## Understanding the small object argument (2008)

Venue: | Applied Categorical Structures |

Citations: | 12 - 0 self |

### BibTeX

@ARTICLE{Garner08understandingthe,

author = {Richard Garner},

title = {Understanding the small object argument},

journal = {Applied Categorical Structures},

year = {2008}

}

### OpenURL

### Abstract

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that

### Citations

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Homotopical algebra
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- 1967
(Show Context)
Citation Context ... system generates its own notion of “equivalence” which respect to which its factorisations are unique. The framework within which this is most readily expressed is that of Quillen’s model categories =-=[20]-=-, which consist in a clever interaction of two w.f.s.’s on a category: but we can make do with a single w.f.s., and for the purposes of this paper, we will. Whilst in many respects, the theory of w.f.... |

251 |
Braided tensor categories
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Citation Context ...ny braided or symmetric monoidal category is two-fold monoidal, with the two monoidal structures coinciding; the maps zA,B,C,D are built from braidings/symmetries and associativity isomorphisms: c.f. =-=[16]-=-. • If V is a cocomplete symmetric monoidal category, then the functor category [X × X, V] has a two-fold monoidal structure. The first monoidal structure (⊗,I) is given by matrix multiplication, whil... |

114 |
Coherence for tricategories
- Gordon, Power, et al.
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Citation Context ...T , and define the category Tricatψ of tricategories and trihomomorphisms to be the co-Kleisli category of Q. The notion of trihomomorphism we obtain in this way cannot be the one we are used to from =-=[12]-=-, since the latter does not admit a strictly associative composition: see [11]. Nonetheless, we can show that our new notion of trihomomorphism is equivalent to the old one, in that we can exhibit a b... |

77 |
Two dimensional monad theory
- Kelly, Blackwell, et al.
- 1989
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Citation Context ...unctor AlgKan ֒→ AlgKan ψ has a left adjoint. It is a corresponding result which forms the cornerstone of two-dimensional monad theory [5, Theorem 3.13]. 7.2 For an example even more in the spirit of =-=[5]-=-, we consider the category C = 2-Cat and the set of maps J given as follows: ∅ • • • • • • ; ; ; • • • • • • •. These maps generate a plain w.f.s. which is one half of the model structure on 2-Cat des... |

76 |
Lokal präsentierbare Kategorien
- Gabriel, Ulmer
- 1971
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Citation Context ...ss property we may consider on C is that: (*) For every X ∈ C, there is an αX for which X is αX-presentable. This is certainly the case for any category C which is locally presentable in the sense of =-=[10]-=-. However, it does not obtain in categories such as the category of topological spaces, the category of Hausdorff topological spaces, or the category of topological groups: and since we would like our... |

73 |
Categories of continuous functors
- Freyd
- 1972
(Show Context)
Citation Context ...detail the various sorts of factorisation system mentioned in the Introduction. 2.1 Most familiar is the notion of strong factorisation system (L, R) on a category C, introduced by Freyd and Kelly in =-=[9]-=-. This is given by two classes of maps L and R in C which are each closed under composition with isomorphisms, and which satisfy the axioms of (factorisation) Every map e: X → Y in C can be written as... |

70 |
Model categories, volume 63 of Mathematical Surveys and Monographs
- Hovey
- 1999
(Show Context)
Citation Context ...e small object argument, and so we may compare the two by comparing the choices of factorisation which they provide. For a detailed account of the small object argument, we refer the reader to [6] or =-=[15]-=-. 6.3 Suppose we are given a category C and a set of maps J as in the Proposition; and let g: C → D be a morphism of C that we wish to factorise. The first step in 33both the small object argument an... |

68 |
Distributive laws
- Beck
- 1969
(Show Context)
Citation Context ...ence of the (co)monad axioms). We will say that a natural w.f.s. satisfies the distributivity axiom if this natural transformation 10∆: LR ⇒ RL defines a distributive law of L over R in the sense of =-=[4]-=-. Note that this is a property of a natural w.f.s., rather than extra structure on it. 2.19 Example: We may check that each of the natural w.f.s.’s given so far satisfies the distributivity axiom. 2.2... |

49 |
A Unified Treatment of Transfinite Constructions for Free Algebras, Free Monoids
- Kelly
- 1980
(Show Context)
Citation Context ...how we should build the factorisations, and for this we are able bring to bear a well-established body of knowledge concerning transfinite constructions in categories, on which the definitive word is =-=[17]-=-. There is a corresponding theory for weak factorisation systems. Again, we suppose ourselves given a well-behaved C and a set of maps J, but this time we take for R the class of maps weakly right ort... |

35 |
Kan extensions in enriched category theory
- Dubuc
- 1970
(Show Context)
Citation Context ...give a reflection along G by giving a reflection along each functor G1, G2 and G3 in turn. For G3, we have the following well-known result, which was first stated at this level of generality by Dubuc =-=[7]-=-; but see also [2]. 4.6 Proposition: Let C be cocomplete, and let U : A → C 2 be a small category over C 2 . Then A admits a reflection along G3: Cmd(C 2 ) → CAT/C 2 . 17Proof. Because A is small and... |

32 |
Iterated monoidal categories
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(Show Context)
Citation Context ...a objects in a category, it is with reference to a symmetric or braided monoidal structure on that category: but here we will need something slightly more general. 4.9 By a two-fold monoidal category =-=[3]-=-, we mean a category V equipped with two monoidal structures (⊗,I,α,λ,ρ) and (⊙, ⊥,α ′ ,λ ′ ,ρ ′ ) in such a way that the functors ⊙: V × V → V and ⊥: 1 → V, together with the natural transformations ... |

27 |
Constructions of factorization systems in categories
- Bousfield
- 1977
(Show Context)
Citation Context ...tem, we must also have factorisation of maps: and for this, we apply a construction known as the small object argument, introduced by Quillen [20], and first given in its full generality by Bousfield =-=[6]-=-. The problem lies in divining the precise nature of the small object argument. It is certainly some kind of transfinite construction: but it is a transfinite construction which does not converge, has... |

24 | On a generalized small-object argument for the injective subcategory problem
- Adámek, Herrlich, et al.
- 2001
(Show Context)
Citation Context ... ρ ′ g . This gives rise to the countable sequence g C λ ′ g D idD K ′ g ρ ′ g D λ ′ ρ ′ g idD K ′ ρ ′ g which we extend transfinitely by taking colimits at limit ordinals. However, as pointed out in =-=[1]-=-, this sequence almost never converges. Instead, the small object argument requires one to choose an arbitrary ordinal at which to stop: or rather, an ordinal which is large enough to ensure that the ... |

23 | Exploring the gap between linear and classical logic
- Lamarche
- 2007
(Show Context)
Citation Context ..., an operad in V whose objects of n-ary operations are comonoids; and whose substitution maps are morphisms of comonoids. Bialgebras in two-fold monoidal categories play a central role in recent work =-=[19]-=- of François Lamarche. 4.13 Let us write FF(C) for the category of functorial factorisations on C, and let us write RNWFS(C) for the category dual to LNWFS(C): so its objects are pairs (F,R) of a func... |

18 | An algebraic theory of tricategories - Gurski - 2006 |

9 |
A Quillen model structure for 2-categories
- Lack
(Show Context)
Citation Context ...category C = 2-Cat and the set of maps J given as follows: ∅ • • • • • • ; ; ; • • • • • • •. These maps generate a plain w.f.s. which is one half of the model structure on 2-Cat described by Lack in =-=[18]-=-. Our purpose here will be to consider the 36sand applying the free functor F to each of them. We now proceed as before: we consider this set J as a discrete subcategory J ֒→ Tricat 2 and let (L,R) be... |

4 | Enrichment over iterated monoidal categories
- Forcey
(Show Context)
Citation Context ...en by (F ⊗ G)(n) = ∑ F(m) ⊗ G(k1) ⊗ · · · ⊗ G(km). m,k1,...,km k1+···+km=n The second monoidal structure (⊙, ⊥) is again given pointwise. Further examples and applications to topology may be found in =-=[3, 8]-=-. 4.11 A two-fold monoidal category (V, ⊗,I, ⊙, ⊥) provides a suitable environment to define a notion of bialgebra. Indeed, because the ⊙-monoidal structure is lax monoidal with respect to the ⊗-struc... |

3 |
Categories with models
- Appelgate, Tierney
- 1969
(Show Context)
Citation Context ...along G by giving a reflection along each functor G1, G2 and G3 in turn. For G3, we have the following well-known result, which was first stated at this level of generality by Dubuc [7]; but see also =-=[2]-=-. 4.6 Proposition: Let C be cocomplete, and let U : A → C 2 be a small category over C 2 . Then A admits a reflection along G3: Cmd(C 2 ) → CAT/C 2 . 17Proof. Because A is small and C 2 cocomplete (s... |

3 |
Natural weak factorisation systems
- Grandis, Tholen
(Show Context)
Citation Context ...d have an equally valid expression in terms of structure: and in the case of w.f.s.’s, a suitable “algebraic” reformulation is given by Tholen and Grandis’ notion of natural weak factorisation system =-=[13]-=-. The extra algebraicity provided by natural w.f.s.’s allows us a clearer view of what is actually going on in the small object argument. We now have a functor from the category of natural w.f.s.’s on... |