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23h

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In the category of sets epimorphisms are surjective  Constructive Proof?
@GarlefWegart So what are you really trying to prove? That every epimorphism is a extremal epimorphism? Or regular? Or split? 
1d

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In the category of sets epimorphisms are surjective  Constructive Proof?
@MichalR.Przybylek Pretoposes are balanced. Do you mean quasitoposes? 
Aug 14 
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Are the pullback functors of adjoint functors also adjoint?
It's an easy exercise if you use the definition of adjunction with the triangle identities. 
Aug 14 
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Existence of Colimits in the Definition of Locally Presentable Categories
It doesn't answer the question that is actually asked, but rather the question that should have been asked. 
Aug 13 
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Atlas of a manifold as a Sheaf
I think what Tom Goodwillie is referring to is the locally ringed space formalism. He seems to imply it when he says "Daniel likes the option of replacing (1) by the logically equivalent". You can also represent manifolds as sheaves on the site of cartesian spaces, but that is very different. 
Aug 13 
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Zigzags and contractibility of categories
Do you have an interesting example where $Z \mathcal{C}$ has an initial object? The only ones I can think of are categories $\mathcal{C}$ with a unique strict initial object such as $\mathbf{Set}$, and those are contractible simply by virtue of having an initial object. 
Aug 12 
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Can we “complete” model categories to compute derived functors in the usual way?
Anyway, let me describe a situation where a derived functor exists by accident. Let $\mathcal{M}$ be any model category. Then $\mathrm{Hom} : \mathcal{M}^\mathrm{op} \times \mathcal{M} \to \mathbf{Set}$ has a right derived functor, where we consider $\mathbf{Set}$ with the discrete model structure, namely $\mathrm{Hom} : \operatorname{Ho} \mathcal{M}^\mathrm{op} \times \operatorname{Ho} \mathcal{M} \to \mathbf{Set}$. It seems very doubtful to me to say that there should be a way to see this in "the usual way" by extending $\mathcal{M}^\mathrm{op} \times \mathcal{M}$. 
Aug 12 
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Can we “complete” model categories to compute derived functors in the usual way?
Well, one can embed model categories satisfying certain smallness conditions into combinatorial model categories, but this procedure can't really be used to turn an arbitrary functor into a left Quillen functor. At minimum the original functor must preserve finite colimits, cofibrations, and trivial cofibrations. 
Aug 12 
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Can we “complete” model categories to compute derived functors in the usual way?
Adding new resolutions seems like a terrible idea. As far as I know, the only reason why model structures do actually compute the classical derived functors of homological algebra is because there are enough resolutions already. 
Aug 10 
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right adjoint functor for closed immersion of topoi
The answer to the nonadditive version of the question is no: for instance, the inclusion of a point in a $T_1$space is a closed immersion, but the induced direct image functor (i.e. the skyscraper sheaf functor) on sheaves of sets doesn't have a right adjoint in general. 
Aug 9 
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Category which has no nontrivial adjoint functors
However, if $C$ is empty, then $C \times A \cong C$. 
Aug 9 
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A general theory of quasifunctors, generalizing from dgcategories to $\mathcal V$categories, with $\mathcal V$ monoidal model category
My impression is that the Bergner model structure on simplicially enriched categories is not cartesian. So how do you construct the derived internal hom? 
Aug 7 
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(Homotopy) limits and colimits in a dgcategory
Of course, as with Kanenriched categories, one must first start with an adequate theory of homotopy limits in the base category (in this case, dgmodules). 
Aug 6 
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Two functorial definitions of schemes
Yes, there are some details to be filled in, but I'm sure it can be done. (You can use the comparison theorem, for example.) One has to use the appropriate Grothendieck topology on $\mathbf{Psh}$, of course. 
Aug 6 
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Two functorial definitions of schemes
Ah. Well, that is just a general fact from topos theory. See here. 
Aug 6 
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Two functorial definitions of schemes
What is the difference between (1) and (2)? You seem to have said the same thing twice. 
Aug 5 
answered  Kan condition in simplicial homotopy theory 
Jul 31 
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Existence of Colimits in the Definition of Locally Presentable Categories
Well, there is also the notion of accessible category, which is basically a locally presentable category without the hypothesis that all colimits exist. 
Jul 30 
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If an abelian category $\mathcal{A}$ has enough injectives then so does $\mathrm{Ch}^{\geq 0}(\mathcal{A})$
The title question is related to this one, but the body question seems to be something different. 
Jul 29 
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Tensor product of arbitrary categories
Your characterisation of tensor products of vector spaces is somewhat convoluted. If you do it the normal way, you will see that the tensor product of categories is precisely the cartesian product. (For an easier version of this, try to work out what the tensor product of partially ordered sets is.) 