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1d
revised Is an open map with open relative diagonal necessarily a local homeomorphism?
edited tags
1d
comment The bifunctoriality of co/limits
The short answer is: derivators.
2d
comment Applications of set theory in physics
Isn't that the controversial Chaos, Solitons & Fractals?
Mar
25
comment is sufficient cohesion equivalent to the connectedness of subobject classifier?
I don't understand your question. The subobject classifier always acts on the partial map representor.
Mar
25
comment Relationship between coherent toposes/coherent logic and coherent sheaves
The definition of coherent topos is built on the definition of coherent object, which is essentially the same as the definition of coherent sheaf.
Mar
23
asked Is an open map with open relative diagonal necessarily a local homeomorphism?
Mar
14
comment Parodies of abstruse mathematical writing
Not quite writing per se, but perhaps relevant.
Mar
13
revised Canonical colimit and cartesian product of simplicial sets
edited tags
Mar
13
answered Canonical colimit and cartesian product of simplicial sets
Mar
3
comment Cartesian products between cofibrant simplicial presheaves
Yes, it does seem to carry over.
Mar
3
comment Definition of internal field objects
@მამუკაჯიბლაძე No, I don't think Martin's definition is especially similar to the notion of geometric field (which nLab calls "discrete field"). It's much closer to the definition via axiom F2.
Mar
3
comment Definition of internal field objects
It appears to me your definition is a translation of "$x$ is invertible in a field if and only if $x \ne 0$", where you have interpreted $x \ne 0$ using the notion of a complementary subobject. This is a natural definition, I suppose.
Feb
23
comment Subdivision of a small category
You are correct: the nerve of the category in question has infinitely many non-degenerate simplices.
Feb
20
answered Cartesian products between cofibrant simplicial presheaves
Feb
20
comment Cartesian products between cofibrant simplicial presheaves
So, you are assuming $\mathcal{C}$ has products?
Feb
19
comment homotopy tensor product of functors and bar construction
Short answer to final question: yes, the bar construction computes homotopy weighted colimits.
Feb
14
comment Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?
Hmmm. I don't think it's necessarily true that $S_\bullet (F \Delta^n)$ is always contractible. That would imply that the coequaliser of $\mathcal{C} (F \Delta^1, F \Delta^n) \rightrightarrows \mathcal{C} (F \Delta^0, F \Delta^n)$ is a singleton, but that's not true in general. (The simplest example is to take $F$ constant with value an object in $\mathcal{C}$ with more than one endomorphism. I don't know how to construct an example where $F \Delta^0$ is terminal.)
Feb
14
comment Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?
Yes, I'd rather think about the "singular simplicial set" than pass directly to chains. Then it's much easier to see that simplicial homotopies go to simplicial homotopies, and everything else is as for simplicial homology. I don't follow what you say about cosimplicial resolutions, though.
Feb
14
comment Does homotopy invariance of homology follow from the structure of the simplex category $\Delta$?
You write $\times$ but work in the category of simplicial abelian groups. Do you mean $\otimes$ instead? Either way, what does $\Delta^1 \times (-)$ mean?
Feb
13
comment The groupoid of algebraic expressions and proofs
Look up higher inductive types in homotopy type theory.