I. The Dimensional Ladder 0. Introduction a. The Ladder of n-Categories A story about climbing the dimensional ladder, without detailed definitions but with pictures of: sets functions diagrams of sets and functions categories functors natural transformations diagrams of categories, functors and natural transformations 2-categories 2-functors 2-natural transformations modifications diagrams of 2-cats, 2-functors, 2-nat trans and modifications 3-categories The categorical imperative: understanding n-categories requires that we use (n+1)-categories. The expository difficulties this causes. For example: a fairly general concept of monoid lives in a monoidal category, but a monoidal category is itself a kind of weakened monoid in Cat. (This is an example of the microcosm principle.) b. The Periodic Table Picture of the periodic table. A category with one object is a monoid: picture. The natural numbers as the free monoid on one generator: picture of arrows - x --->--- x x --->--- x --->--- x x --->--- x --->--- x --->--- x and Poincare dual picture of rocks on a line. A 2-category with one object is a monoidal category: pictures. The free monoidal category on one generator - objects as above, and only rather boring morphisms (e.g. associators). The laws of a monoid hold up to isomorphism. Coherence law for associator. A 2-category with one object and one morphism is a commutative monoid: picture. The natural numbers as the free commutative monoid on one generator: dots in a square. The commutative law arises because we can switch rocks past each other. A 3-category with one object and one morphism is a braided monoidal category. Here we see the process of switching some rocks past each other - the braiding. The coherence law satisfied by the braiding and associator. Warning about subtleties (already in commutative monoid case!) 1. Categories a. Categories of Mathematical Objects Definition of category There are many categories of mathematical gadgets, but we'll consider three: Set, Vect and Top, the latter two because they don't arise very quickly from category theory itself. We will downplay this aspect of category theory because it's traditionally been emphasized to the exclusion of all other aspects. Definition of inverse morphism, isomorphism. Universal properties and how they specify an object up to a specified isomorphism. In each case show what they amount to for Set, Vect and Top. Terminal object, initial object Product, coproduct Concept of opposite category - unifying the above pairs Equalizer, coequalizer Pullback, pushout Limit, colimit Theorem: a category with finite products and equalizers has finite limits. Theorem: a category with pullbacks and a terminal object has finite limits. Definition of decategorification: the class |C| of isomorphism classes of objects in C. |FinSet| = |FinVect| = N, |Set| = Card In Set, initial object and coproduct decategorify to 0 and + In Set, terminal object and product decategorify to 1 and x We shall study this "categorified arithmetic" in much more detail in Part III. Subtraction and division are much subtler. Very often mathematical structures arise via decategorification! b. Categories as Mathematical Objects kinds of categories: groupoids, monoids groups, discrete categories, preorders, partially ordered sets (or posets) presenting a category by listing objects, morphisms and equations. Just like presenting a group but with an extra "layer" - the objects. examples: the free category on an object, a morphism, an isomorphism, an endomorphism (N) and an automorphism (Z). example: the free category on a composable chain of n morphisms is called [n], and it looks like an (n-1)-simplex. exercise: the free category on an object is all of the above the free category on a morphism is a preorder the free category on an iso is a groupoid the free category on an endo is a monoid the free category on an auto is a monoid, a groupoid and a group (make a 5x5 grid to fill out) c. Categories from Spaces The fundamental groupoid of a topological space The fundamental group of a pointed space categories from chain complexes: a 2-term chain complex is a category in AbGp. Homology from homotopy: take the fundamental groupoid, linearize it and impose relations saying that composition of morphisms equals addition to get a 2-category in AbGp. Do this very sketchily! The nerve of a category (done quickly) example: the nerve of [n] is a simplex The classifying space of a group (done quickly) d. Functors Definition Examples: a functor from the free category on an object, a morphism, an isomorphism, an endo, an auto Example: a functor from a group G to Set is a set acted on by G, or G-set. Example: a functor from a monoid M to C is an object of C acted on by M. Examples: the forgetful functor from Vect to Set and its adjoint, with sketchy definition of adjoint using the fact that there's a "natural" one-to-one correspondence between hom(Fx,y) and hom(x,Uy) the forgetful functor from Top to Set and its adjoints left adjoints tend to preserve colimits (and conversely); right adjoints tend to preserve limits (and conversely) - reference to later section. e. The Category of Categories Cat: a slightly mindboggling concept, laying the ground for lots of fun self-reference and potential level slips. A little digression on size issues. The full subcategories Mon, Preord, Poset, Grp, Gpd exercise: what's a functor from a group to Grp? or from a monoid to Mon? More fun with of level slips! Example: the fundamental groupoid functor from Top to Gpd Example: the nerve functor from Gpd to Top. Not quite adjoint to the previous one, but "morally" so. More generally, the nerve functor from Cat to Top. Example: the underlying graph of a category as a functor U: Cat -> Grph the free category on a graph as a functor F: Grph -> Cat Example: the terminal and initial categories Example: the product and coproduct of categories Example: the hom-functor hom: C^{op} x C -> Set f. Natural Transformations Definition with Pictures Examples: a homotopy between maps between spaces gives a natural transformation between functors between their fundamental groupoids! Natural transformations between mathematical constructions: Natural isomorphism between identity functor on Vect and double dual. Natural isomorphism between various "ordered pair" functors from Set^2 -> Set... i.e., various ways of making the product into a functor! Fundamental group versus first homology group of a pointed space. Natural transformations as functors F: I x C -> C where I is the free category on a morphism. g. Equivalence of Categories Definition Examples from duality: Vect is equivalent to Vect^{op} some sort of finite posets are equivalent to distributive lattices^{op} Compact Hausdorff spaces is equivalent to commutative C*-algebras^{op} Locally compact abelian groups is equivalent to itself^{op} Skeletal categories Theorem: a category is equivalent to any of its skeleta. h. Functor Categories We've seen that given categories C and D, there's a set hom(C,D) consisting of functors from C to D. But in fact we can do better: there's a category hom(C,D) whose objects are functors from C to D and whose morphisms are natural transformations between these! A functor F: C -> D gives a "representation" of C in D, which is like a picture of C in D (draw a picture!) and we can understand things about C from this picture. A natural transformation gives a "change of pictures". Example: category of actions of a group G is hom(G, Set). Objects are sets acted on by G, or G-sets. Eventually we'll show how to completely recover a group from all its actions (?). Example: category of representations of a group is hom(G, Vect). Doplicher-Roberts theorem says one can completely recover a finite group from this category (with its extra structure). Example: a functor from a group to Top is a continuous action Example: more generally, a functor from a monoid to C is an action of the monoid on some object of C. example: category of representations of various quivers (free categories on graphs) i. Cat as the Primordial 2-Category Composition of functors, vertical and horizontal composition of natural transformations, and the rules they satisfy, done using diagrams. Or: composition of functors, vertical composition of natural transformations, and left/right whiskering, and the rules they satisfy. Show the two setups are equivalent. 2. 2-Categories a. Strict 2-Categories Nuts-and-bolts definition of strict 2-category based on example of Cat, in both styles. Examples: The 2-category of groupoids, the 2-category of categories with finite products and product-preserving functors, the 2-category of categories with finite limits and finite-limit-preserving functors, and other "doctrines". The strict fundamental 2-groupoid of a space: "taking the pictures seriously". A brief treatment, pointing out nuisances due to strictness. 2-categories from chain complexes: a 2-term chain complex is a category in AbGp, while a 3-term chain complex is a 2-category in AbGp. Homology from homotopy: take the fundamental 2-groupoid, linearize it and impose relations saying that composition (of all forms) equals addition to get a 2-category in AbGp. Do this very sketchily! The 2-category generated by a 2-computad. 2-categories of planar string diagrams - brief treatment, pointing out nuisances due to strictness. Describe the 2-category generated by a 2-computad Definition: a strict monoidal category is a 2-categories with one object. A commutative monoid is a strict monoidal category with one object. Also give nuts-and-bolts definitions! Strict monoidal categories of planar diagrams, pointing out nuisances due to strictness. b. Weak 2-Categories (Bicategories) Point out problems of strictness. For example, how the fundamental 2-groupoid doesn't want to be strict: introduce the associator, left/right uniters, and their coherence laws. Definition of weak 2-category. Example: the weak fundamental 2-groupoid of a space. Weak 2-categories of planar diagrams, pointing out nice features due to weakening. Example of Mod, the weak 2-category of rings, bimodules and bimodule morphisms. We could also use monoids, biactions and biaction morphisms. Example of Span, the weak 2-category of sets, spans and maps of spans. Definition of weak monoidal category as weak 2-category with one object. Also, nuts-and-bolts definition. Examples of Vect, RMod (R commutative) or RBiMod with its tensor product. Examples coming from categories with finite products / coproducts. The "all diagrams commute" version of MacLane's theorem - sketch of proof using associahedron. c. Monads and Adjunctions Monads in a bicategory. Example: monads in Vect are algebras. Adjunctions in a bicategory. Monads from adjunctions. Examples: Adjunctions in Vect are dual vector spaces; they give matrix algebras. Other examples of monoids in a monoidal category: Top gives topological monoids Vect gives algebras AbGp with its tensor product gives rings Cat gives strict monoidal categories! Monads in a bicategory: Span gives categories Adjoint functors in Cat give monads. Algebras of monads Comonads and their coalgebras d. Limits and Colimits How limits and colimits are examples of adjunctions and vice versa Kan extensions; how these are example of adjunctions and vice versa. As much as possible do all this "formally" in a 2-category? e. BiCat as the Primordial 3-Category Weak versus strict functors between bicategories. Weak versus strict natural transformations. f. 2Cat versus BiCat - coherence theorems MacLane's coherence theorem for 2-categories One version: "all diagrams commute". The associahedron. Another version: Every weak 2-category is equivalent to a strict one. Proof via categorified Yoneda??? Another version: the strict 2-category [strict 2-categories, strict 2-functors, pseudonatural transformations] is equivalent to the strict 2-category [weak 2-categories, weak 2-functors, weak natural transformations]? Perhaps be a bit sketchy and save details of proofs for later? g. Enriched Categories The definition of enriched category; examples. 2-Categories as enriched categories The category VCat of V-categories and V-functors The 2-category of V-categories, V-functors and V-natural transformations If V has finite products so does VCat If V is Cartesian closed so is VCat, and V is a V-category If V has finite limits so does VCat? If V is a distributive category so is VCat?? If V is a braided monoidal category we can define a monoidal V-category If V is a symmetric monoidal category we can define a symmetric monoidal V-category Hints of higher patterns... Exercise: under what conditions does a category enriched over commutative monoids get enriched over abelian groups? This is discussed in Freyd's work on abelian categories and Ab-categoriges, and/or MacLane's Cats for the Working Mathematician. h. Internal Categories The definition of internal category; examples double categories as internal categories 2-categories versus double categories 3. 3-Categories a. Strict 3-Categories General definition of strict n-categories... and unravelled in case n = 3. 3-categories from chain complexes A strict 2-category with one object is a strict monoidal 2-category (also a monoid in $(2Cat, \times)$). A strict 2-category with one morphism is a commutative strict monoidal category... a notion too strict! b. Semistrict 3-Categories Gray tensor product of 2-categories; semistrict 3-categories are categories enriched over $(2\Cat, \tensor)$. The fundamental 3-groupoid of a topological space. $\Pi_3(S^2)$ as an example. A semistrict 3-category with one object is a semistrict monoidal 2-category (also a monoid in $(2\Cat, \tensor)$). A semistrict 3-category with one morphism is a strict braided category. Definition of strict symmetric category. A glimpse of the periodic table. c. Tricategories Definition. Note how things are getting unmanageable, but also note some patterns. The 3d associahedron shows up here. A tricategory with one object is a monoidal bicategory. A semistrict 3-category with one morphism is a strict braided category. Examples of strict braided and symmetric categories: every category with finite products or coproducts gives a symmetric monoidal category RMod with R a commutative ring is a symmetric monoidal category d. Weak Monads and Adjunctions The concept of weak monad or "pseudomonad" in a tricategory, or perhaps just in a semistrict 3-category. Pictures! Example: a monoidal category is a weak monad in the monoidal 2-category $(\Cat, \times)$. The concept of weak adjunction. The swallowtail coherence law. How weak adjunctions give weak monads. e. Weak Limits and Colimits f. Enrichment and Internalization? 3 levels of "internalization" for 2-categories? g. The Lax World Lax 2-functors, lax monoidal functors: how a lax functor between bicategories is good enough to transport a monad from one to the other. How a lax functor from the terminal bicategory to C is precisely a monoid in C. (Oplax...) Lax limits. How a lax fixed point for an endomorphism T: x -> x in Cat is an algebra for T. How the lax equalizer (?) of T and 1 is the Eilenberg-Moore category. h. Tricat as the Primordial 4-category strict versus weak 3-functors... etc. i. Coherence Theorems Every tricategory is equivalent to a semistrict one, but not to a strict one. 4. 4-Categories 5. Case Studies a. 2-Braids and 2-Tangles Braids as the free braided monoidal category on one object Tangles as the free braided monoidal category with duals on one object (do 1-braids and 1-tangles in dimensions 1, 2, 3, 4) The issue of framing. b. Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp. braided monoidal, symmetric monoidal) category with duals of representations of a Hopf algebra (resp. quasitriangular Hopf algebra, triangular Hopf algebra) Examples of quasitriangular Hopf algebras and their tangle invariants c. Real Numbers, Complex Numbers, Quaternions and Octonions cross product algebras - the 0,1,3,7 theorem d. Electrical Circuits linear algebra over the rig of costs e. Logic Gates linear algebra over the rig of truth values f. 2-Braids and 2-Tangles (do 2-braids and 2-tangles in dimensions 2,3,4,5,6? g. Quivers and Dynkin Diagrams (move to "Categories as Theories", section on "Category Representations".) Representations of quivers: proof that quivers containing certain "bad" subquivers have wild representation type, by noting that the dimension of the space of indecomposable representations exceeds the dimension of the space of intertwiners. Note that this problem is a categorified version of the problem of (a certain sort of) Laplacians on graphs. It's possible that this whole section should wait until part II, where we may introduce the incidence geometry and symmetry aspects of Dynkin diagrams, and have more to play with. But, to really get some synergy here, we'd have to figure out why the quivers with tame representation type are precisely those of finite-dimensional semisimple Lie algebras! And this remains mysterious to me...

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