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Aug
21
comment Clarification of definition of category
A good example of a category which does not have the required property in practice!
Aug
21
comment Clarification of definition of category
However, in terms of notation that is what we do all the time...
Aug
21
answered Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?
Aug
21
comment Why is an $E_\infty$-operad a kind of ''strictification'' for a non-commutative operation?
The definition of algebra over a symmetric operad includes a compatibility condition on the action of the symmetric group. Is that what you are asking about?
Aug
21
answered Clarification of definition of category
Aug
21
comment Unnecessary hypothesis for a pullback in Hungerford
There is no need to assume $\epsilon$ is monic: this is the pullback pasting lemma.
Aug
20
comment When does a pushout of monics induces a monic arrow?
But BC (in its full generality) is false in $\mathbf{sSet}$ as you have already observed! The pushout–product map you are interested in is a monomorphism because this particular pushout computes the union of subobjects. This is true in any elementary topos.
Aug
20
comment When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product
If the chain complexes are cofibrant in an appropriate model structure then this would follow from Ken Brown's lemma, but cofibrant chain complexes tend to be levelwise projective...
Aug
20
comment When colimit of subobjects is still a subobject?
If you said join then the diagram would have been filtered, and then what you want is often true. For meet, things are harder, but I think what you want is true in, say, a Grothendieck topos.
Aug
20
comment When colimit of subobjects is still a subobject?
What kind of semilattice: join or meet?
Aug
20
answered Epimorphisms and Algebras
Aug
20
comment Why SKI when SK is complete
If you look at the proof, you will see that S, K, and I correspond precisely to each of the structural induction steps. Try proving it on your own!
Aug
20
comment Why SKI when SK is complete
The SKI system is natural in the sense that it is exactly what you need to prove the abstraction elimination theorem.
Aug
19
comment Free $\Phi$-cocompletion of a category
Hmmm. I don't see the connection either, but I haven't studied the paper carefully. The way I would do it is to show that the class of presheaves obtained as colimits of finite diagrams of representables is already closed under finite colimits: for finite coproducts this is obvious, and for coequalisers one builds a diagram that looks a bit like a bipartite graph.
Aug
19
answered Free $\Phi$-cocompletion of a category
Aug
18
answered Localization of modules as adjunction
Aug
18
comment Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?
You should add the assumption that $\mathcal{L}$ only has relation symbols and no function symbols, or otherwise mention that in $\mathcal{L}^+$ we replace all function symbols by relation symbols. Either way, we have to add appropriate axioms asserting that the relation symbols have the appropriate "domains".
Aug
18
comment How to prove logical truth in predicate calculus?
The given formula is false: for a countermodel, take a one-element set whose unique element satisfies $P$ and $Q$ but not $R$.
Aug
18
awarded  Strunk & White
Aug
18
revised Prove $\mathbb{R}^3$ is not the product of two identical topological spaces
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