Erase your whiteboards!
Mar. 12th, 2018 09:55 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
timeletsLast week I completely erased 3 out of 4 of my whiteboards and the empty boards immediately created an urge to develop new ideas. After playing (and struggling) with diagrams on my fresh boards for a couple of days, I succeeded in coming up with a functor that maps two of my favorite categories. Awodey's method of selectively erasing objects and arrows to experiment with forgetful functors really helped.
Cat 1
S<- SxH ->H
S -> H
H ->S
===========
Cat 2
1->2->3
1->3
2->3
3->2
===========
Functor mappings:
1 = F(SxH)
2 = F(S)
3 = F(H)
1->3 = F(SxH)->F(H)
1->2 = F(SxH)->F(S)
2->3 =F(S)->F(H)
3->2 =F(H)->F(S)
======
We can also consider the pair of S->H and H->S as a relationship on SxH; therefore, there must be a coequalizer. The relationship constraints outcomes (coequalizer) and it should be erased in a transition to a new category (1->2->3), which can be modeled by a forgetful functor.
Cat 1
S<- SxH ->H
S -> H
H ->S
===========
Cat 2
1->2->3
1->3
2->3
3->2
===========
Functor mappings:
1 = F(SxH)
2 = F(S)
3 = F(H)
1->3 = F(SxH)->F(H)
1->2 = F(SxH)->F(S)
2->3 =F(S)->F(H)
3->2 =F(H)->F(S)
======
We can also consider the pair of S->H and H->S as a relationship on SxH; therefore, there must be a coequalizer. The relationship constraints outcomes (coequalizer) and it should be erased in a transition to a new category (1->2->3), which can be modeled by a forgetful functor.
