bio  website  dpmms.cam.ac.uk/~zll22 

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12h

comment 
Chain homotopy of nonabelian category
The notion of chain homotopy is induced by an "algebraic" interval, namely the chain complex corresponding to the simplicial complex $\Delta^1$. I don't see how to build that chain complex without using subtraction. 
1d

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Morita equivalence via Kan extension
Assuming $\mathcal{A}$ and $\mathcal{C}$ are small, $[\mathcal{A}^\mathrm{op}, \mathbf{Set}]$ and $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ are equivalent if and only if $\mathcal{A}$ and $\mathcal{C}$ have the same Cauchycompletion. If you study the proof more carefully you should be able to extract the necessary/sufficient conditions you want. 
Apr 15 
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Is continuity of a functor stable under pullback?
"Comma object" is at worst as bad as "comma category". Do you object to that too? 
Apr 14 
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The topologies for which a presheaf is a sheaf?
That's not what I'm asking. Rather, there are two reasonable interpretations of your condition: one only considers $Q (V)$ for open $V \subseteq S$, and another considers all $V \subseteq S$. 
Apr 14 
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The topologies for which a presheaf is a sheaf?
There seems to be a subtlety in the predicate "$Q$ is a sheaf". For instance, if $Q (\emptyset) = 1$, then is $Q$ automatically a sheaf for the indiscrete topology? Or do we also require that $Q (V) \to Q (U)$ be a bijection for all $\emptyset \subsetneq U \subseteq V \subseteq S$? 
Apr 8 
answered  Projectives and Injectives in Functor Categories 
Apr 8 
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Projectives and Injectives in Functor Categories
Even if you could choose a functorial projective cover in $\mathcal{A}$, there is still the fact that diagrams that are componentwise projective need not be projective. For instance, $\mathcal{C}$ could be the category freely generated by one endomorphism and $\mathcal{A}$ could be the category of $k$vector spaces; then a diagram $\mathcal{C} \to \mathcal{A}$ is the same thing as a $k [x]$module. 
Apr 7 
revised 
Why is a braided left autonomous category also right autonomous?
added 6 characters in body 
Apr 4 
awarded  Talkative 
Apr 1 
answered  Simple technical adjunction question 
Mar 31 
comment 
An isomorphism of categories
@MichalR.Przybylek There is lately a bad habit of writing $*$ for terminal objects. There is a unique (up to unique isomorphism) coproductpreserving functor $\mathbf{FinSet} \to \mathcal{C}$ that also preserves the terminal object. 
Mar 31 
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An isomorphism of categories
In the case $\mathcal{C} = \mathbf{Set}$ this is well known and probably due to Grothendieck. 
Mar 31 
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defining a bicategory of realvalued matrices
The $\mathbf{Poset}$enrichment of $\mathbf{Rel}$ is coming from the fact that $\{ 0, 1 \}$ is itself partially ordered. Perhaps you might have better luck with a (tropical) semiring instead of a field? 
Mar 31 
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Given a 2category, is the hammock localization wrt the equivalences equivalent to taking the homwise nerve of the maximal subgroupoids?
@PeterArndt There are some further necessary conditions. For instance, we need $\mathcal{C} [\mathcal{W}^{1}]$ to be isomorphic to the category obtained by identifying isomorphic 1cells. This fails for e.g. the 2category with a unique object, a nontrivial involution, and 2cells making the two 1cells isomorphic. You could probably get around this by introducing some kind of path or cylinder object. 
Mar 30 
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Given a 2category, is the hammock localization wrt the equivalences equivalent to taking the homwise nerve of the maximal subgroupoids?
More precisely, if we have an abelian group $A$, then we can form a 2category $\mathbb{B}^2 A$ with a unique object $*$, unique 1cell $\mathrm{id}_*$, and elements of $A$ as 2cells. The hammock localisation cannot see the automorphisms of the 1cell in this case, so the two simplicial categories are Dwyer–Kan equivalent if and only if $A$ is trivial. 
Mar 30 
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Transporting algebraic structure along adjoint equivalences
If I recall correctly, folklore says you can transport pseudoalgebra structures along (adjoint) equivalences. 
Mar 26 
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Objects are finite sets, arrows are matrices. How is this a category?
The definition is complete. Have you tried verifying the axioms? 
Mar 26 
answered  About reflective full subcategories and smallorthogonality classes 
Mar 23 
comment 
On the foundations for large categories
@arsmath Yes, I am perfectly aware. I did write a whole article on the subject, after all. In my mind, the purpose of the universe axiom is to allow us to treat sets, classes, collections of classes, collections of collections of classes, etc. all on the same basis. 
Mar 23 
comment 
On the foundations for large categories
@FernandoMuro Well, if you want to use Vopěnka's principle, that can be made to fit in the universeful setup as well: just posit that each set is a member of some universe that satisfies the relativised version of Vopěnka's principle. But I prefer not to assume large cardinal axioms to make settheoretical difficulties of that nature go away; the purpose of the universe axiom to repair a deficiency in firstorder logic. 