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1d
comment Explicit computation of a limit of a cosimplicial object
Yes, you can make $F_*$ into a simplicial algebra that way, but do you still mean $\varprojlim_{\mathbf{\Delta}^\mathrm{op}} F_*$? There is a very easy formula for that, namely $F_0$.
1d
comment Explicit computation of a limit of a cosimplicial object
Do you mean $(F_*, d, s)$ is a cosimplicial algebra? Do you mean $\varprojlim_{\mathbf{\Delta}} F_*$?
2d
comment What is the most transparent, rigorous definition of the Univalence Axiom?
You really must understand type theory first – at minimum the notion of identity types. I think it would be enough to read Chapter 1 of the HoTT book, then skip to §2.10.
Jun
29
comment Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?
@EricWofsey Of course, the usefulness of this observation is directly proportional to the expressiveness of the first-order language in question. But I am sure you know that.
Jun
29
awarded  Nice Question
Jun
28
asked “To operate the machine, it is not necessary to raise the bonnet.”
Jun
25
comment Internal Hom on simplicial presheaves and the preservation of cofibrant objects
You have shown that the representable presheaf is a retract of the mapping space; what you need to show is that the mapping space is a retract of a disjoint union of representable presheaves. To me, that seems unlikely.
Jun
24
comment Do flat resolutions guarantee the existence of Tor (without enough projectives)?
Are you asking about Tor in the sense of universal $\delta$-functors? If so, can't you use the effaceability criterion? (I suppose one might have to ask for a functorial flat resolution to ensure that one really gets a functor...)
Jun
24
comment Do smooth manifolds create colimits for complex manifolds?
An easy observation is that the inclusion creates coproducts and coequalisers whose quotient morphism is a local homeomorphism.
Jun
23
comment Van Kampen colimits
@MarcHoyois It appears to me that Michal R. Przybylek is talking about 2-(co)limits in the classical sense of ($\mathbf{Cat}$-)enriched category theory, whereas you are talking about bi(co)limits.
Jun
12
answered smash product of pointed spaces preserve weak equivalences
Jun
9
comment Who needs Replacement anyway?
@DavidRoberts That's not what Thomas means. He is referring to pairing in the sense of ordered pairs.
Jun
9
comment Who needs Replacement anyway?
@DavidRoberts Of course ETCS has a notion of pairing. It's built right into the definition of binary product!
Jun
8
comment Who needs Replacement anyway?
I don't think definability is especially important. It only comes in because the axiom of replacement as formulated classically is a first-order axiom scheme. I do not know of a good way of formulating the notion of a "definable endofunctor", which would be needed to formalise the principle that "endofunctors can be iterated" in first-order logic.
Jun
8
comment Who needs Replacement anyway?
The example with $(X, P (X), P^2 (X), \ldots)$ is more subtle than it looks. For instance, imagine a model of ETCS with non-standard naturals: then you wouldn't even be able to define $P^n (X)$ for all natural numbers $n$, let alone apply replacement to that "function". These two issues – the "large" recursion principle and the axiom of replacement – seem to be intertwined in category-theoretic formulations of set theory.
Jun
5
comment Derived functors - homotopical vs homological approach
I do not think it is possible to compare the two definitions in complete generality. I asked a related question some years ago – it is a fact that universal $\delta$-functors computed using resolutions annihilate injective objects, but I do not know if this is true for all universal $\delta$-functors.
Jun
1
comment Ordinals in constructive mathematics ? (references)
Did you look at the section on ordinals in Homotopy type theory?
May
28
comment Does the property of being a local homeomorphism descend through split surjections?
Thanks! For some reason I was sure that there had to be a counterexample, so the positive answer surprises me. Now I have to rethink my intuitions on this matter...
May
28
accepted Does the property of being a local homeomorphism descend through split surjections?
May
28
revised Does the property of being a local homeomorphism descend through split surjections?
Simplified question