Index

a category

Cat, 186

FLin, 167

Fin, 163, 239

Fun(C,D), 223

Grp, 164

Grpd, 205

Grph, 168

Kls(T), 351

Mon, 163

Oprd, 367

PrO, 164

Prop, 208

Sch, 243

Set, 163

Starn, 265

Top, 203

Vect, 204, 335

Δ, 238, 286

CSet, 230

GrIn, 197

free arrow, 187

terminal, 189

a functor

CatCoreGrpd, 205

CatHomSet, 190

CatObSet, 189, 224, 301

CatGrph, 188, 301

CatOprd, 368

CatSch, 246

FLinPrO, 176

GrpCat, 192

GrpGrpd, 205

GrpMon, 175, 301

GrpdCat, 205

GrphPathsGrph, 183, 187, 221, 242

GrphCat, 187, 301

GrphPrO, 178, 196

GrphSet, 178, 222, 301

Kls(T)Set, 352

MonCoreGrp, 301

MonCat, 191

MonSet, 174, 297

PrOCat, 195, 196, 237, 255, 261

PrOGrph, 176, 196, 302

PrOSet, 178, 301

PrOTop, 342

SchCat, 245

SetDiscCat, 189, 224, 301

SetDiscGrph, 188

SetIndCat, 292, 301

SetListSet, 212

SetMon, 181, 297

SetPrO, 301

TopΠ1Grpd, 206

TopPrOop, 203

TopSet, 203

VectGrp, 204

VectPrO, 204

VectSet, 205

VectTop, 204

ΔFLin, 238

SetDiscCat, 224

a group

E3, 115

GL3, 115

U(1), 117

dihedral, 115, 194

a monad

Paths, 351

exceptions, 350

List, 348

maybe, 345

partial functions, 345

a schema

Loop, 253, 359

department store, 149

indexing graphs, 233

a set

ℝ, 11

⩾0, 11

{☺}, 11

n, 20

ℕ, 10

ℤ, 10

a symbol

(FG), 293

<f, g>, 40

X/∼, 64

[n], 134

{fg, 46

Fun, 223

HomSet, 15

HomC, 162

ℕ, 10

Ob, 162

Ω, 82

ℙ, 78

Path, 125

ℝ, 39

ℤ, 10

art, 101

○, 13, 162

colim, 278

◊, 228

∅, 10

∃, 10, 303

∃!, 10

∀, 11, 303

idX, 16

∫, 287

≌, 17

⊲, 265

lim, 273

⌟, 49

↦, 12

Cop, 285

C/X, 273

CX/, 278

++, 97

》, 33

⊳, 266

∼, 63

≃, 31, 236

⊔, 43

×, 37

⌜, 68

f−1, 54

≔, 11

a warning

“set” of objects in a category, 163

different worldviews, 26

misuse of the, 270

notation for composition, 31

operad functors, 367

operads vs. multicategories, 362

oversimplified science, 6

action

left, 101

of a group, 117

of a monoid, 101

orbit of, 118

right, 101

action table, 108

adjoint functors, 297

adjunct, 299

adjunction, 298

adjunction isomorphism, 299

analogy: babies and adults, 298

counit, 360

unit, 360

algebra

operad, 368

an operad

Sets, 366

little n-cubes, 365

little squares, 365

relations, 370

wiring diagrams, 372

appropriate comparison, 110, 126, 142, 162, 174

arrow, 119

Baez, John, 5

biological classification, 143

canonical, 19

cardinality, 20

category, 162

arithmetic of, 295

as equivalent to schema, 241

cartesian closed, 210

cocomplete, 281

comma, 292

complete, 281

coslice, 278

discrete, 188, 301

equivalence of, 236

free category, 187, 336

indiscrete, 301

Kleisli, 351

non-example, 165, 166

of elements, 287

of functors, 223

opposite, 285

presentation, 200

questionable, 164

slice, 273

small, 163

underlying graph of, 187

CCCs, 210

characteristic function, 82

clunky, 42

coequalizer, 72

colimit, 278

closed under, 319

common ground, 342

commuting diagram, 21

component, 213

composition

classical order, 15, 31

diagrammatic order, 15, 31

of functions, 13

of morphisms, 162

concatenation

of lists, 97

of paths, 126

cone

left, 265

right, 266

congruence, 151

on a monoid, 97

connected component, 67, 186

context, 344

coproduct

inclusion functions, 43

coproducts, 257

of sets, 43

universal property for, 45

core, 205, 206

correspondence

one-to-one, 17

coslice, 278

cospan, 260, 261, 274, 293

currying, 74

as adjunction, 301

via data migration functors, 318

data, 6

valid time, 343

data migration, 308

left pushforward Σ, 312

pullback Δ, 309

right pushforward Π, 315

database

business rules, 149

category of instances on, 230

foreign key, 148

homomorphism, 235

instance, 157, 201

Kleisli, 357

primary key, 148

schema, 149, 153

tables, 147

descent data, 340

diagam

commutes, 21

diagram, 262

in Set, 21

Dolan, James, 5

dynamical system

continuous, 204

discrete, 153

Eilenberg, Samuel, 4

element, 9

represented by a function, 14

Englishification, 33, 158, 290

entry

in list, 97

epimorphism, 321

in Set, 85

equalizer, 62, 275

equivalence relation, 63

as partition, 64

equivalence classes, 63

generated, 65

quotient by, 64

trivial, 66

exceptions, 350

exponentials

evaluation of, 75

exponentials

in Set, 74

fiber product, 49

fiber sum, 68

finite state machine, 106, 291

FQL, 309

function, 11

bijection, 83

codomain, 11

composition, 13

domain, 11

equality of, 15

identity, 16

induced, 40

injection, 83

inverse, 17

isomorphism, 16

representing an element, 14

surjection, 83

functor, 174

adjoint, 298

constant, 221, 306

contravariant, 284

covariant, 284

faithful, 240

forgetful, 176

full, 240

representable, 322

representing an object, 224

functorial query language, 309

gateway, 254

generators, 97

geography, 145, 336

graph, 119

as functor, 197

bipartite, 61

chain, 122

converting to a preorder, 135

discrete, 122

free category on, 187, 336

homomorphism, 126

indiscrete, 122

paths, 125

paths-graph, 183, 351

symmetric, 198

graph homomorphism

as functor, 232

Grothendieck

construction, 286

expanding universes, 163

in history, 4

group, 114

action, 117

as category, 192

homomorphism, 119

of automorphisms, 193

groupoid, 205

fundamental, 206

of material states, 205

hierarchy, 154

hom-set, 162

homomorphism

database, 235

graph, 126

group, 119

monoid, 98, 110

iff, 67

image, 13

in olog, 34

inclusion functions, 43

indexed set, 90, 91

as functor, 232

indexing category, 262

infix notation, 95

information theory, 211

initial object, 267

in CSet, 320

instance, 157, 201

Kleisli, 357

isomorphism, 169

of sets, 16

join, 139

Joyal, André, 5

Kan extension

left, 312

right, 315

Kan, Daniel, 5

Kleisli category, 351

labeled null, 314

Lambek, Joachim, 5

laws

category, 162

functor, 174

monad, 347

monoid, 94

monoid action, 101

natural transformation, 214

operad, 364

operad-functor, 367

Lawvere, William, 4

leaf table, 201

limit, 273

closed under, 319

linear order

finite, 134

list, 96, 348

as functor, 181

concatenation, 97

local-to-global, 4, 211

Mac Lane, Saunders, 4

Markov chain, 359

materials

force extension curves, 74

force-extension curves, 12

meet, 139

Moggi, Eugenio, 5

monad, 344, 347

formalizing context, 344

Kleisli category of, 351

on Grph, 351

on Set, 347

on arbitrary category, 361

monoid, 94

action, 101

additive natural numbers, 95

as category, 190

commutative, 96

cyclic, 100

free, 97, 181

generators, 97

homomorphism, 110

initial, 268

inverse of an element in, 114

multiplication formula, 94

of endomorphisms, 193

olog of, 105

presented, 98

terminal, 268

trivial, 96

trivial homomorphism, 111

unit element of, 94

monomorphism, 321

in Set, 85

morphism, 162

inverse, 169

multicategory, 362

multiset, 88

natural isomorphism, 225

natural transformation, 213

as functor, 276

as refinement of model, 218

for adding functionality, 227

horizontal composition of, 228

interchange, 229

questionable, 214

vertical composition of, 223

whiskering of, 228

object

represented by a functor, 224

olog, 22

as database schema, 155

aspects, 25

facts, 30

facts in English, 32

images, 34

invalid aspects, 25

path in, 30

relational, 357

rules, 24, 29, 153

sheaf of, 341

types, 23

underlying graph, 120

one-to-one correspondence, 17

open cover, 339

operad

algebra of, 368

colored, 362

morphism of, 367

orbit, 118

rotating earth, 117

order, 132

linear order, 132

morphism, 142

opposite, 141

partial order, 132

preorder, 132

tree, 140

partial function, 345

partial functions, 345

path, 125

PED, 151

permutation, 116

power-set, 78

as poset, 136

preimage, 54, 303

preorder

as category, 194

clique in, 137

converting to graph, 134

discrete, 142

generated, 137

indiscrete, 142

join, 139

meet, 139

presheaf, 338

product

as grid, 38

projection functions, 38

products, 250, 254, 271

as not always existing, 255

of sets, 37

universal property for, 40

projection functions, 38

pullback, 274

of sets, 49

pushout, 278

of topological spaces, 282

RDF, 286

as category of elements, 288

relation

binary, 129

equivalence, 63

graph of, 130

relative set, 89, 90

as slice category, 277

representable functor, 322

representation theory, 335

representative

of an equivalence class, 64

restriction of scalars, 113

retraction, 73

RNA transcription, 17

schema, 153

as category presentation, 199, 200

as equivalent to category, 241

as syntax, 199

congruence, 151

fact table, 310

leaf table, 150, 310

morphism, 243

of a database, 149

Path equivalence declaration (PED), 151

schematically implied reference spread, 323

security, 144

set, 9

arithmetic of, 76

Lawvere is description of, 210

numeral, 20

permutation of, 116

set builder notation, 10

sheaf

condition, 339

descent data, 340

sheaves, 337

simplex, 79

simplicial complex, 79, 342

simplicial set, 286

Skolem, 323

Skolem variable, 314

slice, 273

space, 145, 202

topological, 202

space group, 116

span, 58

composite, 59

stereotype, 26

subcategory

full, 166, 291

subobject classifier

in CSet, 331

in Set, 81

subset, 10

as function, 12

characteristic function of, 82

complement, 82

subway, 283

symmetry, 115

terminal object, 267

in CSet, 320

in Set, 62

topological space, 203

topology, 202

topos, 331

tree, 140

root, 140

trivial homomorphism

of monoids, 111

universal property, 254

products, 40

pullback, 274

vector field, 168, 207

conservative, 207

vector space, 204, 335

vertex, 119

wiring diagram, 372

Yoneda’s lemma, 327