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Nov
15
answered Symmetric groups and the “field with one element”
Nov
15
comment The twisted cubic is an affine variety.
Show that the quotient is an integral domain by exhibiting an explicit isomorphism with the polynomial ring in one variable.
Nov
15
comment What really is a colimit of sets?
I wouldn't say "some". I cannot recall a single example of "direct limit" which does not mean "directed colimit".
Nov
15
comment Characteristic properties for topological pushouts and pullbacks
It's exactly the reverse, actually.
Nov
15
comment Morita theory for simplicial rings
That sounds like what I imagined might exist!
Nov
15
comment ETCS set theory: Are empty sets isomorphic?
@magma That is not true. ETCS is formulated in an equivalence-invariant way, so it is perfectly possible to have more than one empty set.
Nov
15
comment Morita theory for simplicial rings
I have not seen a proof, but it seems reasonable.
Nov
15
comment Morita theory for simplicial rings
I think the version with simplicially enriched equivalences can be proved in exactly the same way. But perhaps the real question is whether there is a homotopical version of the Morita theorem.
Nov
14
reviewed Approve $g : Z \rightarrow W$ a regular covering map where $Z$ and $W$ are path-connected, and $\pi_1(Z)$
Nov
14
comment Why is it worth spending time on type theory?
That's because extensional type theory is so natural that it needs no introduction! This is what we mean when we say that type theory is closer to how working mathematicians think.
Nov
14
revised ETCS set theory: Are empty sets isomorphic?
added 166 characters in body
Nov
14
answered ETCS set theory: Are empty sets isomorphic?
Nov
14
comment ETCS set theory: Are empty sets isomorphic?
@Svinepels It is an axiom that there is an initial object. Are you asking whether a set that has no elements is initial?
Nov
14
revised ETCS set theory: Are empty sets isomorphic?
edited tags
Nov
14
comment ETCS set theory: Are empty sets isomorphic?
Any two initial objects are isomorphic, in a unique way. So it suffices to show that empty sets are initial.
Nov
14
comment Sheaffication using a $\lambda$-transfinite colimit
I am referring to Quillen's small object argument, which constructs factorisations with certain lifting properties. Sheafification is similar but not identical.
Nov
14
comment Why is it worth spending time on type theory?
Type theory is not new! Homotopy type theory is just a version of Martin-Löf type theory, which was created in the 1980s.
Nov
14
comment Sheaffication using a $\lambda$-transfinite colimit
It's a kind of small object argument.
Nov
14
comment Why is it worth spending time on type theory?
That bit of homotopy type theory is a little bit oversold and is hardly unique to homotopy type theory (qua type theories). Coq, Agda etc. have been around a long time!
Nov
14
answered what's wrong with this categorical proof that maps between two covering spaces are unique?