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Mirrors > Home > MPE Home > Th. List > yoneda | Structured version Visualization version GIF version |
Description: The Yoneda Lemma. There is a natural isomorphism between the functors π and πΈ, where π(πΉ, π) is the natural transformations from Yon(π) = Hom ( β , π) to πΉ, and πΈ(πΉ, π) = πΉ(π) is the evaluation functor. Here we need two universes to state the claim: the smaller universe π is used for forming the functor category π = πΆ op β SetCat(π), which itself does not (necessarily) live in π but instead is an element of the larger universe π. (If π is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set π = π in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.) |
Ref | Expression |
---|---|
yoneda.y | β’ π = (YonβπΆ) |
yoneda.b | β’ π΅ = (BaseβπΆ) |
yoneda.1 | β’ 1 = (IdβπΆ) |
yoneda.o | β’ π = (oppCatβπΆ) |
yoneda.s | β’ π = (SetCatβπ) |
yoneda.t | β’ π = (SetCatβπ) |
yoneda.q | β’ π = (π FuncCat π) |
yoneda.h | β’ π» = (HomFβπ) |
yoneda.r | β’ π = ((π Γc π) FuncCat π) |
yoneda.e | β’ πΈ = (π evalF π) |
yoneda.z | β’ π = (π» βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) |
yoneda.c | β’ (π β πΆ β Cat) |
yoneda.w | β’ (π β π β π) |
yoneda.u | β’ (π β ran (Homf βπΆ) β π) |
yoneda.v | β’ (π β (ran (Homf βπ) βͺ π) β π) |
yoneda.m | β’ π = (π β (π Func π), π₯ β π΅ β¦ (π β (((1st βπ)βπ₯)(π Nat π)π) β¦ ((πβπ₯)β( 1 βπ₯)))) |
yoneda.i | β’ πΌ = (Isoβπ ) |
Ref | Expression |
---|---|
yoneda | β’ (π β π β (ππΌπΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yoneda.r | . . 3 β’ π = ((π Γc π) FuncCat π) | |
2 | 1 | fucbas 17922 | . 2 β’ ((π Γc π) Func π) = (Baseβπ ) |
3 | eqid 2726 | . 2 β’ (Invβπ ) = (Invβπ ) | |
4 | yoneda.y | . . . . . . 7 β’ π = (YonβπΆ) | |
5 | yoneda.b | . . . . . . 7 β’ π΅ = (BaseβπΆ) | |
6 | yoneda.1 | . . . . . . 7 β’ 1 = (IdβπΆ) | |
7 | yoneda.o | . . . . . . 7 β’ π = (oppCatβπΆ) | |
8 | yoneda.s | . . . . . . 7 β’ π = (SetCatβπ) | |
9 | yoneda.t | . . . . . . 7 β’ π = (SetCatβπ) | |
10 | yoneda.q | . . . . . . 7 β’ π = (π FuncCat π) | |
11 | yoneda.h | . . . . . . 7 β’ π» = (HomFβπ) | |
12 | yoneda.e | . . . . . . 7 β’ πΈ = (π evalF π) | |
13 | yoneda.z | . . . . . . 7 β’ π = (π» βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) | |
14 | yoneda.c | . . . . . . 7 β’ (π β πΆ β Cat) | |
15 | yoneda.w | . . . . . . 7 β’ (π β π β π) | |
16 | yoneda.u | . . . . . . 7 β’ (π β ran (Homf βπΆ) β π) | |
17 | yoneda.v | . . . . . . 7 β’ (π β (ran (Homf βπ) βͺ π) β π) | |
18 | 4, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17 | yonedalem1 18235 | . . . . . 6 β’ (π β (π β ((π Γc π) Func π) β§ πΈ β ((π Γc π) Func π))) |
19 | 18 | simpld 494 | . . . . 5 β’ (π β π β ((π Γc π) Func π)) |
20 | funcrcl 17820 | . . . . 5 β’ (π β ((π Γc π) Func π) β ((π Γc π) β Cat β§ π β Cat)) | |
21 | 19, 20 | syl 17 | . . . 4 β’ (π β ((π Γc π) β Cat β§ π β Cat)) |
22 | 21 | simpld 494 | . . 3 β’ (π β (π Γc π) β Cat) |
23 | 21 | simprd 495 | . . 3 β’ (π β π β Cat) |
24 | 1, 22, 23 | fuccat 17933 | . 2 β’ (π β π β Cat) |
25 | 18 | simprd 495 | . 2 β’ (π β πΈ β ((π Γc π) Func π)) |
26 | yoneda.i | . 2 β’ πΌ = (Isoβπ ) | |
27 | yoneda.m | . . 3 β’ π = (π β (π Func π), π₯ β π΅ β¦ (π β (((1st βπ)βπ₯)(π Nat π)π) β¦ ((πβπ₯)β( 1 βπ₯)))) | |
28 | eqid 2726 | . . 3 β’ (π β (π Func π), π₯ β π΅ β¦ (π’ β ((1st βπ)βπ₯) β¦ (π¦ β π΅ β¦ (π β (π¦(Hom βπΆ)π₯) β¦ (((π₯(2nd βπ)π¦)βπ)βπ’))))) = (π β (π Func π), π₯ β π΅ β¦ (π’ β ((1st βπ)βπ₯) β¦ (π¦ β π΅ β¦ (π β (π¦(Hom βπΆ)π₯) β¦ (((π₯(2nd βπ)π¦)βπ)βπ’))))) | |
29 | 4, 5, 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15, 16, 17, 27, 3, 28 | yonedainv 18244 | . 2 β’ (π β π(π(Invβπ )πΈ)(π β (π Func π), π₯ β π΅ β¦ (π’ β ((1st βπ)βπ₯) β¦ (π¦ β π΅ β¦ (π β (π¦(Hom βπΆ)π₯) β¦ (((π₯(2nd βπ)π¦)βπ)βπ’)))))) |
30 | 2, 3, 24, 19, 25, 26, 29 | inviso1 17720 | 1 β’ (π β π β (ππΌπΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βͺ cun 3941 β wss 3943 β¨cop 4629 β¦ cmpt 5224 ran crn 5670 βcfv 6536 (class class class)co 7404 β cmpo 7406 1st c1st 7969 2nd c2nd 7970 tpos ctpos 8208 Basecbs 17151 Hom chom 17215 Catccat 17615 Idccid 17616 Homf chomf 17617 oppCatcoppc 17662 Invcinv 17699 Isociso 17700 Func cfunc 17811 βfunc ccofu 17813 Nat cnat 17902 FuncCat cfuc 17903 SetCatcsetc 18035 Γc cxpc 18130 1stF c1stf 18131 2ndF c2ndf 18132 β¨,β©F cprf 18133 evalF cevlf 18172 HomFchof 18211 Yoncyon 18212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-hom 17228 df-cco 17229 df-cat 17619 df-cid 17620 df-homf 17621 df-comf 17622 df-oppc 17663 df-sect 17701 df-inv 17702 df-iso 17703 df-ssc 17764 df-resc 17765 df-subc 17766 df-func 17815 df-cofu 17817 df-nat 17904 df-fuc 17905 df-setc 18036 df-xpc 18134 df-1stf 18135 df-2ndf 18136 df-prf 18137 df-evlf 18176 df-curf 18177 df-hof 18213 df-yon 18214 |
This theorem is referenced by: (None) |
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