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Definition df-cat 15652
 Description: A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated with those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" (), the morphisms "hom" ( ) and the composition law "o" (comp). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 15653. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
df-cat comp
Distinct variable group:   ,,,,,,,,,,

Detailed syntax breakdown of Definition df-cat
StepHypRef Expression
1 ccat 15648 . 2
2 vg . . . . . . . . . . . . . . 15
32cv 1451 . . . . . . . . . . . . . 14
4 vf . . . . . . . . . . . . . . 15
54cv 1451 . . . . . . . . . . . . . 14
6 vy . . . . . . . . . . . . . . . . 17
76cv 1451 . . . . . . . . . . . . . . . 16
8 vx . . . . . . . . . . . . . . . . 17
98cv 1451 . . . . . . . . . . . . . . . 16
107, 9cop 3965 . . . . . . . . . . . . . . 15
11 vo . . . . . . . . . . . . . . . 16
1211cv 1451 . . . . . . . . . . . . . . 15
1310, 9, 12co 6308 . . . . . . . . . . . . . 14
143, 5, 13co 6308 . . . . . . . . . . . . 13
1514, 5wceq 1452 . . . . . . . . . . . 12
16 vh . . . . . . . . . . . . . 14
1716cv 1451 . . . . . . . . . . . . 13
187, 9, 17co 6308 . . . . . . . . . . . 12
1915, 4, 18wral 2756 . . . . . . . . . . 11
209, 9cop 3965 . . . . . . . . . . . . . . 15
2120, 7, 12co 6308 . . . . . . . . . . . . . 14
225, 3, 21co 6308 . . . . . . . . . . . . 13
2322, 5wceq 1452 . . . . . . . . . . . 12
249, 7, 17co 6308 . . . . . . . . . . . 12
2523, 4, 24wral 2756 . . . . . . . . . . 11
2619, 25wa 376 . . . . . . . . . 10
27 vb . . . . . . . . . . 11
2827cv 1451 . . . . . . . . . 10
2926, 6, 28wral 2756 . . . . . . . . 9
309, 9, 17co 6308 . . . . . . . . 9
3129, 2, 30wrex 2757 . . . . . . . 8
329, 7cop 3965 . . . . . . . . . . . . . . . 16
33 vz . . . . . . . . . . . . . . . . 17
3433cv 1451 . . . . . . . . . . . . . . . 16
3532, 34, 12co 6308 . . . . . . . . . . . . . . 15
363, 5, 35co 6308 . . . . . . . . . . . . . 14
379, 34, 17co 6308 . . . . . . . . . . . . . 14
3836, 37wcel 1904 . . . . . . . . . . . . 13
39 vk . . . . . . . . . . . . . . . . . . 19
4039cv 1451 . . . . . . . . . . . . . . . . . 18
417, 34cop 3965 . . . . . . . . . . . . . . . . . . 19
42 vw . . . . . . . . . . . . . . . . . . . 20
4342cv 1451 . . . . . . . . . . . . . . . . . . 19
4441, 43, 12co 6308 . . . . . . . . . . . . . . . . . 18
4540, 3, 44co 6308 . . . . . . . . . . . . . . . . 17
4632, 43, 12co 6308 . . . . . . . . . . . . . . . . 17
4745, 5, 46co 6308 . . . . . . . . . . . . . . . 16
489, 34cop 3965 . . . . . . . . . . . . . . . . . 18
4948, 43, 12co 6308 . . . . . . . . . . . . . . . . 17
5040, 36, 49co 6308 . . . . . . . . . . . . . . . 16
5147, 50wceq 1452 . . . . . . . . . . . . . . 15
5234, 43, 17co 6308 . . . . . . . . . . . . . . 15
5351, 39, 52wral 2756 . . . . . . . . . . . . . 14
5453, 42, 28wral 2756 . . . . . . . . . . . . 13
5538, 54wa 376 . . . . . . . . . . . 12
567, 34, 17co 6308 . . . . . . . . . . . 12
5755, 2, 56wral 2756 . . . . . . . . . . 11
5857, 4, 24wral 2756 . . . . . . . . . 10
5958, 33, 28wral 2756 . . . . . . . . 9
6059, 6, 28wral 2756 . . . . . . . 8
6131, 60wa 376 . . . . . . 7
6261, 8, 28wral 2756 . . . . . 6
63 vc . . . . . . . 8
6463cv 1451 . . . . . . 7
65 cco 15280 . . . . . . 7 comp
6664, 65cfv 5589 . . . . . 6 comp
6762, 11, 66wsbc 3255 . . . . 5 comp
68 chom 15279 . . . . . 6
6964, 68cfv 5589 . . . . 5
7067, 16, 69wsbc 3255 . . . 4 comp
71 cbs 15199 . . . . 5
7264, 71cfv 5589 . . . 4
7370, 27, 72wsbc 3255 . . 3 comp
7473, 63cab 2457 . 2 comp
751, 74wceq 1452 1 comp
 Colors of variables: wff setvar class This definition is referenced by:  iscat  15656
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