nLab Joyal-Tierney calculus

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

$$f \,⧄\, g$$

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

$$f \,⧄\, S$$

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

$$S \,⧄\, f \,.$$

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

$$f \,⧄\, S \;\;\Leftrightarrow\;\; f \,⧄\, (S \to *)$$

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

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

Accordingly, for $Q \subset Mor(\mathcal{E})$ write ${}^{⧄}Q$ and $Q^{⧄}$ 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

$$f \,⧄\, R(g) \;\; \Leftrightarrow \;\; L(f) \,⧄\, g$$

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. $R = L^{⧄}$ and $L = {}^{⧄} R$.

Tensoring

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

Definition

A functor

$$\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$$

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

$$f \,⧄\, (X/g) \;\; \Leftrightarrow \;\; g \,⧄\, (f \backslash X) \,.$$

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

$$A \otimes Y \overset {A \otimes X} {\amalg} B \otimes X \longrightarrow B \otimes Y$$

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

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

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 $f$.

Related concepts

• pushout-product axiom

• pullback-power

• Quillen negation/weak orthogonal class

References

• André Joyal, Myles Tierney, Quasi-categories vs Segal spaces (arXiv:0607820)