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12h
comment Chain homotopy of non-abelian 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
comment 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 Cauchy-completion. If you study the proof more carefully you should be able to extract the necessary/sufficient conditions you want.
Apr
15
comment 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
comment 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
comment 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
comment 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) coproduct-preserving functor $\mathbf{FinSet} \to \mathcal{C}$ that also preserves the terminal object.
Mar
31
comment An isomorphism of categories
In the case $\mathcal{C} = \mathbf{Set}$ this is well known and probably due to Grothendieck.
Mar
31
comment defining a bicategory of real-valued 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
comment Given a 2-category, is the hammock localization wrt the equivalences equivalent to taking the hom-wise 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 1-cells. This fails for e.g. the 2-category with a unique object, a non-trivial involution, and 2-cells making the two 1-cells isomorphic. You could probably get around this by introducing some kind of path or cylinder object.
Mar
30
comment Given a 2-category, is the hammock localization wrt the equivalences equivalent to taking the hom-wise nerve of the maximal subgroupoids?
More precisely, if we have an abelian group $A$, then we can form a 2-category $\mathbb{B}^2 A$ with a unique object $*$, unique 1-cell $\mathrm{id}_*$, and elements of $A$ as 2-cells. The hammock localisation cannot see the automorphisms of the 1-cell in this case, so the two simplicial categories are Dwyer–Kan equivalent if and only if $A$ is trivial.
Mar
30
comment Transporting algebraic structure along adjoint equivalences
If I recall correctly, folklore says you can transport pseudo-algebra structures along (adjoint) equivalences.
Mar
26
comment 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 small-orthogonality 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 universe-ful 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 set-theoretical difficulties of that nature go away; the purpose of the universe axiom to repair a deficiency in first-order logic.