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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

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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

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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 
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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 firstorder 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 
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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 
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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 
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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 
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Who needs Replacement anyway?
@DavidRoberts That's not what Thomas means. He is referring to pairing in the sense of ordered pairs. 
Jun 9 
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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 
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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 firstorder 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 firstorder logic. 
Jun 8 
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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 nonstandard 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 categorytheoretic formulations of set theory. 
Jun 5 
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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 
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Ordinals in constructive mathematics ? (references)
Did you look at the section on ordinals in Homotopy type theory? 
May 28 
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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 