nLab Joyal-Tierney calculus




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


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


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

fg f \,⧄\, g

if ff has the left lifting property against gg, or equivalently if gg has the right lifting property against ff.

For SS \in \mathcal{E} an object, write

fS f \,⧄\, S

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

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

is surjective, dually for

Sf. S \,⧄\, f \,.

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

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

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

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

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


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

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

A pair of classes of morphisms (L,R)(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 LL followed by a morphism in RR;

  2. R=L R = L^{⧄} and L= RL = {}^{⧄} R.


Let 1\mathcal{E}_1, 2\mathcal{E}_2, 3\mathcal{E}_3 be three categories.


A functor

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

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

    ()/A: 3 1; (-)/ A : \mathcal{E}_3 \to \mathcal{E}_1 \,;

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

ABX A \otimes B \to X


BA\X B \to A\backslash X


AX/B. A \to X / B \,.

For every fMor( 1)f \in Mor(\mathcal{E}_1), gMor( 2)g \in Mor(\mathcal{E}_2) and X 3X \in \mathcal{E}_3 we have

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

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 [,]: op×[-,-] : \mathcal{E}^{op} \times \mathcal{E} \to \mathcal{E}

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


Let now 3\mathcal{E}_3 have finite colimits and let : 1× 2 3\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3 be a functor.


for f:ABf : A \to B in 1\mathcal{E}_1 and g:XYg : X \to Y in 2\mathcal{E}_2, write

AY⨿AXBXBY 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

AX BX AY BY. \array{ A \otimes X &\to& B \otimes X \\ \downarrow && \downarrow \\ A \otimes Y &\to& B \otimes Y } \,.

The pushout-product extends to a functor

¯: 1 I× 2 I 3 I, \bar \otimes : \mathcal{E}_1^I \times \mathcal{E}_2^I \to \mathcal{E}_3^I \,,

where C IC^I denotes the arrow category of CC.


If in the above situation 1\mathcal{E}_1 and 2\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:ABf : A \to B in 1\mathcal{E}_1 and g:XYg : X \to Y in 3\mathcal{E}_3, the left quotient is

    f\¯g:B\XB\Y× A\YA\X; f \bar \backslash g \;\colon\; B \backslash X \to B \backslash Y \times_{A \backslash Y} A \backslash X \,;
  2. for f:STf : S \to T in 2\mathcal{E}_2 and g:XYg : X \to Y in 3\mathcal{E}_3, the right quotient is

    g/¯f:X/TY/T× Y/SX/S; 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:


In the above situation, let 1\mathcal{E}_1, 2\mathcal{E}_2, 3\mathcal{E}_3 have all finite limits and colimits. For all uMor( 1)u \in Mor(\mathcal{E}_1), vMor( 2)v \in Mor(\mathcal{E}_2), fMor( 3)f \in Mor(\mathcal{E}_3) we have

(u¯v)fuf/¯vvu\¯f. (u \bar \otimes v) \,⧄\, f \;\; \Leftrightarrow \;\; u \,⧄\, f \bar /v \;\; \Leftrightarrow \;\; v \,⧄\, u \bar \backslash f \,.


Reedy theory

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



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

for the functor given by

(SX):nS nX. (S \Box X) : n \mapsto S_n \cdot X \,.


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

for the functor given by the coend

SX= nΔS nX n. 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 nX sS nXS_n \cdot X \simeq \coprod_{s \in S_n} X.)


The functor \Box is divisible on both sides.

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

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

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

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

  • the relative matching morphism of ff at stage nn is

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

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

  • the morphism (Δ cΔ)¯f(\partial \Delta^c \to \Delta) \bar \otimes f is the relative latching morphism of ff.


Last revised on April 23, 2023 at 11:36:05. See the history of this page for a list of all contributions to it.