category theory

# Contents

## Idea

In the context of factorization systems such as they appear notably in enriched model category one frequently needs to handle iterated lifting problems. In the appendix of (Joyal–Tierney, 06) a symbolic calculus is introduced to facilitate these computations.

A central point of it is to have the statement of prop. 6 below be easily expressible in terms of “division on both sides”-operations.

## The calculus

### Lifting

Let $ℰ$ be a category (locally small).

###### Notation

For $f,g\in \mathrm{Mor}\left(ℰ\right)$, write

$f⋔g$f \pitchfork g

if $f$ has the left lifting property against $g$, or equivalently if $g$ has the right lifting property against $f$.

For $S\in ℰ$ an object, write

$f⋔S$f \pitchfork S

to indicate that for the morphism $f:X\to Y$ the induced hom set morphism

$ℰ\left(f,S\right):ℰ\left(S,Y\right)\to ℰ\left(S,X\right)$\mathcal{E}(f, S) : \mathcal{E}(S,Y) \to \mathcal{E}(S,X)

is surjective, dually for

$S⋔f\phantom{\rule{thinmathspace}{0ex}}.$S \pitchfork f \,.

In the case that $ℰ$ has a terminal object $*$ we have equivalently

$f⋔S\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}f⋔\left(S\to *\right)$f \pitchfork S \;\;\Leftrightarrow\;\; f \pitchfork (S \to *)

and if $ℰ$ has an initial object $epmtyset$ we have equivalently

$S⋔f\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\varnothing \to S\right)⋔f\phantom{\rule{thinmathspace}{0ex}}.$S \pitchfork f \;\;\Leftrightarrow \;\; (\emptyset \to S) \pitchfork f \,.

Accordingly, for $Q\subset \mathrm{Mor}\left(ℰ\right)$ write ${}^{⋔}Q$ and ${ℰ}^{Q}$ for the class of morphisms with left or right lifting property against all elements of $Q$, respectively.

###### Observation

If $\left(L⊣R\right):ℰ\to ℱ$ is a pair of adjoint functors, then

$f⋔R\left(g\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}L\left(F\right)⋔g$f \pitchfork R(g) \;\; \Leftrightarrow \;\; L(F) \pitchfork g
###### Definition

A pair of classes of morphisms $\left(L,R\right)$ in $ℰ$ is a weak factorization system precisely if

1. every morphism in $ℰ$ follows as a morphism in $L$ followed by a morphism in $R$;

2. $R={L}^{⋔}$ and $L={}^{⋔}R$.

### Tensoring

Let ${ℰ}_{1}$, ${ℰ}_{2}$, ${ℰ}_{3}$ be three categories.

###### Definition
$\otimes :{ℰ}_{1}×{ℰ}_{2}\to {ℰ}_{3}$\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3
1. is called divisible on the left if for every $A\in {ℰ}_{1}$ the functor $A\otimes \left(-\right)$ has a right adjoint, to be denoted

$A\\left(-\right):{ℰ}_{3}\to {ℰ}_{2}\phantom{\rule{thinmathspace}{0ex}};$A \backslash (-) : \mathcal{E}_3 \to \mathcal{E}_2 \,;
2. is called divisible on the right if for every $A\in {ℰ}_{2}$ the functor $\left(-\right)\otimes A$ has a right adjoint, to be denoted

$\left(-\right)/A:{ℰ}_{3}\to {ℰ}_{1}\phantom{\rule{thinmathspace}{0ex}};$(-)/ A : \mathcal{E}_3 \to \mathcal{E}_1 \,;
###### Observation

If $\otimes$ is divisble on both sides, then there are natural isomorphisms between the collections of morphisms

$A\otimes B\to X$A \otimes B \to X

and

$B\to A\X$B \to A\backslash X

and

$A\to X/B\phantom{\rule{thinmathspace}{0ex}}.$A \to X / B \,.
###### Observation

For every $f\in \mathrm{Mor}\left({ℰ}_{1}\right)$, $g\in \mathrm{Mor}\left({ℰ}_{2}\right)$ and $X\in {ℰ}_{3}$ we have

$f⋔\left(X/g\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}g⋔\left(f\X\right)\phantom{\rule{thinmathspace}{0ex}}.$f \pitchfork (X/g) \;\; \Leftrightarrow \;\; g \pitchfork (f \backslash X) \,.
###### Example

If $ℰ$ is a closed symmetric monoidal category, then its tensor product functor $\otimes :ℰ×ℰ\to ℰ$ is divisible on both sides, the two divisions coincide and are given by the internal hom $\left[-,-\right]:{ℰ}^{\mathrm{op}}×ℰ\to ℰ$

$X/A\simeq \left[A,X\right]\simeq A\X\phantom{\rule{thinmathspace}{0ex}}.$X/A \simeq [A,X] \simeq A\backslash X \,.

### Pushout-tensoring

Let now ${ℰ}_{3}$ have finite colimits and $\otimes :{ℰ}_{1}×{ℰ}_{2}\to {ℰ}_{3}$ a functor.

###### Definition

for $f:A\to B$ in ${ℰ}_{1}$ and $g:X\to Y$ in ${ℰ}_{2}$, write

$A\otimes Y\coprod _{A\otimes X}B\otimes Y\to B\otimes Y$A \otimes Y \coprod_{A \otimes X} B \otimes Y \to B \otimes Y

for the induced pushout-product morphism, the canonical morphism out of the pushout induced from the commutativity of the diagram

$\begin{array}{ccc}A\otimes X& \to & B\otimes X\\ ↓& & ↓\\ B\otimes X& \to & B\otimes Y\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ A \otimes X &\to& B \otimes X \\ \downarrow && \downarrow \\ B \otimes X &\to& B \otimes Y } \,.
###### Observation

The pushout-product extends to a functor

$\overline{\otimes }:{ℰ}_{1}^{I}×{ℰ}_{2}^{I}\to {ℰ}_{3}^{I}\phantom{\rule{thinmathspace}{0ex}},$\bar \otimes : \mathcal{E}_1^I \times \mathcal{E}_2^I \to \mathcal{E}_3^I \,,

where ${C}^{I}$ denotes the arrow category of $C$.

###### Observation

If in the above situation ${ℰ}_{1}$ and ${ℰ}_{2}$ have finite limits and $\otimes$ is divisble on both sides, def. 3, then also $\overline{\otimes }$ is divisible on both sides:

1. for $f:A\to B$ in ${ℰ}_{1}$ and $g:X\to Y$ in ${ℰ}_{3}$, the left quotient is

$f\overline{\}g:B\X\to B\Y{×}_{A\Y}A\X\phantom{\rule{thinmathspace}{0ex}};$f \bar \backslash g : B \backslash X \to B \backslash Y \times_{A \backslash Y} A \backslash X \,;
2. for $f:S\to T$ in ${ℰ}_{2}$ and $g:X\to Y$ in ${ℰ}_{3}$, the right quotient is

$g\overline{/}f:X/T\to Y/T{×}_{Y/S}X/S\phantom{\rule{thinmathspace}{0ex}};$g \bar / f : X / T \to Y / T \times_{Y / S} X / S \,;
###### Proposition

In the above situation, let ${ℰ}_{1}$, ${ℰ}_{2}$, ${ℰ}_{3}$ have all finite limits and colimits. For all $u\in \mathrm{Mor}\left({ℰ}_{1}\right)$, $v\in \mathrm{Mor}\left({ℰ}_{2}\right)$, $f\in \mathrm{Mor}\left({ℰ}_{3}\right)$ we have

$\left(u\overline{\otimes }v\right)⋔f\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}U⋔f\overline{/}v\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}v⋔u\overline{\}f\phantom{\rule{thinmathspace}{0ex}}.$(u \bar \otimes v) \pitchfork f \;\; \Leftrightarrow \;\; U \pitchfork f \bar /v \;\; \Leftrightarrow \;\; v \pitchfork u \bar \backslash f \,.

## Applications

### Reedy theory

Let $ℰ$ be a model category. Write $\Delta$ for the simplex category and sSet for the category of simplicial sets. In the Reedy model structure on the presheaf category $\left[{\Delta }^{\mathrm{op}},ℰ\right]$ the following constructions are central.

###### Definition

Write

$\square :\mathrm{sSet}×ℰ\to \left[{\Delta }^{\mathrm{op}},ℰ\right]$\Box : sSet \times \mathcal{E} \to [\Delta^{op}, \mathcal{E}]

for the functor given by

$\left(S\square X\right):n↦{S}_{n}\cdot X\phantom{\rule{thinmathspace}{0ex}}.$(S \Box X) : n \mapsto S_n \cdot X \,.

Write

$\otimes \left[\Delta ,\mathrm{Set}\right]×\left[{\Delta }^{\mathrm{op}},ℰ\right]\to ℰ$\otimes [\Delta, Set] \times [\Delta^{op}, \mathcal{E}] \to\mathcal{E}

for the functor given by the coend

$S\otimes X={\int }^{n\in \Delta }{S}_{n}\cdot {X}_{n}\phantom{\rule{thinmathspace}{0ex}}.$S \otimes X = \int^{n \in \Delta} S_n \cdot X_n \,.

(Here on the right we have the canonical tensoring of $ℰ$ over Set, where ${S}_{n}\cdot X\simeq {\coprod }_{s\in {S}_{n}}X$.)

###### Observation

The functor $\square$ is divisible on both sides.

Let $X\in \left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$. Then

• the object $\partial \Delta \left[n\right]\X$ is the matching object of $X$ at stage $n$;

• the morphism $\left(\partial \Delta \left[n\right]↪\Delta \left[n\right]\right)\X$ is the canonical morphism from ${X}_{n}$ into the $n$-matching object.

Let $f:X\to Y$ be a morphism in $\left[{\Delta }^{\mathrm{op}},\mathrm{sSet}\right]$. Then

• the relative matching morphism of $f$ at stage $n$ is

$\left(\partial \Delta \left[n\right]↪\Delta \left[n\right]\right)\overline{\}f\phantom{\rule{thinmathspace}{0ex}};$(\partial \Delta[n] \hookrightarrow \Delta[n]) \bar \backslash f \,;
• the object $\left(\partial {\Delta }^{c}\right)\otimes X$ is the latching object at stage $n$;

• the morphism $\left(\partial {\Delta }^{c}\to \Delta \right)\otimes X$ is the canonical morphism out of the latching object into ${X}_{n}$;

• the morphism $\left(\partial {\Delta }^{c}\to \Delta \right)\overline{\otimes }f$ is the relative latching morphism of $f$.

## References

Revised on March 23, 2012 12:00:06 by Toby Bartels (98.16.172.63)