Contents

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. below be easily expressible in terms of “division on both sides”-operations.

## The calculus

### Lifting

Let $\mathcal{E}$ be a category (locally small).

###### Notation

For $f, g \in Mor(\mathcal{E})$, write

$f \pitchfork g$

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

For $S \in \mathcal{E}$ an object, write

$f \pitchfork S$

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

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

is surjective, dually for

$S \pitchfork f \,.$

In the case that $\mathcal{E}$ has a terminal object $*$ we have equivalently

$f \pitchfork S \;\;\Leftrightarrow\;\; f \pitchfork (S \to *)$

and if $\mathcal{E}$ has an initial object $\emptyset$ we have equivalently

$S \pitchfork f \;\;\Leftrightarrow \;\; (\emptyset \to S) \pitchfork f \,.$

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

###### Proposition

If $(L \dashv R) : \mathcal{E} \to \mathcal{F}$ is a pair of adjoint functors, then

$f \pitchfork R(g) \;\; \Leftrightarrow \;\; L(f) \pitchfork g$
###### Definition

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

1. every morphism in $\mathcal{E}$ factors as the composition of a morphism in $L$ followed by a morphism in $R$;

2. $R = L^\pitchfork$ and $L = {}^\pitchfork R$.

### Tensoring

Let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ be three categories.

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

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

$(-)/ A : \mathcal{E}_3 \to \mathcal{E}_1 \,;$
###### Proposition

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

$A \otimes B \to X$

and

$B \to A\backslash X$

and

$A \to X / B \,.$
###### Proposition

For every $f \in Mor(\mathcal{E}_1)$, $g \in Mor(\mathcal{E}_2)$ and $X \in \mathcal{E}_3$ we have

$f \pitchfork (X/g) \;\; \Leftrightarrow \;\; g \pitchfork (f \backslash X) \,.$
###### Example

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

$X/A \simeq [A,X] \simeq A\backslash X \,.$

### Pushout-tensoring

Let now $\mathcal{E}_3$ have finite colimits and let $\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$ be a functor.

###### Definition

for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_2$, write

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

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

$\array{ A \otimes X &\to& B \otimes X \\ \downarrow && \downarrow \\ A \otimes Y &\to& B \otimes Y } \,.$
###### Proposition

The pushout-product extends to a functor

$\bar \otimes : \mathcal{E}_1^I \times \mathcal{E}_2^I \to \mathcal{E}_3^I \,,$

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

###### Proposition

If in the above situation $\mathcal{E}_1$ and $\mathcal{E}_2$ have finite limits and $\otimes$ is divisble on both sides, def. , then also $\bar{\otimes}$ is divisible on both sides:

1. for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_3$, the left quotient is

$f \bar \backslash g \;\colon\; B \backslash X \to B \backslash Y \times_{A \backslash Y} A \backslash X \,;$
2. for $f : S \to T$ in $\mathcal{E}_2$ and $g : X \to Y$ in $\mathcal{E}_3$, the right quotient is

$g \bar / f \;\colon\; X / T \to Y / T \times_{Y / S} X / S \,;$

A key statement now is the following, characterizing the right lifting property again pushout product morphisms:

###### Proposition

In the above situation, let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ have all finite limits and colimits. For all $u \in Mor(\mathcal{E}_1)$, $v \in Mor(\mathcal{E}_2)$, $f \in Mor(\mathcal{E}_3)$ we have

$(u \bar \otimes v) \pitchfork f \;\; \Leftrightarrow \;\; u \pitchfork f \bar /v \;\; \Leftrightarrow \;\; v \pitchfork u \bar \backslash f \,.$

## Applications

### Reedy theory

Let $\mathcal{E}$ 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 $[\Delta^{op}, \mathcal{E}]$ the following constructions are central.

###### Definition

Write

$\Box : sSet \times \mathcal{E} \to [\Delta^{op}, \mathcal{E}]$

for the functor given by

$(S \Box X) : n \mapsto S_n \cdot X \,.$

Write

$\otimes : [\Delta, Set] \times [\Delta^{op}, \mathcal{E}] \to\mathcal{E}$

for the functor given by the coend

$S \otimes X = \int^{n \in \Delta} S_n \cdot X_n \,.$

(Here on the right we have the canonical tensoring of $\mathcal{E}$ over Set, where $S_n \cdot X \simeq \coprod_{s \in S_n} X$.)

###### Proposition

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

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

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

• the morphism $(\partial \Delta[n] \hookrightarrow \Delta[n]) \backslash X$ is the canonical morphism from $X_n$ into the $n$-matching object.

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

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

$(\partial \Delta[n] \hookrightarrow \Delta[n]) \bar \backslash f \,;$
• the object $(\partial \Delta^c) \otimes X$ is the latching object at stage $n$;

• the morphism $(\partial \Delta^c \to \Delta)\otimes X$ is the canonical morphism out of the latching object into $X_n$;

• the morphism $(\partial \Delta^c \to \Delta) \bar \otimes f$ is the relative latching morphism of $f$.