-
Notifications
You must be signed in to change notification settings - Fork 7
Expand file tree
/
Copy path007_category-properties.sql
More file actions
1631 lines (1614 loc) · 39.6 KB
/
007_category-properties.sql
File metadata and controls
1631 lines (1614 loc) · 39.6 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
INSERT INTO category_properties (
category_id,
property_id,
reason
)
VALUES
-- basic categories
(
'Set',
'locally small',
'The collection of maps between two sets $X,Y$ is a subset of $X \times Y$ and therefore a set.'
),
(
'Set',
'Grothendieck topos',
'It is equivalent to the category of sheaves on a one-point space.'
),
(
'Set',
'strongly connected',
'Every nonempty set is weakly terminal.'
),
(
'Set',
'finitary algebraic',
'Use the empty algebraic theory.'
),
(
'Ab',
'locally small',
'There is a forgetful functor $\mathbf{Ab} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Ab',
'abelian',
'standard'
),
(
'Ab',
'finitary algebraic',
'Take the algebraic theory of a commutative group.'
),
(
'Top',
'locally small',
'There is a forgetful functor $\mathbf{Top} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Top',
'complete',
'Take the limit of the underlying sets and endow it with the coarsest topology making all projections continuous.'
),
(
'Top',
'cocomplete',
'Take the colimit of the underlying sets and endow it with the finest topology making all inclusions continuous.'
),
(
'Top',
'well-powered',
'This is clear from the classification of monomorphisms as injective continuous maps.'
),
(
'Top',
'well-copowered',
'This is clear from the classification of epimorphisms as surjective continuous maps.'
),
(
'Top',
'infinitary distributive',
'The canonical continuous map $\coprod_i (X \times Y_i) \to X \times \coprod_{i \in I} Y_i$ is bijective since $\mathbf{Set}$ is infinitary distributive, and one easily checks with the definitions of product and coproduct topologies that it maps open sets to open sets.'
),
(
'Top',
'generator',
'The one-point space is a generator since it represents the forgetful functor $\mathbf{Top} \to \mathbf{Set}$.'
),
(
'Top',
'cogenerator',
'It is easily checked that the indiscrete two-point space is a cogenerator.'
),
(
'Top',
'disjoint coproducts',
'This follows easily from the corresponding fact for sets.'
),
(
'Top',
'strongly connected',
'Every nonempty space is weakly terminal.'
),
(
'Grp',
'locally small',
'There is a forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Grp',
'pointed',
'The trivial group is a zero object.'
),
(
'Grp',
'finitary algebraic',
'Take the algebraic theory of a group.'
),
(
'Grp',
'mono-regular',
'See Prop. 4.2 at the <a href="https://ncatlab.org/nlab/show/regular+monomorphism#Examples" target="_blank">nLab</a>.'
),
(
'Grp',
'disjoint coproducts',
'It follows from the normal form of the elements of coproducts of groups that the inclusions $G \to G \sqcup H \leftarrow H$ are injective (hence, monomorphisms) and that their intersection is trivial.'
),
(
'Grp',
'epi-regular',
'This holds since every epimorphism is surjective, and surjective homomorphism $A \to B$ is the coequalizer of its kernel pair $A \times_B A \rightrightarrows A$.'
),
(
'Grp',
'Malcev',
'See Example 2.2.4 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>.'
),
(
'Vect',
'locally small',
'There is a forgetful functor $\mathbf{Vect} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Vect',
'split abelian',
'It is a fact from linear algebra that every subspace has a complement.'
),
(
'Vect',
'finitary algebraic',
'Take the algebraic theory of a vector space.'
),
(
'Ring',
'locally small',
'There is a forgetful functor $\mathbf{Ring} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Ring',
'finitary algebraic',
'Take the algebraic theory of a ring.'
),
(
'Ring',
'strict terminal object',
'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.'
),
(
'Ring',
'Malcev',
'follows in the same way as for (additive) groups'
),
(
'Alg(R)',
'locally small',
'There is a forgetful functor $\mathbf{Alg}(R) \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Alg(R)',
'finitary algebraic',
'Take the algebraic theory of an $R$-algebra.'
),
(
'Alg(R)',
'strict terminal object',
'If $f : 0 \to A$ is an algebra homomorphism, then $A$ satisfies $1=f(1)=f(0)=0$, so that $A=0$.'
),
(
'Alg(R)',
'Malcev',
'follows in the same way as for (additive) groups'
),
(
'CRing',
'locally small',
'There is a forgetful functor $\mathbf{CRing} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'CRing',
'finitary algebraic',
'Take the algebraic theory of a commutative ring.'
),
(
'CRing',
'strict terminal object',
'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.'
),
(
'CRing',
'Malcev',
'follows in the same way as for (additive) groups'
),
(
'CAlg(R)',
'locally small',
'There is a forgetful functor $\mathbf{CAlg(R)} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'CAlg(R)',
'finitary algebraic',
'Take the algebraic theory of a commutative ring.'
),
(
'CAlg(R)',
'strict terminal object',
'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.'
),
(
'CAlg(R)',
'Malcev',
'follows in the same way as for (additive) groups'
),
(
'Rng',
'locally small',
'There is a forgetful functor $\mathbf{Rng} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Rng',
'finitary algebraic',
'Take the algebraic theory of a rng.'
),
(
'Rng',
'pointed',
'The zero ring is a zero object.'
),
(
'Rng',
'disjoint coproducts',
'The coproduct of two rngs $R,S$ has as underlying additive group $R \oplus S \oplus (R \otimes S) \oplus (S \otimes R) \oplus \cdots$ from which the claim easily follows.'
),
(
'Rng',
'Malcev',
'follows in the same way as for (additive) groups'
),
(
'Set*',
'locally small',
'There is a forgetful functor $\mathbf{Set}_* \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Set*',
'pointed',
'The singleton set is a zero object.'
),
(
'Set*',
'finitary algebraic',
'Take the algebraic theory with just one constant.'
),
(
'Set*',
'subobject classifier',
'The pointed set $(\{0,1\},1)$ is a subobject classifier.'
),
(
'Set*',
'cogenerator',
'The pointed set $(\{0,1\},1)$ is a cogenerator.'
),
(
'Set*',
'disjoint coproducts',
'This follows easily from the description of coproducts as wedge sums.'
),
(
'Set*',
'epi-regular',
'Every epimorphism is surjective for this category, and in an algebraic category every surjective homomorphism is a regular epimorphism.'
),
(
'Mon',
'locally small',
'There is a forgetful functor $\mathbf{Mon} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Mon',
'pointed',
'The trivial monoid is a zero object.'
),
(
'Mon',
'finitary algebraic',
'Take the algebraic of a monoid.'
),
(
'Mon',
'disjoint coproducts',
'It follows from the normal form of the elements of coproducts of monoids that the inclusions $G \to G \sqcup H \leftarrow H$ are injective (hence, monomorphisms) and that their intersection is trivial.'
),
(
'CMon',
'locally small',
'There is a forgetful functor $\mathbf{CMon} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'CMon',
'pointed',
'The trivial monoid is a zero object.'
),
(
'CMon',
'finitary algebraic',
'Take the algebraic of a commutative monoid.'
),
(
'CMon',
'disjoint coproducts',
'This follows from the description of the coproduct as a form of direct sum (very similar to $\mathbf{Ab}$).'
),
(
'Pos',
'locally small',
'There is a forgetful functor $\mathbf{Pos} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Pos',
'locally finitely presentable',
'See <a href="https://ncatlab.org/nlab/show/Locally+Presentable+and+Accessible+Categories" target="_blank">Adamek-Rosicky</a>, Example 1.10.'
),
(
'Pos',
'cartesian closed',
'For partial orders $P,Q$ we endow $\hom(P,Q)$ with the partial order in which $f \leq g$ holds iff $f(p) \leq g(p)$ for all $p \in P$. The universal evaluation map is $\hom(P,Q) \times P \to Q$, $(f,p) \mapsto f(p)$, it is order-preserving, and it satisfies the universal property.'
),
(
'Pos',
'cogenerator',
'We prove that the poset $\{0 < 1 \}$ is a cogenerator: Let $P$ be a poset and $a,b \in P$ be two elements such that $f(a) = f(b)$ for all order-preserving maps $f : P \to \{0 < 1 \}$. This means that $a$ and $b$ lie in the same upper sets. In particular, $b$ lies in the upper set generated by $a$, meaning %$a \leq b$, and similarly we deduce $b \leq a$. Thus, $a = b$.'
),
(
'Pos',
'disjoint coproducts',
'This follows easily from the corresponding fact for sets, since the coproduct of posets is built using the disjoint union.'
),
(
'Pos',
'strongly connected',
'Every nonempty preorder is weakly terminal.'
),
(
'M-Set',
'locally small',
'There is a forgetful functor $M{-}\mathbf{Set} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'M-Set',
'Grothendieck topos',
'It is the category of sheaves on the opposite of the one-object category associated to $M$.'
),
(
'M-Set',
'finitary algebraic',
'Take the algebraic theory of an $M$-sets (having a unary operation for each $m \in M$).'
),
(
'R-Mod',
'locally small',
'There is a forgetful functor $R{-}\mathbf{Mod} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'R-Mod',
'abelian',
'standard'
),
(
'R-Mod',
'finitary algebraic',
'Take the algebraic theory of an $R$-module (given by the algebraic theory of an abelian group and for each $r \in R$ a unary operation).'
),
(
'Rel',
'locally small',
'The set of morphisms $X \to Y$ is the set $P(X \times Y)$.'
),
(
'Rel',
'self-dual',
'There is an isomorphism $\mathbf{Rel} \to \mathbf{Rel}^{\mathrm{op}}$ that is the identity on objects (sets) and maps a relation $R \subseteq X \times Y$ to the opposite relation $R^o \subseteq Y \times X$ defined by $R^o := \{(y,x) : (x,y) \in R \}$.'
),
(
'Rel',
'pointed',
'The empty set is clearly both initial and terminal.'
),
(
'Rel',
'generator',
'One checks that the the one-point set is a generator.'
),
(
'Rel',
'disjoint coproducts',
'It is an easy exercise to deduce this from the corresponding fact for sets and that sets form a distributive category.'
),
(
'Rel',
'well-powered',
'Any relation $R : A \to B$ induces an injective map $P(A) \to P(B)$, so that in particular there is an injective map $A \to P(B)$.'
),
(
'Rel',
'balanced',
'See <a href="https://math.stackexchange.com/questions/5030393" target="_blank">MSE/5030393</a>.'
),
-- categories of "finite" objects
(
'FinSet',
'locally small',
'There is a forgetful functor $\mathbf{FinSet} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'FinSet',
'essentially small',
'Every finite set is isomorphic to some $[n] = \{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.'
),
(
'FinSet',
'elementary topos',
'This follows easily from the fact that sets form an elementary topos.'
),
(
'FinSet',
'generator',
'The one-point set is a generator since it represents the forgetful functor $\mathbf{FinSet} \to \mathbf{Set}$.'
),
(
'FinSet',
'strongly connected',
'Every nonempty finite set is weakly terminal.'
),
(
'FinSet',
'cogenerator',
'The two-element set is a cogenerator.'
),
(
'FinAb',
'locally small',
'There is a forgetful functor $\mathbf{FinAb} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'FinAb',
'essentially small',
'The underlying set of a finite structure can be chosen to be a subset of $\mathbb{N}$.'
),
(
'FinAb',
'abelian',
'follows from the fact for abelian groups'
),
(
'FinAb',
'self-dual',
'Pontrjagin duality'
),
(
'Abfg',
'locally small',
'There is a forgetful functor $\mathbf{FinAb} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Abfg',
'essentially small',
'Every finitely generated abelian group is isomorphic to a group of the form $\mathbb{Z}^n / U$, where $n \in \mathbb{N}$ and $U$ is a subgroup of $\mathbb{Z}^n$. And these constitute a set.'
),
(
'Abfg',
'abelian',
'This follows from the fact for abelian groups and the fact that subgroups of finitely generated abelian groups are also finitely generated.'
),
(
'Abfg',
'generator',
'The group $\mathbb{Z}$ is a generator since it represents the forgetful functor $\mathbf{Abfg} \to \mathbf{Set}$.'
),
(
'B',
'locally small',
'There is a forgetful functor $\mathbb{B} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'B',
'essentially small',
'Every finite set is isomorphic to some $[n] = \{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.'
),
(
'B',
'groupoid',
'trivial'
),
(
'B',
'inhabited',
'trivial'
),
(
'FI',
'locally small',
'There is a forgetful functor $\mathbf{FI} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'FI',
'initial object',
'Take the empty set.'
),
(
'FI',
'left cancellative',
'trivial'
),
(
'FI',
'essentially small',
'Every finite set is isomorphic to some $[n] = \{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.'
),
(
'FI',
'generator',
'The one-point set is a generator since it represents the forgetful functor $\mathbf{FI} \to \mathbf{Set}$.'
),
(
'FI',
'connected limits',
NULL
),
(
'FI',
'mono-regular',
NULL
),
(
'FI',
'epi-regular',
NULL
),
(
'FS',
'locally small',
'There is a forgetful functor $\mathbf{FS} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'FS',
'essentially small',
'Every finite set is isomorphic to some $[n] = \{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.'
),
(
'FS',
'right cancellative',
'trivial'
),
(
'FS',
'cogenerator',
NULL
),
(
'FS',
'wide pushouts',
NULL
),
(
'FS',
'coequalizers',
NULL
),
(
'FS',
'mono-regular',
NULL
),
(
'FS',
'epi-regular',
NULL
),
-- tiny categories
(
'0',
'preadditive',
'This is vacuously true.'
),
(
'0',
'discrete',
'trivial'
),
(
'0',
'binary products',
'This is vacuously true.'
),
(
'0',
'finite',
'trivial'
),
(
'0',
'locally cartesian closed',
'This is vacuously true.'
),
(
'1',
'trivial',
'trivial'
),
(
'1',
'finite',
'trivial'
),
(
'1',
'discrete',
'trivial'
),
(
'2',
'discrete',
'trivial'
),
(
'2',
'finite',
'trivial'
),
(
'2',
'inhabited',
'trivial'
),
(
'walking_morphism',
'finitary algebraic',
'Take the algebraic theory with no operations but with the equation $x=y$ that is supposed to hold for all elements $x,y$. The algebras for this theory are the empty set and the singleton set, hence we get the equivalence to $\{0 \to 1\}$.'
),
(
'walking_morphism',
'self-dual',
'trivial'
),
(
'walking_morphism',
'finite',
'trivial'
),
(
'walking_morphism',
'infinitary distributive',
'Clearly, this category is (co)complete. Also, the functors $X \mapsto X \times 1 = X$ and $X \mapsto X \times 0 = 0$ are clearly cocontinuous.'
),
(
'walking_morphism',
'skeletal',
'The two objects are not isomorphic'
),
(
'walking_morphism',
'strongly connected',
'trivial'
),
(
'walking_pair',
'finite',
'trivial'
),
(
'walking_pair',
'self-dual',
'trivial'
),
(
'walking_pair',
'strongly connected',
'trivial'
),
(
'walking_pair',
'generator',
'It is easy to check that $0$ is a generator.'
),
(
'walking_pair',
'filtered colimits',
'Every filtered diagram can be reduced to one of the three trivial diagrams $0 \to 0$, $0 \to 1$, $1 \to 1$, which have colimits $0$, $1$, resp. $1$.'
),
(
'walking_pair',
'left cancellative',
'The two morphisms $0 \to 1$ are clearly monomorphisms.'
),
(
'walking_pair',
'skeletal',
'The two objects are not isomorphic.'
),
(
'walking_isomorphism',
'trivial',
'The inclusion $\{0\} \hookrightarrow \{0 \to 1\}$ is clearly an equivalence.'
),
(
'walking_isomorphism',
'finite',
'trivial'
),
(
'walking_composable_pair',
'finite',
'trivial'
),
(
'walking_composable_pair',
'strongly connected',
'trivial'
),
(
'walking_composable_pair',
'skeletal',
'trivial'
),
(
'walking_composable_pair',
'products',
'trivial'
),
(
'walking_composable_pair',
'coproducts',
'trivial'
),
-- geometric categories
(
'Sh(X)',
'locally small',
'easy'
),
(
'Sh(X)',
'Grothendieck topos',
'by definition of a Grothendieck topos'
),
(
'sSet',
'locally small',
'This follows from the general fact that $[\mathcal{C},\mathcal{D}]$ is locally small when $\mathcal{C}$ is small and $\mathcal{D}$ is locally small, here applied to $\mathcal{C} = \Delta^{\mathrm{op}}$ and $\mathcal{D} = \mathbf{Set}$.'
),
(
'Sh(X,Ab)',
'locally small',
'easy'
),
(
'Sh(X,Ab)',
'Grothendieck abelian',
'This is standard, see for example Theorem 18.1.6. in <a href="https://ncatlab.org/nlab/show/Categories+and+Sheaves" target="_blank">Kashiwara-Schapira</a>.'
),
(
'sSet',
'Grothendieck topos',
'obvious'
),
(
'sSet',
'locally finitely presentable',
'This follows from the fact that the free cocompletion of a small category is locally finitely presentable.'
),
(
'Met',
'locally small',
'There is a forgetful functor $\mathbf{Met} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Met',
'well-powered',
'This follows since monomorphisms are injective.'
),
(
'Met',
'equalizers',
'Just restrict the metric to the equalizer built from the sets.'
),
(
'Met',
'binary products',
'The product of two metric spaces $(X,d)$, $(Y,d)$ is $(X \times Y,d)$ with $d((x_1,y_1),(x_2,x_2)) := \sup(d(x_1,x_2),d(y_1,y_2))$.'
),
(
'Met',
'terminal object',
'The one-point metric space is terminal.'
),
(
'Met',
'coequalizers',
NULL
),
(
'Met',
'filtered colimits',
'Given a filtered diagram $(X_i)$ of metric spaces, take the filtered colimit $X$ of the underlying sets with the following metric: If $x,y \in X$, let $d(x,y)$ be infimum of all $d(x_i,y_i)$, where $x_i,y_i \in X_i$ are some preimages of $x,y$ in some $X_i$. This is only a pseudo-metric, so finally take the associated metric space (Kolmogorov quotient). The definition ensures that each $X_i \to X$ is non-expansive, and the universal property is easy to check.'
),
(
'Met',
'strict initial object',
'This is because the initial object is the empty metric space.'
),
(
'Met',
'generator',
'The one-point metric space is a generator since it represents the forgetful functor $\mathbf{Met} \to \mathbf{Set}$.'
),
(
'Met',
'cogenerator',
'We claim that $\mathbb{R}$ with the usual metric is a cogenerator. Let $a,b \in X$ be two points of a metric space such that $f(a)=f(b)$ for all non-expansive maps $f : X \to \mathbb{R}$. This applies in particular to $f(x) := d(a,x)$ and shows that $0=d(a,a)=d(a,b)$, so that $a=b$.'
),
(
'Met_oo',
'locally small',
'There is a forgetful functor $\mathbf{Met}_{\infty} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Met_oo',
'locally ℵ₁-presentable',
'Example 4.5 in <a href="https://arxiv.org/abs/1504.02660" target="_blank">this preprint</a>'
),
(
'Met_oo',
'infinitary distributive',
'This follows from the corresponding fact for topological spaces. Here, the coproduct metric takes value $\infty$ for points in distinct spaces.'
),
(
'Met_oo',
'cogenerator',
'The proof is similar to $\mathbf{Met}$, a cogenerator is given by $\mathbb{R} \cup \{\infty\}$ with the metric in which $d(a,\infty)=\infty$ for $a \in \mathbb{R}$. Then one checks that the maps $d(a,-) : X \to \mathbb{R} \cup \{\infty\}$ are non-expansive and finishes as for $\mathbf{Met}$.'
),
(
'Met_oo',
'disjoint coproducts',
'This easily follows from the corresponding fact for sets, since coproducts are built using disjoint unions.'
),
(
'Met_c',
'locally small',
'There is a forgetful functor $\mathbf{Met}_c \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Met_c',
'equalizers',
'Just restrict the metric to the equalizer built from the sets.'
),
(
'Met_c',
'disjoint coproducts',
'This easily follows from the corresponding fact for sets, since coproducts are built using disjoint unions.'
),
(
'Met_c',
'well-powered',
'follows easily from the fact that monomorphisms here are injective'
),
(
'Met_c',
'generator',
'The one-point metric space is a generator since it represents the forgetful functor $\mathbf{Met}_c \to \mathbf{Set}$.'
),
(
'Met_c',
'cogenerator',
'The same proof as for $\mathbf{Met}$ shows that $\mathbb{R}$ with the usual metric is a cogenerator.'
),
(
'Met_c',
'infinitary distributive',
'This follows from the corresponding fact for topological spaces.'
),
(
'Met_c',
'countable products',
'For finite products, we take the cartesian product with, say, the sup-metric. The product of countably many metric spaces $(X_i,d_i)_{i \geq 0}$ is given by the cartesian product $\prod_{i \geq 0} X_i$ with the metric $d(x,y) := \sum_{i \geq 0} d_i(x_i,y_i)/(1 + d_i(x_i,y_i))$. See Engelking''s book <i>General Topology</i>.'
),
(
'Man',
'locally small',
'There is a forgetful functor $\mathbf{Man} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.'
),
(
'Man',
'well-powered',
'follows from the fact that monomorphisms are injective here'
),
(
'Man',
'distributive',
'This follows from the corresponding fact for topological spaces.'
),
(
'Man',
'generator',
'The $0$-dimensional one-point manifold is a generator since it represents the forgetful functor $\mathbf{Top} \to \mathbf{Set}$.'
),
(
'Man',
'disjoint finite coproducts',
'This follows from the corresponding fact for topological spaces.'
),
(
'Man',
'countable coproducts',
'Take the usual disjoint union, which is clearly a smooth manifold, still second-countable. (Without that condition, all coproducts would exist.)'
),
(
'Man',
'Cauchy complete',
'See Theorem 2.1 at the <a href="https://ncatlab.org/nlab/show/smooth+manifold" target="_blank">nLab</a>.'
),
(
'LRS',
'locally small',
'For two ringed spaces $(X,O_X), (Y,O_Y)$, the collection of morphisms $X \to Y$ of ringed spaces is the collection $\prod_{f \in \hom(X,Y)} \hom(O_Y,f_* O_X)$, which is a set since $\hom(X,Y)$ is a set and each $\hom(O_Y,f_* O_X)$ is a set.'
),
(
'LRS',
'complete',
'See <a href="https://arxiv.org/abs/1103.2139" target="_blank">Localization of ringed spaces</a> by W. Gillam. See also <a href="https://math.stackexchange.com/questions/1033675" target="_blank">MSE/1033675</a>.'
),
(
'LRS',
'cocomplete',
'See Demazure-Gabriel''s "Groupes algébriques", I. §1. 1.6. Specifically, the forgetful functor to ringed spaces preserves colimits, and colimits of ringed spaces are built from colimits of topological spaces and limits of commutative rings, see <a href="https://math.stackexchange.com/questions/1646202" target="_blank">MSE/1646202</a>.'
),
(
'LRS',
'well-powered',
'Let $f : X \to Y$ be a monomorphism of locally ringed spaces. I claim that $f$ is injective, from which the claim will follow. The diagonal $\Delta : X \to X \times_Y X$ is an isomorphism. By the explicit construction of fiber products, $X \times_Y X$ consists of triples $(x_1,x_2,\mathfrak{p})$ where $x_1,x_2 \in X$, $y := f(x_1) = f(x_2)$ and $\mathfrak{p}$ is a prime ideal in $k(x_1) \otimes_{k(y)} k(x_2)$. The map $\Delta$ is given by $\Delta(x) = \bigl(x,x,\ker(k(x) \otimes_{k(f(x))} k(x) \to k(x)\bigr)$, and it is bijective. This clearly implies that $f$ is injective (and that each tensor product $k(x) \otimes_{k(f(x))} k(x)$ has a unique prime ideal, namely the kernel mentioned).'
),
(
'LRS',
'well-copowered',
'It is enough to prove that every epimorphism of locally ringed spaces is surjective. The forgetful functor $\mathbf{LRS} \to \mathbf{RS}$ has a right adjoint (see <a href="https://arxiv.org/abs/1103.2139" target="_blank">Localization of ringed spaces</a> by W. Gillam), so it preserves epimorphisms. The forgetful functor $\mathbf{RS} \to \mathbf{Top}$ also has a right adjoint, namely $X \mapsto (X,\underline{\mathbb{Z}})$, so it also preserves epimorphisms.'
),
(
'LRS',
'infinitary distributive',
'See <a href="https://math.stackexchange.com/questions/5036488/" target="_blank">MSE/5036488</a>.'
),
(
'LRS',
'disjoint coproducts',
'This can easily be derived from the explicit construction of coproducts: The topological space is the disjoint union, the sheaf is the "product".'
),
(
'Sch',
'locally small',
'There is a forgetful functor $\mathbf{Sch} \to \mathbf{LRS}$ and $\mathbf{LRS}$ is locally small.'
),
(
'Sch',
'terminal object',
'The scheme $\mathrm{Spec}(\mathbf{Z})$ is terminal.'
),
(
'Sch',
'pullbacks',
'This is the well-known construction of the fiber product of schemes, see e.g. EGA I, Chap. I, Thm. 3.2.1.'
),
(
'Sch',
'disjoint coproducts',
'This follows from the corresponding fact for locally ringed spaces.'
),
(
'Sch',
'well-powered',
'This is because every monomorphism of schemes is injective, so that the size of the underlying set of a subobject is bounded.'
),
(
'Sch',