- 4. Substitution This section gives a treatment of the syntactic operation of substitution for multitopes and
- Avoiding the axiom of choice in general category theory by M. Makkai!(McGill University)
- [Be1] J. Benabou, Introduction to bicategories. In: Reports of the Midwest Category Seminar, Lecture Notes in Math. 47, SpringerVerlag, 1967; pp. 177.
- 2. Multitopes and multitopic sets Here, I will recall the notions in the title; they are dealt with in [H/M/P] in complete detail.
- Multitopic sets are the same as many--to--one computads by Victor Harnik# (University of Haifa),
- The multitopic w--category of all multitopic w--categories by M. Makkai!(McGill University)
- 3. Colored multitopes. The question how to define the concept of weak ncategory, for finite n , and in the second
- Q a RA @Q a { a RM NQ a R Q a RA @Q a { a RM NQ a R
- 4. Substitution This section gives a treatment of the syntactic operation of substitution for multitopes and
- 8. Kinds of completeness For our purposes here, we will have to be specific about sizerestrictions. In our previous
- [B2] M. A. Batanin, Computads for Finitary Monads on Globular Sets. In: Contemporary Mathematics 230 (1998), AMS; pp. 3757.
- The first triangle identity Let R be any ms. We want to show that the diagram
- 7. The category of multitopes Suppose S=(C , C , D , d ) , S=(C , C , D , d ) are multitopic sets. A
- We prove the if direction of 3.(1). We use the notation introduced in and before the statement of 3.(1). On the basis of the data # , C , F defined on # , and q , satisfying the
- Part 4. Simultaneous composition in the multicategories associated with M=[X] . As in 2, we have X , M=[X] , etc. We use the notation of 2.
- On comparing definitions of "weak ncategory" by M. Makkai, McGill University
- Assignment 9/MATH 247/Winter 2010 Due: Wednesday, April 14
- 5. The adjunction The counit map
- 4. Specifying operations defined as adjoints F. W. Lawvere's discovery that underlies categorical logic is that "all logical operations arise
- 5. Omega--dimensional universal properties The fundamental idea of multitopic categories, borrowed from the Baez/Dolan "opetopic"
- 2. Multicategories For # ={0, 1, 2, ...} , we write [1, # ] for the set {1, 2, ..., # } ; [1, # ]=
- McGill University, Department of Mathematics and Statistics Honours Applied Linear Algebra, MATH 247/Winter 2010
- On weak higher dimensional categories by Claudio Hermida#,# and Michael Makkai#,#
- The multitopic w--category of all multitopic w--categories by M. Makkai!(McGill University)
- Honour List of Mathematical Logic Karl Weierstrass (1815-1897): made analysis rigorous by the use of logic; by using
- 1. Anafunctors Let X and A be categories. An anafunctor F with domain X and codomain A , in
- On comparing definitions of "weak n--category" by M. Makkai, McGill University
- 6. Multitopic sets An multitopic set S , by definition, consists of data (i) to (iii), subject to conditions (iv) to
- 1. An informal description 1.1. n--graphs and multitopic sets.
- 9. Sketch--semantics versus Tarskian semantics In this section, I explain how Tarskitype semantics can be related to sketchsemantics. This
- 2. The multitopic nerve of an w--category The purpose of this section is to define a functor
- commutes. One has a functionreplacement multicategory (D, W, j, y) associated with (C, , r, s) , and another one, (D, W, j, y) , associated with (C, , r, s) .
- Assignment 3/MATH 247/Winter, 2010 Due: Friday, January 29
- [B] M. Batanin, Monoidal globular categories as a natural environment for the theory of weak ncategories. Advances in Mathematics 136 (1998), 39103 .
- 6. Many--to--one computads Let n , X be an (n1)category. Let U be a set, and u @du, u @cu two functions
- 1 The multicategory of function replacement This section is an essentially selfcontained reworking of the construction of the entity named
- The word problem for computads by M. Makkai
- 3. Colored multitopes. The question how to define the concept of weak ncategory, for finite n , and in the second
- McGill University, Department of Mathematics and Statistics History and Philosophy of Mathematics, MATH 338, Fall 2009
- 3. Anabicategories In a twodimensional category, for a given pair of objects (0cells), the totality of arrows
- 5. Exactness properties Exactness properties of structured categories abound in category theory; "coproducts are stable
- Sample questions for exams: second installment (See the "first installment" for general remarks.)
- Rearranging colimits: A categorical lemma due to Jacob Lurie 1 Statement of the result
- On comparing definitions of "weak ncategory" by M. Makkai, McGill University
- 7. The sketch--specification of monoidal categories The difficulty in sketchspecifying the concept of monoidal category is with arranging that the
- The associative law for the W--composition Let us verify the associative law
- 1. FOLDS--signatures and FOLDS--equivalence FOLDS is an acronym for "first order logic with dependent sorts"; see [M2]. FOLDS has a
- The unit map We fix an arbitrary ms R ; let Y = R as constructed in 4 ; and let M = [Y] as
- 4. The free w--category on a multitopic set We construct Y= R , the free wcat on the arbitrary ms R .
- 2. Multitopes and multitopic sets Here, I will recall the notions in the title; they are dealt with in [H/M/P] in complete detail.
- 6. Equivalence of two virtual concepts of category. I want to elaborate on the natural consequence of the presence of FOLDS equivalence for an
- First Order Logic with Dependent Sorts, with Applications to Category Theory
- The multitopic category of all multitopic categories by M. Makkai (McGill University)
- The word problem for computads by M. Makkai
- Computads and 2 dimensional pasting diagrams (April 23, 2007)
- 11 RRReeeggguuulllaaarrr cccaaarrrdddiiinnnaaalllsss In what follows, , , , , always denote cardinals.
- Introduction The papers [H/M/P] and [M8] describe a concept of "weak higher dimensional category", the
- (October 27) 1. We define an axiomatic system, called the First-Order Theory of Abstract Sets
- Assignment 1/MATH 247/Winter, 2010 Due: Monday, January 11
- Assignment 2/MATH 247/Winter 2010 Due: Thursday, January 21
- Assignment 7/MATH 247/Winter, 2010 Due: Friday, March 19
- Assignment 8/MATH 247/Winter, 2010 Part 1. Notes on systems of differential equations
- Assignment 8/MATH 247/Winter 2010 Due: Tuesday, March 30
- MATH 247/Winter, 2007 A calculation of a Fourier series
- MATH 247/Winter 2010 Notes on the adjoint and on normal operators.
- McGill University, Department of Mathematics and Statistics Mathematical Logic 2, MATH 592/Winter 2010
- Take-home final for Mathematical Logic 2, MATH 592, Winter 2010 Part 1. Model theory
- MATH 338/History and Philosophy of Mathematics/Fall 2009 Assignment 1
- Assignment 3/Math 338/Fall, 2009 Due: Monday, October 5
- Sample questions for exams: first installment (More will be added later. The present list covers only a part of the geometry we
- McGill University/Department of Mathematics and Statistics History and Philosophy of Mathematics/Midterm examination
- 6. Equivalence of two virtual concepts of category. I want to elaborate on the natural consequence of the presence of FOLDS equivalence for an
- Assignment 7/MATH 338/Fall 2009 Due: Wednesday, November 11
- 1. Categories of sketches Let G be a category, and K= K a (small) indexed set of objects K of G . We
- On Gabbay's proof of the Craig interpolation theorem for intuitionistic predicate logic
- 4. The anabicategory of saturated anafunctors The saturated composition of two saturated anafunctors (or sanafunctors, for short) is obtained
- [1] J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series 189, Cambridge University Press, 1994.
- We turn to the proofs of the multicategory laws in E . Remember: n is fixed, and we work with the induction hypothesis that "all is well up to n1 ".
- 5. A 2--level multicategory with non--standard amalgamation Let # be a language, and C a (not necessarily standard, but 1level) multicategory free over
- [B/D1] J. Baez and J. Dolan, Higherdimensional algebra and topological quantum field theory, Journal of Mathematical Physics 36 (1995), 60736105.
- Assignment 5/MATH 247/Winter 2010 Due: Friday, February 19 in class (!)
- Egyptian fractions revisited by M. Makkai
- Assignment 9/MATH 338/Fall 2009 Due: Thursday, December 3
- (October 14/2010) 1. Well-foundedness
- [B1] M. A. Batanin, Monoidal Globular Categories As a Natural Environment for the Theory of Weak n-Categories. Advances in Mathematics 136 (1998), 39-103.
- 6. Further examples In this section, we will see that the method of specifying a doctrine by sketches is not
- 5. The effects of weak versions of the axiom of choice. In the previous parts of the paper, we left open whether the bicategory AnaCat is Cartesian
- 3. Formal deductions of sketch--entailments For motivation, we first consider the simple example of a possible way the category S
- T(b)=T( a (p)) . (5) The intuitive idea behind the operation W , called functionreplacement, is that aW b is the
- 5. Omega--dimensional universal properties The fundamental idea of multitopic categories, borrowed from the Baez/Dolan "opetopic"
- Introduction A completeness theorem asserts the equality of the formal deducibility relation and the
- Assignment 4/MATH 247/Winter 2010 Due: Tuesday, February 9
- 6. The transfors Having defined what the 0cells of the (large) multitopic set MltwCat are, we now
- Introduction In [B/D2] and [B/D3], John C. Baez and James Dolan have introduced a concept of weak
- 3. Simultaneous composition Part 1: Simultaneous composition in a general multicategory
- Assignment 2/MATH 338/Fall 2009 Due: Wednesday, September 23
- Assignment 8/MATH 338/Winter, 2009 Due: Friday, November 20
- 2. Sketch semantics Let S be an arbitrary category; we talk about "sketches" when referring to objects of S since
- Assignment 6/MATH 247/Winter 2010 Due: Thursday, March 4
- Introduction The papers [H/M/P] and [M8] describe a concept of "weak higher dimensional category", the
- 6. The transfors Having defined what the 0cells of the (large) multitopic set MltwCat are, we now
- Invariance of (W, y) under renaming transformations Although it is trivial to prove, perhaps it's worth pointing out that the definition of the
- Part 3: Simultaneous composition for pasting diagrams in a multitopic set Now, let's make the context even more special. Let R be an arbitrary ms; we will use the