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In Seven sketches, categorical schemas are talked about a lot. They're given by a graph and a path equivalence relation on that graph. It seems very much like the construction of the quotient category (https://en.wikipedia.org/wiki/Quotient_category), which identifies sets of morphisms as well.

Is there perhaps a subtle difference between these two? Seven sketches doesn't seem to mention quotient categories at any point

## Comments

There is also a phrase "finitely-presented category" mentioned in Seven sketches, which seems like a very similar thing. I'm wondering if there are any subtle differences

`There is also a phrase "finitely-presented category" mentioned in Seven sketches, which seems like a very similar thing. I'm wondering if there are any subtle differences`

"formally" word in wikepedia points a difference in my opinion..

`"formally" word in wikepedia points a difference in my opinion..`

https://ncatlab.org/nlab/show/quotient+category

strict localization...

`https://ncatlab.org/nlab/show/quotient+category strict localization...`

Do you mean the "Formally, it is a quotient object in the category of (small) categories, analogous to a quotient group or quotient space, but in the categorical setting." sentence? Doesn't a path equivalence relation on Free(G) exactly the congruence R they talk about on the Wikipedia page? Can't we define the quotient category Free(G)/~ in very much the same way, such that there's a quotient functor Q which equates certain paths?

`Do you mean the "Formally, it is a quotient object in the category of (small) categories, analogous to a quotient group or quotient space, but in the categorical setting." sentence? Doesn't a path equivalence relation on Free(G) exactly the congruence R they talk about on the Wikipedia page? Can't we define the quotient category Free(G)/~ in very much the same way, such that there's a quotient functor Q which equates certain paths?`

yes this "formally"...regarding Free(G) i dont know if you are right or not..

`yes this "formally"...regarding Free(G) i dont know if you are right or not..`