## Generic morphisms, parametric representations and weakly cartesian monads (2004)

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Venue: | THEORY APPL. CATEG |

Citations: | 4 - 3 self |

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@TECHREPORT{Weber04genericmorphisms,,

author = {Mark Weber},

title = {Generic morphisms, parametric representations and weakly cartesian monads },

institution = { THEORY APPL. CATEG},

year = {2004}

}

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### Citations

236 |
Categories for the working mathematician. Graduate Texts
- Lane
- 1998
(Show Context)
Citation Context ...) is complete. Proof. It suffices to show that Fact(f) has (arbitrary) products and pullbacks, and these are easily constructed directly using the hypotheses of this lemma.s216 MARK WEBER Recall from =-=[Mac71]-=-, that a solution set for a category A is a set X of objects of A that is jointly weakly initial, in the sense that for any A ∈A,thereisanS ∈X and a morphism S→A. Recall also that the initial object t... |

196 |
Locally Presentable and Accessible Categories
- Adámek, Rosicky
- 1994
(Show Context)
Citation Context ...also that the automorphism groups of the S ∈Sare finite. If T admits generic factorisations relative to the objects of S then T is cofinitary. Proof. By the dual of theorem(1.5) and corollary(1.7) of =-=[AR94]-=-, it suffices to show that T preserves limits of diagrams of the form ... ∂2 �� X2 ∂1 �� X1 ∂0 �� X0 Let cn : X→Xn be the components of a universal cone. We must show that the cone for the diagram ...... |

141 |
The system F of variable types, fifteen years later. Theoretical Computer Science 45
- Girard
- 1986
(Show Context)
Citation Context ...a generic morphism f : A→TX for an endofunctor T of an arbitrary category. A stricter notion of generic morphism arose in the PhD thesis of Lamarche [Lam88], which itself builds on the work of Girard =-=[Gir86]-=- on qualitative domains. Qualitative domains give a semantics for variable types, and generic morphisms arise in two ways in this semantics. Namely, to express the normal form characterisation for sta... |

125 |
The formal theory of monads
- Street
- 1972
(Show Context)
Citation Context ...onoids in End(A). The objects of this category are of course monads, and we shall refer to its arrows ass194 MARK WEBER monad morphisms. While there are other important notions of monad morphism, see =-=[Str72]-=- for example, they will not be discussed here. 2.1. Definition. Let A be a category with finite limits, and T ∈ End(A). T -Coll is the full subcategory of End(A)↓T consisting of the cartesian transfor... |

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...ad T on Glob (the category of globular sets) described in section (9) is cartesian and, as pointed out in [Lei00], a T -operad is an operad in the monoidal globular category Span(Set) in the sense of =-=[Bat98]-=-. Among these operads is one whose algebras are weak ω-categories in the sense of Michael Batanin. 3. In [Web01] a cartesian monad F on Glob is described whose algebras are strict monoidal strict ω-ca... |

79 |
Foncteurs analytiques et espèces de structures
- Joyal
- 1985
(Show Context)
Citation Context ...duction Combinatorial species and analytic functors were introduced by André Joyal as a unifying conceptual notion for enumerative combinatorics. In characterising the analytic endofunctors of Set in =-=[Joy86]-=-, Joyal was led to the technical notion of generic element of an endofunctor T of Set. Viewing the elements of T as functions 1→TX, the property of genericness does not rely on the domain of these fun... |

32 | Operads in higher-dimensional category theory
- Leinster
- 2000
(Show Context)
Citation Context ...complete category A for which T preserves pullbacks and µ and η are cartesian. A T -operad in our sense, is precisely a T -multicategory in the sense of Burroni [Bur71], Hermida [Her00b] and Leinster =-=[Lei00]-=-, whose underlying object is the terminal object of A. 2.5. Definition. The category of algebras for an operad φ : S⇒T is the category of Eilenberg-Moore algebras for the monad S. 2.6. Proposition. [K... |

22 | An abstract approach to coherence - KELLY |

21 | A cellular nerve for higher categories
- Berger
(Show Context)
Citation Context ...ting diagrams. The proof that this corresponds with strict-ω-categories defined by successive enrichment can be found in [Lei00]. The following characterisation of Ω, originally due to Clemens Berger =-=[Ber02]-=-, follows from (4.8), (7.3) and the above generic description of the tree monad. 9.2. Proposition. Ω is isomorphic to the dual of the following subcategory of the Kleisli category of T : • objects are... |

20 |
T-catégories (catégories dans un triple), Cahiers de Topologie et Géométrie Différentielle XII(3
- Burroni
- 1971
(Show Context)
Citation Context ...acks. That is, a monad T on a finitely complete category A for which T preserves pullbacks and µ and η are cartesian. A T -operad in our sense, is precisely a T -multicategory in the sense of Burroni =-=[Bur71]-=-, Hermida [Her00b] and Leinster [Lei00], whose underlying object is the terminal object of A. 2.5. Definition. The category of algebras for an operad φ : S⇒T is the category of Eilenberg-Moore algebra... |

18 |
The universal property of the multitude of trees
- Batanin, Street
(Show Context)
Citation Context ...T (δ) ◦ f = α. We call such a δ a T -fill for this square. We say that f is a strict T -generic, when there is a unique T -fill for any α, β, andγ as above. 5 Another important viewpoint described in =-=[BS00]-=- and [Her00a] is to regard Ω as a monoidal globular category which plays the role of a globular monoid classifier.sGENERIC MORPHISMS, PARAMETRIC REPRESENTATIONS AND WEAKLY CARTESIAN MONADS 207 5.3. Ex... |

13 | From coherent structures to universal properties
- Hermida
(Show Context)
Citation Context ... α. We call such a δ a T -fill for this square. We say that f is a strict T -generic, when there is a unique T -fill for any α, β, andγ as above. 5 Another important viewpoint described in [BS00] and =-=[Her00a]-=- is to regard Ω as a monoidal globular category which plays the role of a globular monoid classifier.sGENERIC MORPHISMS, PARAMETRIC REPRESENTATIONS AND WEAKLY CARTESIAN MONADS 207 5.3. Example. Let T ... |

5 |
Some remarks on free monoids in a topos
- Bénabou
- 1991
(Show Context)
Citation Context ... strict monoidal categories, symmetric monoidal categories, and symmetric strict monoidal categories are examples of categorical structures that arise as algebras of S-clubs. 2 In fact it is shown in =-=[Bén91]-=- that this is true far more generally, that is, when Set is replaced by an elementary topos with a natural numbers object.sGENERIC MORPHISMS, PARAMETRIC REPRESENTATIONS AND WEAKLY CARTESIAN MONADS 197... |

5 |
Connected limits, familial representability and
- Carboni, Johnstone
- 1995
(Show Context)
Citation Context ... are the subject of on-going research, but will not be discussed any further in this paper. In his PhD thesis [Die77], Diers considered functors into Set with a family of representing objects, and in =-=[CJ95]-=- the theory of these familially representable functors was developed further. However, in higher dimensional category theory, there arise endofunctors of presheaf categories that are of a similar form... |

5 |
Symmetric Operads for Globular Sets
- Weber
- 2001
(Show Context)
Citation Context ...], a T -operad is an operad in the monoidal globular category Span(Set) in the sense of [Bat98]. Among these operads is one whose algebras are weak ω-categories in the sense of Michael Batanin. 3. In =-=[Web01]-=- a cartesian monad F on Glob is described whose algebras are strict monoidal strict ω-categories. There is an F-operad whose algebras are monoidal weak ω-categories. These are one-object Batanin weak ... |

3 |
Catégories Localisables
- Diers
- 1977
(Show Context)
Citation Context ...udy of stable domains and variable types, and their connections to higher category theory, are the subject of on-going research, but will not be discussed any further in this paper. In his PhD thesis =-=[Die77]-=-, Diers considered functors into Set with a family of representing objects, and in [CJ95] the theory of these familially representable functors was developed further. However, in higher dimensional ca... |

2 |
Modelling Polymorphism with Categories
- Lamarche
- 1988
(Show Context)
Citation Context ...of sets. In this way, one arrives at the notion of a generic morphism f : A→TX for an endofunctor T of an arbitrary category. A stricter notion of generic morphism arose in the PhD thesis of Lamarche =-=[Lam88]-=-, which itself builds on the work of Girard [Gir86] on qualitative domains. Qualitative domains give a semantics for variable types, and generic morphisms arise in two ways in this semantics. Namely, ... |

2 |
The trace factorisation of stable functors
- Taylor
- 1988
(Show Context)
Citation Context ...converse result that the preservation of wide pullbacks by an endofunctor implies that it admits strict generic factorisations. This result is not new – versions of it appear in [Die77], [Lam88], and =-=[Tay88]-=-, although I know of no published proof. The second result, Theorem(6.8), is new, generalising André Joyal’s observation that analytic endofunctors of Set preserve cofiltered limits. The strict generi... |

1 |
functorial calculus I
- Many-variable
- 1972
(Show Context)
Citation Context ... n �� � n ��� � x �� y �� ��� ������ f �� X where φ ∈ Sym n and f is a natural transformation as shown. This monad is also cartesian, and S-operads are the clubs originally considered by Max Kelly in =-=[Kel72b]-=- and [Kel72a]. Monoidal categories, strict monoidal categories, braided monoidal categories, braided strict monoidal categories, symmetric monoidal categories, and symmetric strict monoidal categories... |

1 |
operads, higher categories, lecture note series
- Higher
- 2003
(Show Context)
Citation Context ...nd concepts for the manipulation and investigation of such endofunctors. The inadequacy of our present state of knowledge of such endofunctors and monads is expressed most forcefully in appendix C of =-=[Lei03]-=-. Generic morphisms and related notions enable us to generalise the theory of familially representable functors, so as to include familially representable endofunctors of presheaf categories. Abstract... |