coproduct in set

1. Proposition

Given category set and a collection of sets 20230226161248-koprodukt_af773f204019917a22fb1e343acbd7b20d384f5a.svg, then the categorical coproduct is given as disjoint union

20230226161248-koprodukt_af0f226c1be27a6a1b5559bf89e096ec3ab7e7fa.svg

with the canonical inclusion

20230226161248-koprodukt_1e348e1f49a49cbd34598e9d9d89317d5a531ca4.svg

2. Proof

2.1. construction of a map

Let 20230226161248-koprodukt_9c8bdb9f6ad11c13881249483fa93b024273b477.svg be a set and

20230226161248-koprodukt_18321bfc6144a84302be3d9a45cdbff72b66b5b9.svg

be a collection of maps

Then we define

20230226161248-koprodukt_df6a880148f2c69ec304004b3a10c3eeba3cf4e3.svg

Then by construction

20230226161248-koprodukt_b7f3558b0f6621780b507526dd0a17b10f85e2e6.svg

commutes

2.2. uniqueness

Suppose there exists a map 20230226161248-koprodukt_5f965d0a56374955f8a17da7129fa95cde5a6603.svg making

20230226161248-koprodukt_8efad4d940fb18426f786bf7cda7447f04057ac0.svg

commute Then for 20230226161248-koprodukt_e39f86ad101ef45066115d10be7a2c9528148632.svg, it follows, that

20230226161248-koprodukt_d82fe1348c8ee130e085e9ce6b071953d964ba0b.svg

therefore it follows, that

20230226161248-koprodukt_ccf113d492f357ef018849c1c75854512b8c891f.svg

and since 20230226161248-koprodukt_b9c3e97619b8af7ea5a89d4c6c0090d2eff85b8e.svg was arbitrary,

20230226161248-koprodukt_2591a301b16ef2a20ffb6f3129ed74fb91072e70.svg

Date: nil

Author: Anton Zakrewski

Created: 2024-10-11 Fr 22:02