization of an arrow $f\colon X\to Y$ ($im_i f$ and $coim_i f$ are a bit misleading to my eye). I agree that a $\alpha$-stage FS in $\mathcal{C}$ should consist of a functor $\alpha\to FS(\mathcal{C})$ (where I'm regarding both the poset/ordinal $\alpha$ and $FS(\mathcal{C})$ as categories). This entails that for any $i\in\alpha$ there is a factorization $$\begin{array}{cccc} X \quad &\xrightarrow{\quad f\quad} & \quad Y\\ {}_{L_{i}(f)}\searrow && \nearrow_{R_{i}(f)} \\ & F_i (f) \end{array}$$ Now we would like to impose four "convergence conditions" on these factorizations, which play the role of $(1_X, f)\colon X = F_0(f) \to Y$ and $(f, 1_Y)\colon X\to F_k(f)= Y$: $$\begin{array}{c} \underset{i\in\alpha}{\lim}\; R_i(f) = \underset{i\in\alpha}{\lim}\; \left[\begin{smallmatrix} F_i(f) \\ \downarrow \\ Y \end{smallmatrix}\right] = \left[\begin{smallmatrix} X\\ \downarrow f\\ Y\end{smallmatrix}\right] ; \qquad \underset{i\in\alpha}{\colim} \;L_i(f) = \underset{i\in\alpha}{\colim}\; \left[\begin{smallmatrix} X \\ \downarrow \\ F_i(f) \end{smallmatrix}\right] = \left[\begin{smallmatrix} X\\ \downarrow f\\ Y\end{smallmatrix}\right] \\ \; \\ \underset{i\in\alpha}{\colim}\; R_i(f) = \underset{i\in\alpha}{\colim}\; \left[\begin{smallmatrix} F_i(f) \\ \downarrow \\ Y \end{smallmatrix}\right] = 1_Y ; \qquad \underset{i\in\alpha}{\lim}\; L_i(f) = \underset{i\in\alpha}{\lim}\; \left[\begin{smallmatrix} X \\ \downarrow \\ F_i(f) \end{smallmatrix}\right] = 1_X \end{array}$$ (all the limits and colimits are taken over the various slice/coslice categories). Now, an asimmetry arises: it's rather easy to compute colimits in a slice category [resp. limits in a coslice category], since [they are computed as colimits](http://ncatlab.org/nlab/show/overcategory#LimitsAndColimits) (resp., limits) [in the underlying category](http://ncatlab.org/nlab/show/overcategory#LimitsAndColimits): the natural forgetful functor $p\colon A/\mathcal{C}\to\mathcal{C}$ (resp., $q\colon \mathcal{C}/A\to\mathcal{C}$) reflects limits (resp., colimits). Conversely, limits in a slice category (resp., colimits in a coslice category) tend to be more complicated to compute. In fact, not *so* complicated: it can be shown that the above forgetful functors reflect also *connected* limits/colimits (depending on the side where you slice). My first question is: the nLan page is rather obscure in this part. Is *connectedness* the main reason why we choose ordinals as indices for $\alpha$-ary factorization systems? When the author says > Fix any ordinal number (or opposite thereof, or any poset, really) does (s?)he mean, instead, any *connected* poset, really? If not, arguing in general, I don't see how it is possible to resolve the "skewness" between the limit of a diagram $F$ computed in a slice category, and the limit of the functor projected to the base. But maybe it's my fault! Please, follow my computation...

   I’m trying to expand a little bit on the page about $\kappa$-ary factorization systems. In particular I want to understand the “convergence condition” given by

   • each morphism $flip_p(\rho) & flip_p(\rho) &f\colon X \to Y$ is both the inverse limit $\underset{i \to \infty}\lim \im_i f$ in the slice category $C/Y$ and the direct limit $\underset{i \to -\infty}\colim \coim_i f$ in the coslice category $X/C$, and
   • for each $f\colon X \to Y$, $\id_Y$ is $\underset{i \to -\infty}\colim \im_i f$ and $\id_X$ is $\underset{i \to \infty}\lim \coim_i f$.

   I can understand that this is an abstraction of what happens in the finite case, where every $k$-ary factorization system (for $2\le k \,<\, \omega$) can be thought “completed” by

   $$\big(\mathbb{F}_0=(Iso, All) \subset \! \big)\; \mathbb{F}_1 \subset\dots\subset \mathbb{F}_{k-1}\;\big(\!\subset (All, Iso) = \mathbb{F}_k\big)$$

   Nevertheless it seems to me that the whole situation deserves a more subtle analysis. Let me switch to a less peculiar notation to denote the factorization of an arrow $f\colon X\to Y$ ($im_i f$ and $coim_i f$ are a bit misleading to my eye).

   I agree that a $\alpha$-stage FS in $\mathcal{C}$ should consist of a functor $\alpha\to FS(\mathcal{C})$ (where I’m regarding both the poset/ordinal $\alpha$ and $FS(\mathcal{C})$ as categories). This entails that for any $i\in\alpha$ there is a factorization

   $$\begin{array}{cccc} X \quad &\xrightarrow{\quad f\quad} & \quad Y\\ {}_{L_{i}(f)}\searrow && \nearrow_{R_{i}(f)} \\ & F_i (f) \end{array}$$

   Now we would like to impose four “convergence conditions” on these factorizations, which play the role of $(1_X, f)\colon X = F_0(f) \to Y$ and $(f, 1_Y)\colon X\to F_k(f)= Y$:

   $$\begin{array}{c} \underset{i\in\alpha}{\lim}\; R_i(f) = \underset{i\in\alpha}{\lim}\; \left[\begin{smallmatrix} F_i(f) \\ \downarrow \\ Y \end{smallmatrix}\right] = \left[\begin{smallmatrix} X\\ \downarrow f\\ Y\end{smallmatrix}\right] ; \qquad \underset{i\in\alpha}{\colim} \;L_i(f) = \underset{i\in\alpha}{\colim}\; \left[\begin{smallmatrix} X \\ \downarrow \\ F_i(f) \end{smallmatrix}\right] = \left[\begin{smallmatrix} X\\ \downarrow f\\ Y\end{smallmatrix}\right] \\ \; \\ \underset{i\in\alpha}{\colim}\; R_i(f) = \underset{i\in\alpha}{\colim}\; \left[\begin{smallmatrix} F_i(f) \\ \downarrow \\ Y \end{smallmatrix}\right] = 1_Y ; \qquad \underset{i\in\alpha}{\lim}\; L_i(f) = \underset{i\in\alpha}{\lim}\; \left[\begin{smallmatrix} X \\ \downarrow \\ F_i(f) \end{smallmatrix}\right] = 1_X \end{array}$$

   (all the limits and colimits are taken over the various slice/coslice categories). Now, an asimmetry arises: it’s rather easy to compute colimits in a slice category [resp. limits in a coslice category], since they are computed as colimits (resp., limits) in the underlying category: the natural forgetful functor $p\colon A/\mathcal{C}\to\mathcal{C}$ (resp., $q\colon \mathcal{C}/A\to\mathcal{C}$) reflects limits (resp., colimits). Conversely, limits in a slice category (resp., colimits in a coslice category) tend to be more complicated to compute.

   In fact, not so complicated: it can be shown that the above forgetful functors reflect also connected limits/colimits (depending on the side where you slice).

   My first question is: the nLan page is rather obscure in this part. Is connectedness the main reason why we choose ordinals as indices for $\alpha$-ary factorization systems? When the author says

   Fix any ordinal number (or opposite thereof, or any poset, really)

   does (s?)he mean, instead, any connected poset, really? If not, arguing in general, I don’t see how it is possible to resolve the “skewness” between the limit of a diagram $F$ computed in a slice category, and the limit of the functor projected to the base. But maybe it’s my fault! Please, follow my computation…

2. • CommentRowNumber2.
   • CommentAuthorFosco
   • CommentTimeMay 27th 2014
   • PermaLink
   Author: Fosco
   Format: MarkdownItexIn general, it is a matter of verifying universal property the proof that $$ \underset{j\in\mathcal{J}}{colim}\; F\cong \underset{j\in\mathcal{J}^\lhd}{colim}\; (F^\lhd) $$ where $\mathcal{J}^\lhd$ is the category $\mathcal{C}$ which is [forced](http://ncatlab.org/nlab/show/over-%28infinity%2C1%29-category#definition) to have an (additional) initial object, and $F^\lhd$ is the extension of the functor $F\colon \mathcal {J}\to A/\mathcal{C}$ with value $A$ on the unique object of $[0]\subset \mathcal{J}^\lhd = [0]\star \mathcal{J}$. The RHS colimit inherits a canonical arrow $F[0]=A\to \underset{j\in\mathcal{J}^\lhd}{colim}\; (F^\lhd)$, which turns it into an object of $A/\mathcal{C}$. A completely dual result characterizes limits in a slice category: $$ \underset{j\in\mathcal{J}}{lim}\; F\cong \underset{j\in\mathcal{J}^\rhd}{lim}\; (F^\rhd) $$ Now, I would like to know if one can say something more in this general case. In the diagram $$ \begin{array}{ccc} \underset{i\in\alpha}{lim}\; F_i(f) &\overset{?}\dashrightarrow & \underset{i\in\alpha}{lim}\; R_i(f)\\ \uparrow && \downarrow\\ X &\xrightarrow{\qquad f\qquad} & Y \\ \downarrow && \uparrow\\ \underset{i\in\alpha}{colim}\; L_i(f) & \underset{?}\dashrightarrow & \underset{i\in\alpha}{colim}\; F_i(f) \end{array} $$ the limit arrows have no reason to be adjacent, and more in general it is not possible to compare ${colim}_{j^\lhd}\; (F^\lhd)$ and ${colim}_j\; p F$ in the right direction (but there is an arrow ${colim}_j\; p F\to {colim}_{j^\lhd}\; F^\lhd$ --dually, ${lim}_{j^\lhd}\; F^\lhd\to {lim}_j\; q F$--). In order to fix this, one should pray either for the existence of a (hopefully canonical) arrow $$ lim_j \; F_j(f) \xrightarrow{\qquad} colim_j \; F_j(f) $$ such that the composition $X \xrightarrow{ } lim_j \; F_j(f) \xrightarrow{\;} colim_j \; F_j(f) \xrightarrow{} Y$ equals the $f$ we started with, or for the existence of a similar comparison $colim_j\; L_j(f) \to lim_j\; R_j(f)$. Let's concentrate on the first case. How about the existence of such an arrow? Well, it relies on a rather peculiar property of the index category $\mathcal{J}$: * It doesn't exist in general, since for example if $\mathcal{J}=\varnothing$, $\mathcal{C}= Set$ this should entail a canonical arrow $1\to \varnothing$, which would entail that $Set$ has a zero object. * A sufficient condition for it to exist is (sort of) a connectedness request: > $(*)$ any two objects $i,j\in \mathcal{J}$ can be connected by an arrow, either $i\to j$ or $j\to i$ (which is stronger than $\mathcal{J}$ being connected), since if this happens we can define an arrow $lim_j \; A_j\to colim_j \; A_j$ as $lim_j \; A_j \to A_i \to colim_j \; A_j$ for some $i\in \mathcal{J}$: this choice doesn't affect the definition of $lim_j \; A_j\to colim_j \; A_j$ since the diagram $$ \begin{array}{ccc} lim_j \; A_j &\xrightarrow{\quad p_j\quad} & A_j\\ \downarrow &\;\;\;\;\;\;\;\;\;\;\;\underset{A(i\to j)}\nearrow & \downarrow\\ {}^{p_i}\downarrow &\nearrow& \downarrow^{h_j}\\ A_i &\xrightarrow{\quad h_i\quad} & colim_j \; A_j \end{array} $$ commutes for any two $i,j$. A total order is a typical (easy) example of a category satisfying this property ("every two objects can be compared"), so in the case of a totally ordered family of factorization systems $\alpha\to FS(\mathcal{C})$ I'm left with checking that the composition $$ X \xrightarrow{\quad} lim_j \; F_j(f)\to colim_j \; F_j(f) \xrightarrow{\quad}Y $$ coincides with $f$; fortunately this is a trivial exercise in pasting commutative triangles. This is where I stopped. Am I going in the right direction? Is condition $(*)$ also necessary, or rather we fall again in the realm of connected categories?

   In general, it is a matter of verifying universal property the proof that

   $$\underset{j\in\mathcal{J}}{colim}\; F\cong \underset{j\in\mathcal{J}^\lhd}{colim}\; (F^\lhd)$$

   where $\mathcal{J}^\lhd$ is the category $\mathcal{C}$ which is forced to have an (additional) initial object, and $F^\lhd$ is the extension of the functor $F\colon \mathcal {J}\to A/\mathcal{C}$ with value $A$ on the unique object of $[0]\subset \mathcal{J}^\lhd = [0]\star \mathcal{J}$. The RHS colimit inherits a canonical arrow $F[0]=A\to \underset{j\in\mathcal{J}^\lhd}{colim}\; (F^\lhd)$, which turns it into an object of $A/\mathcal{C}$.

   A completely dual result characterizes limits in a slice category:

   $$\underset{j\in\mathcal{J}}{lim}\; F\cong \underset{j\in\mathcal{J}^\rhd}{lim}\; (F^\rhd)$$

   Now, I would like to know if one can say something more in this general case. In the diagram

   $$\begin{array}{ccc} \underset{i\in\alpha}{lim}\; F_i(f) &\overset{?}\dashrightarrow & \underset{i\in\alpha}{lim}\; R_i(f)\\ \uparrow && \downarrow\\ X &\xrightarrow{\qquad f\qquad} & Y \\ \downarrow && \uparrow\\ \underset{i\in\alpha}{colim}\; L_i(f) & \underset{?}\dashrightarrow & \underset{i\in\alpha}{colim}\; F_i(f) \end{array}$$

   the limit arrows have no reason to be adjacent, and more in general it is not possible to compare ${colim}_{j^\lhd}\; (F^\lhd)$ and ${colim}_j\; p F$ in the right direction (but there is an arrow ${colim}_j\; p F\to {colim}_{j^\lhd}\; F^\lhd$ –dually, ${lim}_{j^\lhd}\; F^\lhd\to {lim}_j\; q F$–).

   In order to fix this, one should pray either for the existence of a (hopefully canonical) arrow

   $$lim_j \; F_j(f) \xrightarrow{\qquad} colim_j \; F_j(f)$$

   such that the composition $X \xrightarrow{ } lim_j \; F_j(f) \xrightarrow{\;} colim_j \; F_j(f) \xrightarrow{} Y$ equals the $f$ we started with, or for the existence of a similar comparison $colim_j\; L_j(f) \to lim_j\; R_j(f)$.

   Let’s concentrate on the first case. How about the existence of such an arrow? Well, it relies on a rather peculiar property of the index category $\mathcal{J}$:

   • It doesn’t exist in general, since for example if $\mathcal{J}=\varnothing$, $\mathcal{C}= Set$ this should entail a canonical arrow $1\to \varnothing$, which would entail that $Set$ has a zero object.
   • A sufficient condition for it to exist is (sort of) a connectedness request:

   $(*)$ any two objects $i,j\in \mathcal{J}$ can be connected by an arrow, either $i\to j$ or $j\to i$

   (which is stronger than $\mathcal{J}$ being connected), since if this happens we can define an arrow $lim_j \; A_j\to colim_j \; A_j$ as $lim_j \; A_j \to A_i \to colim_j \; A_j$ for some $i\in \mathcal{J}$: this choice doesn’t affect the definition of $lim_j \; A_j\to colim_j \; A_j$ since the diagram

   $$\begin{array}{ccc} lim_j \; A_j &\xrightarrow{\quad p_j\quad} & A_j\\ \downarrow &\;\;\;\;\;\;\;\;\;\;\;\underset{A(i\to j)}\nearrow & \downarrow\\ {}^{p_i}\downarrow &\nearrow& \downarrow^{h_j}\\ A_i &\xrightarrow{\quad h_i\quad} & colim_j \; A_j \end{array}$$

   commutes for any two $i,j$. A total order is a typical (easy) example of a category satisfying this property (“every two objects can be compared”), so in the case of a totally ordered family of factorization systems $\alpha\to FS(\mathcal{C})$ I’m left with checking that the composition

   $$X \xrightarrow{\quad} lim_j \; F_j(f)\to colim_j \; F_j(f) \xrightarrow{\quad}Y$$

   coincides with $f$; fortunately this is a trivial exercise in pasting commutative triangles.

   This is where I stopped. Am I going in the right direction? Is condition $(*)$ also necessary, or rather we fall again in the realm of connected categories?