CATEGORIES, ALLEGORIES

Peter J. FREYD Andre SCEDROV University of Pennsylvania Philadelphia, PA 19104-6395, USA

m 1990

NORTH-HOLLAND AMSTERDAM • NEW YORK • OXFORD • TOKYO CONTENTS (and introduced notions) In the list of notions, alternate words appear in brackets.

Chapter One: CATEGORIES

1.1. Basic definitions 3 1.1 , , source, target, composition 3 1.11 ESSENTIALLY ALGEBRAIC THEORY 3 1.12 directed equality 3 1.13 IDENTITY MORPHISM 3 1.14 MONOID 4 1.15 DISCRETE CATEGORY 4 1.17 LEFT-INVERTIBLE, RIGHT-INVERTIBLE, ISOMORPHISM, INVERSE, GROUPOID, GROUP 5 1.18 FUNCTOR, separating functions 5 1.182 CONTRAVARIANT FUNCTOR, OPPOSITE CATEGORY, COVARIANT FUNCTOR 6 1.1(10) ISOMORPHISM OF CATEQORIES 6

1.2. Basic examples and constructions 7 1.2 object, proto-morphism, SOURCE-TARGET PREDICATE [ARROW PREDICATE] 7 1.22 category of. . . , category composed of . . . 8 1.241 CATEGORY OF SETS 9 1.242 CATEGORY OF GROUPS 9 1.243 FOUNDED (one category on another), FORGETFUL FUNCTOR, CONCRETE CATEGORY, UNDERLYING SET FUNCTOR 9 1.244 underlying set 10 1.245 PRE-ORDERING 10 1.246 group as a category, POSET 10 1.251 ARROW NOTATION, puncture mark 10 1.26 SLICE CATEGORY 11 1.261 category of rings, category of augmented rings 12 1.262 LOCAL HOMEOMORPHISMS, LAZARD SHEAVES 12 1.263 counter-slice category, category of pointed sets, category of pointed Spaces 12 1.27 SM ALL CATEGORY, FUNCTOR CATEGORY, NATURAL TRANSFORMATION, CONJUGATE 13 1.271 CATEGORY OF M-SETS, RIGHT A-SET 13 1.272 CAYLEY REPRESENTATION 14 1.273 LEFT A-SET 14 1.274 NATURAL EQUIVALENCE 15 x CONTENTS

1.28 IDEMPOTENT 15 1.281 SPLIT IDEMPOTENT 15 1.283 STRONGLY CONNECTED 16 1.284 PRE-FUNCTOR 16

1.3. Equivalence of categories 17 1.31 EMBEDDING, FÜLL FUNCTOR, FÜLL , REPRESENTATIVE IMAGE, EQUIVALENCE FUNCTOR 17 1.32 STRONG EQUIVALENCE 17 1.33 REFLECTS (properties by functors), FAITHFUL FUNCTOR 17 1.332 contravariant Cayley representation, functor 18 1.34 ISOMORPHIC OBJECTS 18 1.35 FORGETFUL FUNCTOR, grounding, foundation functor 19 1.36 INFLATION, INFLATION CROSS-SECTION 19 1.363 EQUIVALENT CATEGORIES 20 1.364 SKELETAL, SKELETON, COSKELETON, support of a permutation, transposition 20 1.366 EQUIVALENCE KERNEL 21 1.372 ideal, downdeal, updeal 22 1.373 SECTION OF A SHEAF, PRE-SHEAF, GERM, STALK, ADJOINT PAIR, LEFT ADJOINT, RIGHT ADJOINT, ASSOCIATED SHEAF FUNCTOR 23 1.374 consistent, realizable (subsets of a pre-sheaf), complete pre-sheaf 25 1.38 DUALITY 26 1.381 category composed of finite lists 26 1.384 category of rings, category of augmented rings 26 1.389 STONE DUALITY, STONE SPACE 28 1.39 linearly ordered category 28 1.392 FINITE PRESENTATION 29 1.395 ß-SEQUENCE, SATISFIES (a ß-sequence), COMPLEMENTARY ß-SEQUENCE 30 1.398 tree, rooted tree, root, length of a tree, sprouting, ß-tree 32 1.399 mapping-cylinder 33 1.3(10)4 good, nearly-good, stable, coextensive, 5-coextensive (ß-trees) 35 1.3(10)6 C-stability (of ß-trees) 36

1.4. Cartesian categories 37 1.41 MONIC [, mono, injection, inclusion, monic morphism] 37 1.412 monic family, TABLE, COLUMN, TOP, FEET, RELATION, , VALUE [SUBTERMINATOR] 38 1.413 CONTAINMENT (of tables) 39 1.415 tabulation, tabulates a relation 39 1.421 TERMINATOR [final object, terminal object] 39 1.423 binary PRODUCT diagram, has binary products 40 1.425 product of a family 41 1.427 support of a functor 42 1.428 EQUALIZER, has equalizers 42 1.43 CARTESIAN CATEGORY [finitely complete, left exact] 43 1.431 PULLBACK diagram, has pullbacks 44 1.437 REPRESENTATION OF CARTESIAN CATEGORIES 46 1.442 REPRESENTABLE FUNCTOR 47 CONTENTS xi

1.444 HÖRN SENTENCE 48 1.451 INVERSE IMAGE 48 1.452 SEMI-LATTICE, entire subobject 49 1.454 LEVEL [kernel-pair, congruence], DIAGONAL, diagonal subobject 50 1.461 fiber, fiber-product 50 1.462 EVALUATION FUNCTORS 50 1.463 conjugate functors 50 1.464 YONEDA REPRESENTATION 51 1.47 special cartesian category 51 1.48 DENSE MONIC, RATIONAL CATEGORY 52 1.49 SHORT COLUMN (of a table), COMPOSITION (of tables) AT (a column) 54 1.491 T-CATEGORY 54 1.492 SUPPORTING (sequence of columns), PRUNING (of a column) 54 1.493 category of ordinal lists 55 1.494 RESURFACING (of a table) 55 1.498 CANONICAL CARTESIAN STRUCTURE 56 1.49(11) AUSPICIOUS (sequence of columns) 58 1.4(10) FREE T-CATEGORY 59 1.4(10)1 WELL-MADE, WELL-MADE PART 59 1.4(11) CANONICAL SLICE 62 1.4(11)4 POINT, GENERIC POINT 64

1.5. Regulär categories 68 1.51 ALLOWS, IMAGE, has images, ADJOINT PAIR (of functions between posets), LEFT ADJOINT, RIGHT ADJOINT 68 1.512 COVER 68 1.514 EPIC [epimorphism] 69 1.52 REGULÄR CATEGORY, PRE- 69 1.521 STALK-FUNCTOR 70 1.522 SUPPORT, WELL-SUPPORTED 70 1.523 WELL-POINTED 70 1.524 PROJECTIVE 71 1.525 CAPITAL 72 1.53 SLICE LEMMA for regulär categories, DIAGONAL FUNCTOR 72 1.54 CAPITALIZATION LEMMA 74 1.541 equivalence condition, slice condition, union condition, directed union 74 1.545 relative capitalization 75 1.55 HENKIN-LUBKIN THEOREM [representation theorem for regulär categories] 77 1.552 special pre-regular category 77 1.56 78 1.561 RECIPROCAL 79 1.563 MODULAR IDENTITY 79 1.564 GRAPH (of a morphism), MAP, ENTIRE, SIMPLE 80 1.565 PUSHOUT 81 1.566 COEQUALIZER 81 1.567 EQUIVALENCE RELATION, EFFECTIVE EQUIVALENCE RELATION, EFFECTIVE REGULÄR CATEGORY 82 1.568 QUOTIENT-OB JECT 82 1.56(10) CONSTANT MORPHISM 83 1.57 CHOICE OBJECT, AC REGULÄR CATEGORY, Axiom of Choice 83 xii CONTENTS

1.572 category composed of recursive functions 84 1.573 category composed of primitive recursive functions 84 1.58 BICARTESIAN CATEGORY, COCARTESIAN CATEGORY, COTERMINATOR [initial object, coterminal object], COPRODUCT, STRICT COTERMINATOR 85 1.581 representation of bicartesian categories 85 1.587 bicartesian characterization of the set of natural numbers 86 1.59 ABELIAN CATEGORY 87 1.591 ZERO OBJECT, ZERO MORPHISM, category with zero, middle- two interchange law, HALF-ADDITIVE CATEGORY, ADDITIVE CATEGORY 87 1.592 KERNEL, COKERNEL 89 1.593 NORMAL SUBOBJECT 89 1.595 abelian group object, homomorphism 90 1.597 EXACT CATEGORY 92 1.598 left-normal, right-normal, normal (categories with zero) 95 1.599 EXACT SEQUENCE, five lemma, snake lemma 96

1.6. Pre-logoi 98 1.6 PRE-LOGOS 98 1.612 DISTRIBUTIVE LATTICE 99 1.614 REPRESENTATION OF PRE-LOGOI 99 1.62 PASTING LEMMA 100 1.623 POSITIVE PRE-LOGOS 102 1.63 slice lemma for pre-logoi 103 1.631 COMPLEMENTED SUBOBJECT, COMPLEMENTED SUBTERMINATOR 103 1.632 GENERATING SET, BASIS 104 1.634 PRE-FILTER, FILTER 105 1.635 REPRESENTATION THEOREM FOR PRE-LOGOI, BOOLEAN ALGEBRA, ULTRA-FILTER 106 1.637 special pre-logos 107 1.638 well-joined category 108 1.64 BOOLEAN PRE-LOGOS 109 1.644 ULTRA-PRODUCT FUNCTOR, ULTRA-POWER FUNCTOR 109 1.645 properness of a subobject 110 1.648 COMPLETE MEASURE, ATOMIC MEASURE 110 1.65 PRE- 111 1.651 AMALGAMATION LEMMA 111 1.658 DECIDABLE OBJECT 114 1.662 DIACONESCU BOOLEAN THEOREM 115

1.7. Logoi 117 1.7 LOGOS 117 1.712 LOCALLY COMPLETE CATEGORY 117 1.72 HEYTING ALGEBRA 118 1.723 LOCALE, category of complete Heyting algebras, category of locales 118 1.727 NEGATION 121 1.728 LAW OF EXCLUDED MIDDLE 121 1.72(10) scone of a Heyting algebra 122 1.72(11) free Heyting algebra, RETRACT 123 1.732 slice lemma for logoi 123 CONTENTS

1.733 COPRIME OBJECT, CONNECTED OBJECT, FOCAL LOGOS 124 1.734 FOCAL REPRESENTATION 124 1.74 GEOMETRIC REPRESENTATION THEOREM FOR LOGOI 125 1.744 DOMINATES, LEFT-FULL 127 1.74(10) FREYD CURVE 129 1.75 STONE REPRESENTATION THEOREM FOR LOGOI 129 1.751 ATOM, ATOMICALLY BASED, ATOMLESS, periodic power 130 1.752 STONE SPACE, CLOPEN 130 1.76 MICRO-SHEAF 132 1.77 TRANSITIVE CLOSURE, TRANSITIVE-REFLEXIVE CLOSURE, TRANSITIVE (PRE-)LOGOS 133 1.772 cr-TRANSITIVE LOGOS, o--TRANSITIVE PRE-LOGOS 133 1.775 EQUIVALENCE CLOSURE, E-STANDARD PRE-LOGOS 134 1.776 representation theorem for countable cr-transitive (pre-)logoi 135

1.8. , Grothendieck topoi, and exponential categories 138 1.81 ADJOINT PAIR OF FUNCTORS, LEFT ADJOINT, RIGHT ADJOINT 138 1.813 REFLECTIVE SUBCATEGORY, REFLECTION 138 1.815 CLOSURE OPERATION 139 1.816 COREFLECTIVE INCLUSION 139 1.818 ADJOINT ON THE RIGHT (LEFT), Galois connection 140 1.82 DIAGONAL FUNCTOR 140 1.821 diagram in one category modelled on another, lower bound, compatibility condition, greatest lower bound 140 1.822 LIMIT, COLIMIT 141 1.823 COMPLETE, COCOMPLETE (category) 141 1.827 CONTINUOUS, COCONTINUOUS (functor) 142 1.828 weak-, WEAK-LIMIT, WEAKLY-COMPLETE 142 1.82(10) PRE-LIMIT, PRE-COMPLETE 143 1.83 PRE-ADJOINT, PRE-REFLECTION, PRE-ADJOINT FUNCTOR, GENERAL ADJOINT FUNCTOR THEOREM 143 1.831 UNIFORMLY CONTINUOUS (functor), MORE GENERAL ADJOINT FUNCTOR THEOREM 144 1.832 POINTWISE CONTINUOUS (functor) 144 1.833 functor generated by the elements, PETTY-FUNCTOR 145 1.834 GENERAL REPRESENTABILITY THEOREM, category of elements 145 1.838 WELL-POWERED CATEGORY, minimal object 146 1.839 cardinality , generated by A through G 147 1.83(10) COGENERATING SET, SPECIAL ADJOINT FUNCTOR THEOREM 148 1.84 GIRAUD DEFINITION OF A GROTHENDIECK TOPOS 148 1.85 EXPONENTIAL CATEGORY [cartesian-closed], EVALUATION MAP 150 1.853 bifunctor 152 1.857 EXPONENTIAL IDEAL, REPLETE SUBCATEGORY 155 1.858 KURATOWSKI INTERIOR OPERATION, open elements, LAWVERE-TIERNEY CLOSURE OPERATION [L-T], Kuratowski closure Operation, closed elements 156 1.859 BASEABLE 156 xiv CONTENTS

.9. Topoi 157 1.9 UNIVERSAL RELATION, POWER-OBJECT, TOPOS 157 1.912 SUBOBJECT CLASSIFIER, universal subobject, CHARACTERISTIC MAP 158 1.919 g-large subobject 161 1.92 SINGLETON MAP 162 1.921 elementary topos 162 1.93 slice lemma for topoi 165 1.931 FUNDAMENTAL LEMMA OF TOPOI 166 1.94 family of NAMED BY, INTERNALLY DEFINED INTERSECTION 168 1.942 NAME OF a subobject 168 1.944 topos has a strict coterminator 170 1.945 topos is regulär 170 1.946 topos is a logos 171 1.947 topos is a transitive logos 171 1.949 INTERNALLY DEFINED UNION, permanent lower (upper) bound 172 1.94(10) WELL-POINTED PART, SOLVABLE TOPOS 172 1.95 topos is a pre-topos 173 1.952 topos is positive 173 1.954 topos has coequalizers 174 1.961 INJECTIVE, INTERNALLY INJECTIVE 174 1.964 VALUE-BASED 175 1.965 INTERNALLY COGENERATES 175 1.966 PROGENITOR 176 1.969 LAWVERE DEFINITION, TIERNEY DEFINITION (of a Grothendieck topos) 177 1.96(11) slice lemma for Grothendieck topoi 178 1.97 BOOLEAN TOPOS 178 1.971 small object 178 1.973 IAC [Internal Axiom of Choice] 179 1.978 ETENDUE 181 1.98 NATURAL NUMBERS OBJECT in a topos 181 1.987 PEANO PROPERTY 185 1.98(10) bicartesian characterization of a natural numbers object 187 1.98(12) ,4-ACTION, FREE A-ACTION 188

1.(10). Sconing 190 1.(10) EXACTING CATEGORY 190 1.(10)1 SCONE 190 1.(10)3 free categories, RETRACT 192 1.(10)4 SMALL PROJECTIVE 192

Chapter Two: ALLEGORIES

2.1. Basic definitions 195 2.1 RECIPROCATION, COMPOSITION, INTERSECTION, semi distributivity, law of modularity 195 2.11 ALLEGORY 196 2.111 r-VALUED RELATION 197 2.113 MODULAR LATTICE 197 CONTENTS XV

2.12 REFLEXIVE, SYMMETRIC, TRANSITIVE, COREFLEXIVE, EQUIVALENCE RELATION 198 2.122 DOMAIN 198 2.13 ENTIRE, SIMPLE, MAP 199 2.14 TABULATES (a morphism), TABULAR (morphism), TABULAR ALLEGORY, connected locale 200 2.15 PARTIAL UNIT, UNIT, UNITARY ALLEGORY 202 2.153 ASSEMBLY, CAUCUS, modulus 202 2.154 (UNITARY) REPRESENTATION OF ALLEGORIES, representation theorem for unitary tabular allegories 204 2.156 partition representation [combinatorial representation], geometric representation (of modular lattices) 205 2.157 projective plane, Desargues' theorem 205 2.158 representable allegory 207 2.165 PRE-TABULAR ALLEGORY 211 2.167 tabular reflection 212 2.169 EFFECTIVE ALLEGORY, EFFECTIVE REFLECTION 213 2.16(10) SEMI-SIMPLE morphism, SEMI-SIMPLE ALLEGORY 213 2.16(11) neighbors (pair of idempotents) 214 2.16(12) r-VALUED SETS 215

2.2. Distributive allegories 216 2.21 DISTRIBUTIVE ALLEGORY 216 2.215 POSITIVE ALLEGORY 218 2.216 POSITIVE REFLECTION 218 2.218 representation theorem for distributive allegories 220 2.22 LOCALLY COMPLETE DISTRIBUTIVE ALLEGORY 221 2.221 downdeal, LOCAL COMPLETION 221 2.222 ideal 221 2.223 GLOBALLY COMPLETE 221 2.224 GLOBAL COMPLETION 222 2.226 SYSTEMIC COMPLETION 222 2.227 C(y)-valued sets and sheaves on Y 223

2.3. Division allegories 225 2.31 DIVISION ALLEGORY 225 2.331 representation theorem for division allegories 228 2.35 SYMMETRIC DIVISION 231 2.351 STRAIGHT (morphism) 231 2.357 SIMPLE PART, DOMAIN OF SIMPLICITY 234

2.4. Power allegories 235 2.41 POWER ALLEGORY, THICK (morphism) 235 2.415 POWER-OBJECT, SINGLETON MAP 237 2.418 REALIZABILITY TOPOS 238 2.42 SPLITTING LEMMAS 238 2.43 PRE-POWER ALLEGORY 240 2.436 Cantor's diagonal proof 242 2.437 recursively enumerable sets which are not recursive 243 2.438 Peano axioms, Gödel-numbers, inconsistency 243 2.441 PRE-POSITIVE ALLEGORY, well-joined category 244 2.442 LAW OF METONYMY 246 XVI CONTENTS

2.445 stilted relations 249 2.451 FREE BOOLEAN ALGEBRA 250 2.453 Continuum Hypothesis 252 2.454 WELL-POINTED 253

2.5. Quotient allegories 255 2.5 CONGRUENCE (on an allegory), QUOTIENT ALLEGORY 255 2.521 BOOLEAN QUOTIENT 255 2.522 CLOSED QUOTIENT 255 2.542 faithful bicartesian representation in a boolean topos 257 2.53 AMENABLE CONGRUENCE, AMENABLE QUOTIENT 256 2.55 quotients of complete allegories 258 2.56 Axiom of Choice, independence of 258 2.563 SEPARATED OBJECT, DENSE RELATION 260

APPENDICES

Appendix A 267 countable dense linearly ordered set, Cantor's back-and-forth argument,

complete metric on a Gs set, countable power of 2, Cantor space, countable power of the natural numbers, Baire space, countable atomless boolean algebras

Appendix B 270 B.l SORT, SORT WORD, VARIABLE, SORT ASSIGNMENT, PREDICATE SYMBOL, SORT TYPE ASSIGNMENT [arity], EQUALITY SYMBOL, CONNECTIVES, QUANTIFIERS, PUNCTUATORS 270 B.ll FORMULA, FREE, BOUND, INDEX (occurrences of a variable), SCOPE (of a quantifler), ASSERTION, TOLERATES 270 B.12 PRIMITIVE FUNCTIONAL SEMANTICS, VALID (assertion), MODEL, THEORY, ENTAILS IN PRIMITIVE FUNCTIONAL SEMANTICS 270 B.21 RULES OF INFERENCE, FIRST ORDER LOGIC, SYNTACTICALLY ENTAILS 271 B.211 COHERENT LOGIC, REGULÄR LOGIC, HÖRN LOGIC, HIGHER ORDER LOGIC, propositional theories 272 B.22 DERIVED RULES 273 B.3 DERIVED PREDICATE TOKEN, INSTANTIATION (of a variable), DERIVED PREDICATE 275 B.31 FREE ALLEGORY (on a theory) 276 B.315 FREE (REGULÄR CATEGORY, PRE-LOGOS, LOGOS, TOPOS) 277 B.316 ARITHMETIC (theories of), NUMERICAL SORT, NUMERICAL CONSTANT, FUNCTION SYMBOL, term, INDUCTION, PEANO AXIOMS, HIGHER ORDER ARITHMETIC 278 B.317 free topos with a natural numbers object 279 B.318 numerical coding of inference and inconsistency 281 B.32 DISJUNCTION PROPERTY, EXISTENCE PROPERTY, NUMERICAL EXISTENCE PROPERTY 281 B.41 SEMANTICALLY ENTAILS IN A UNITARY ALLEGORY 281 B.411 tarskian semantics, BOOLEAN THEORY 282 B.421 GÖDEL'S COMPLETENESS THEOREM 282 CONTENTS

B.5 ZERMELO-FRAENKEL SET THEORY 283 B.51 FOURMAN-HAYASHI INTERPRETATION, well-founded part, SCOTT-SOLOVAY BOOLEAN-VALUED MODEL 283 B.52 Continuum Hypothesis, independence of 285 B.53 Axiom of Choice, independence of 285

Subject Index 287