glueing lemma for pushouts

1. Proposition

Let 20231102-glueing_lemma_for_pushouts_760824120891fc757bc0445145b027d09b237baa.svg be a category, and

20231102-glueing_lemma_for_pushouts_06bfb024aa1086db2f0444b64f011fcc52af23fa.svg

be a commutative diagram, such that

20231102-glueing_lemma_for_pushouts_b21ddcf6f8796ea284ad280eab160eb5d6a6be57.svg

and

20231102-glueing_lemma_for_pushouts_452d631e760d6fd94b7252e86b445754bb7d9753.svg

are pushouts Then

20231102-glueing_lemma_for_pushouts_074e5c199ea78f6b7190e58d00bcf248bb009bf7.svg

is also a pushout

2. Proof

2.1. existence

Suppose there exists an object 20231102-glueing_lemma_for_pushouts_b06875b149404e1d0544220eb101c48b12dd52f2.svg and morphisms

20231102-glueing_lemma_for_pushouts_d782b83a0a33b71e42a4926dfb702f7fa9d8acc7.svg

Then by universal property

20231102-glueing_lemma_for_pushouts_b60114394de671b52837d291c19d76125811e777.svg

Furthermore, by universal property

20231102-glueing_lemma_for_pushouts_340e3dfbc9ba083a0a81a6cd5dc4809b838f7af4.svg

2.2. uniqueness

TODO

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 18:59