## On comparing definitions of "weak n–category" (2001)

### BibTeX

@MISC{Makkai01oncomparing,

author = {M. Makkai},

title = {On comparing definitions of "weak n–category"},

year = {2001}

}

### OpenURL

### Abstract

1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand-- and this is more relevant here--, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally about technical matters that are written down formally elsewhere, where they can be studied further. [The text in square brackets [-] is either some technical explanation, or a digression.] 2. Terminology First, some terminological conventions. I will use the word "category " in its most general sense: weak ω-category. This is completely inclusive: all sorts of "categories " are categories ow. here are two extensions of the original meaning: "weak", and "omega-dimensional". Weak " signifies an indeterminate notion; there are several different specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would

### Citations

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...nt specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would ike to have. There are specific kinds of category, such as "Batanin category" =-=[B1]-=-, "multitopic ategory" [HMP1,2,3,4,5], [M8]. When one wants to talk about the "ordinary, strict" version of he notion, one says "strict". Thus, "strict category" is my term for an ordinary, strict ω-c... |

74 | Higher-dimensional algebra III: n-categories and the algebra of opetopes
- Baez, Dolan
- 1998
(Show Context)
Citation Context ...specific. But, " 2-category" is an indeterminate notion; "strict 2-category" is ne specific version; "bicategory" (which can also be called "Benabou 2-category"), Baez-Dolan, or opetopic, 2-category" =-=[BD2]-=-, "Batanin 2-category" [B1], "multitopic 2-category" (see below) are further specific concepts. he best thing about this terminology is that it makes good sense of talking about "the ategory of all ca... |

66 | An Intuitionstic Theory of Types - Martin-Löf - 1995 |

28 | The petit topos of globular sets - Street |

26 |
Avoiding the axiom of choice in general category theory
- Makkai
(Show Context)
Citation Context ...it is an incomplete notion; and the FOLDS quivalence detects this incompleteness. "Yes" because, complete or incomplete, "anafunctor" n general is a good notion (actually, very useful for the work in =-=[M4]-=-), and it is a good notion f equivalence to say that two (general) anafunctors give the same (equivalent) ordinary unctors. t may very well be that it is here that Daniel Quillen's model categories sh... |

21 |
duality and θ-categories
- Joyal, Disks
- 1997
(Show Context)
Citation Context ...aces and with standard amalgamation. I may add that the change o explicitly abstract places is no real change: any multicategory in the sense of [HMZ] is somorphic to one according to [HMP1,2,3,4]. n =-=[J]-=-, dating from the Fall of 1997, Andre Joyal defined "theta category" as a proposal for weak, omega-)category. n many, maybe even most, ways, theta categories may be the best of the competing notions e... |

21 |
Categorical structures, in: Handbook of algebra 1
- Street
- 1996
(Show Context)
Citation Context ... just mentioned contain a complicated one; you'll see that the complicated definition is not at all superfluous though. The simple definition in [HMZ] relies on Ross Street's notion of computad [S1], =-=[S2]-=-. I will now use " ω-category" for what I may call "strict category"; this is ordinary strict ω-category. (Small) ω-categories form a nice 1-category ωCat .GivenA∈ωCat , and a amily X={x :d A i i @c i... |

19 | On weak higher dimensional categories - Hermida, Makkai, et al. - 1997 |

18 |
The universal property of the multitude of trees
- Batanin, Street
(Show Context)
Citation Context ...t have a proof, although he had suspected the result was true. 29The "simple" categories of the previous paragraph have an important connection with Michael Batanin's and Ross Street's work in [B1], =-=[BS]-=-, [S]. In particular, Batanin's construction in [B1] of the free ω-category on a globular set plays an important role in [MZ]. [Digression. I would like to mention the encouraging fact that, it seems,... |

10 | Computads for finitary monads on globular sets, Higher category theory
- Batanin
- 1997
(Show Context)
Citation Context ...ay category Mlt such that Comp JSet . m/1 do not know how to prove this result, even with "one-way" removed, in a way that avoids alking about multitopic sets in the original, constructive, sense. In =-=[B2]-=-, Michael Batanin tated a general result in this connection; but the proof, as he himself pointed out to Marek awadowski, is incorrect. As a matter of fact, the total category Comp is not apresheaf at... |

4 |
Equivalence between approaches to the theory of opetopes
- Cheng
- 2000
(Show Context)
Citation Context ...the concept of multitopic category was inspired by the Baez/Dolan opetopic ategory, as a matter of fact, the exact relationship of the two concepts is still obscure to me despite Eugenia Cheng's work =-=[C]-=-). I believe that the multitopic concept is now worked out in ufficient detail (see also below) to stand on its own feet, and therefore I am not actively rying to relate it to the opetopic approach --... |

3 |
letter to
- Baez, Dolan
- 1995
(Show Context)
Citation Context ...n via universal properties, but universal properties will remain 2the main and preferred source for them. [My work [M1,2,3,4,5,7] on "virtual operations" predates [BD2], even the famous announcement =-=[BD1]-=-; of course, I will say more about the work below.] A definition of a (virtual) operation via a universal property is good because an equivalence of categories automatically respects/preserves the ope... |

3 |
Multitopic sets are the same as many-to-one computads
- Harnik, Makkai, et al.
- 2002
(Show Context)
Citation Context ...so it should appear soon). This work deals with multitopic sets; it does not contain the part on virtual composition. At this point of time, we have a simple definition of "multitopic set", thanks to =-=[HMZ]-=- (results of this paper were announced at the Toronto meeting of the AMS in September 2000); the papers just mentioned contain a complicated one; you'll see that the complicated definition is not at a... |

1 |
Coherence for Tricategories. Memoirs of the American Mathematical Society, vol 117 (no. 558
- Gordon, Power, et al.
- 1995
(Show Context)
Citation Context ...ras of the monad is not the same as the intended notion of morphism (homomorphism of bicategories); the former is a special case, the strict case, of the latter. The Gordon/Power/Street "tricategory" =-=[GPS]-=- is also honest-algebraic, and so is Michael Batanin's concept of category [B1]. Virtual operations, an expression that I learned from John Baez and James Dolan [BD2], are in fact almost as well-known... |

1 |
Limits indexed by category-valued 2-fucntors
- Algebra
- 1976
(Show Context)
Citation Context ...papers just mentioned contain a complicated one; you'll see that the complicated definition is not at all superfluous though. The simple definition in [HMZ] relies on Ross Street's notion of computad =-=[S1]-=-, [S2]. I will now use " ω-category" for what I may call "strict category"; this is ordinary strict ω-category. (Small) ω-categories form a nice 1-category ωCat .GivenA∈ωCat , and a amily X={x :d A i ... |