nLab functorial factorization




A functorial factorization is a structure on a category that factors any morphism into a composite of two morphisms, in a way that depends functorially on commutative squares.

Functorial factorizations play a prominent role in model category theory. On the one hand, many constructions there do rely on the factorizations into (acyclic) cofibrations and (acyclic) fibrations to be functorial, while, on the other hand, via the small object argument many examples of model categories do in fact carry a functorial factorization. (As a result, some authors include functorial factorization in the axioms of a model category right away.)

Functorial factorizations also play an important role as an ingredient in algebraic weak factorization systems.


Concrete explicit


A functorial factorization on a category ๐’ž\mathcal{C} is a way of assigning to any arrow ff in ๐’ž\mathcal{C} a pair of composable arrows f L,f Rf_L, f_R such that f=f Rโˆ˜f Lf = f_R \circ f_L, together with for any commuting square

โŸถh fโ†“ โ†“ g โŸถk \array{ & \overset{h}{\longrightarrow} \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g}} \\ & \overset{k}{\longrightarrow} }

a morphism E(h,k)E(h,k) completing their factorizations f=f Rโˆ˜f Lf = f_R \circ f_L and g=g Rโˆ˜g Lg = g_R \circ g_L to a further commuting diagram

โŸถh f Lโ†“ โ†“ g L โŸถE(h,k) f Rโ†“ โ†“ g R โŸถk, \array{ & \overset{h}{\longrightarrow} \\ {}^{\mathllap{f_L}}\downarrow && \downarrow^{\mathrlap{g_L}} \\ & \overset{E(h,k)}{\longrightarrow} \\ {}^{\mathllap{f_R}}\downarrow && \downarrow^{\mathrlap{g_R}} \\ & \overset{k}{\longrightarrow} } \,,

in a way that depends functorially on the given commutative square, i.e. E(h 1โˆ˜h 2,k 1โˆ˜k 2)=E(h 1,k 1)โˆ˜E(h 2,k 2)E(h_1\circ h_2,k_1\circ k_2) = E(h_1,k_1) \circ E(h_2,k_2).

Abstract version

Write ฮ”[1]={0โ†’1}\Delta[1] = \{0 \to 1\} and ฮ”[2]={0โ†’1โ†’2}\Delta[2] = \{0 \to 1 \to 2\} for the ordinal numbers, regarded as posets and hence as categories. The arrow category Arr(๐’ž)Arr(\mathcal{C}) is equivalently the functor category ๐’ž ฮ”[1]โ‰”Funct(ฮ”[1],๐’ž)\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C}), while ๐’ž ฮ”[2]โ‰”Funct(ฮ”[2],๐’ž)\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C}) has as objects pairs of composable morphisms in ๐’ž\mathcal{C}. There are three injective functors ฮด i:[1]โ†’[2]\delta_i \colon [1] \rightarrow [2], where ฮด i\delta_i omits the index ii in its image. By precomposition, this induces functors d i:๐’ž ฮ”[2]โŸถ๐’ž ฮ”[1]d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}. Here

  • d 1d_1 sends a pair of composable morphisms to their composition;

  • d 2d_2 sends a pair of composable morphisms to the first morphism;

  • d 0d_0 sends a pair of composable morphisms to the second morphism.


For ๐’ž\mathcal{C} a category, a functorial factorization of the morphisms in ๐’ž\mathcal{C} is a functor

fact:๐’ž ฮ”[1]โŸถ๐’ž ฮ”[2] fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}

which is a section of the composition functor d 1:๐’ž ฮ”[2]โ†’๐’ž ฮ”[1]d_1 \;\colon\; \mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}.



In a combinatorial model category there exists functorial factorizations into

  1. acyclic cofibrations followed by fibrations,

  2. cofibrations followed by acyclic fibrations,

hence in particular there exists functoral fibrant replacement and functorial cofibrant replacement.

This is due to unpublished results by Jeff Smith, see Dugger (2001), Prop. 2.3 and ยง7 and Barwick (2010), Prop. 2.5.


All orthogonal factorization systems are automatically functorial.


Not all weak factorization systems are functorial, although most are. This includes those produced by the small object argument, with due care, and also all algebraic weak factorization systems.


Enriched functorial factorizations

Sufficient conditions in enriched category theory (in particular enriched model category-theory) for functorial factorization to exist as an enriched functor is discussed in Hirschhorn 02, Theorem 4.3.8, Shulman 06, Prop. 24.2

Equivalence to pointed endofunctors


The following are equivalent:

  1. A functorial factorization on ๐’ž\mathcal{C}.
  2. A pointed endofunctor RR of ๐’ž ฮ”[1]\mathcal{C}^{\Delta[1]} such that cod:๐’ž ฮ”[1]โ†’๐’žcod : \mathcal{C}^{\Delta[1]} \to \mathcal{C} is a strict morphism of pointed endofunctors from RR to Id ๐’žId_{\mathcal{C}}, i.e. codโˆ˜R=codcod \circ R = cod and codcod maps the point Id ๐’ž ฮ”[1]โ†’RId_{\mathcal{C}^{\Delta[1]}}\to R to the identity map of Id ๐’žId_{\mathcal{C}}. (This is called a pointed endofunctor over codcod.)
  3. Dually, a copointed endofunctor LL of ๐’ž ฮ”[1]\mathcal{C}^{\Delta[1]} under domdom.

A endofunctor RR of ๐’ž ฮ”[1]\mathcal{C}^{\Delta[1]} maps every morphism ff in ๐’ž\mathcal{C} to a morphism f Rf_R, in a functorial way. A point of such an endofunctor assigns to each ff a pair of morphisms f L,f Mf_L, f_M such that f Rโˆ˜f L=f Mโˆ˜ff_R \circ f_L = f_M\circ f, naturally with respect to commutative squares. To say codโˆ˜R=codcod \circ R = cod means that f Rf_R has the same codomain as ff, and to say that codcod respects the points says that f Mf_M is an identity. Thus, f=f Rโˆ˜f Lf = f_R \circ f_L is a factorization. For functoriality, the functoriality of RR gives the commutative square kโˆ˜f=f Rโˆ˜E(h,k)k \circ f = f_R \circ E(h,k) and the functoriality of E(โˆ’,โˆ’)E(-,-), while the naturality of (โˆ’) L(-)_L gives the commutative square E(h,k)โˆ˜f=f Lโˆ˜hE(h,k) \circ f = f_L \circ h. The converse and dual are straightforward.

An algebraic weak factorization system is a functorial factorization together with compatible enhancements of these endofunctors to a monad and comonad. This can often be detected with the help of a composition law for factorizations.


Existence of functorial factorization in combinatorial model categories (following unpublished results by Jeff Smith):

Further discussion for enriched functors:

Last revised on May 14, 2023 at 15:28:04. See the history of this page for a list of all contributions to it.