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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have expanded slightly at coalgebra – Properties – As filtered colimits of finite dimensional pieces.
And I have added and cross-linked with corresponding remarks at dg-coalgebra, at pro-object, at L-infinity algebra and at model structure for L-infinity algebras.
The axioms for the coalgebra don’t appear to show.
Your chance to practice your editing skills on an elementary example!
Best to use \tikzcd for the diagrams. If you get stuck, let me know and I’ll lend a hand.
It is definitely not true that an arbitrary associative algebra is a pro-limit of finite associative algebras. For example, the polynomial ring k[x] is not equivalent to a pro-finite algebra as a functor from associative algebras to sets, since it is not the case that every element of an associative algebra is contained in a finite-dimensional subalgebra.
It is true that an associative algebra in pro-finite vector spaces is a pro-limit of finite-dimensional algebras, though.
Joshua Mundinger
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