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Apr
26
comment Injective Resolutions in $\mathfrak{Ab}(X)$
Injective resolutions are not easy things to find; that is why we have Čech cohomology!
Apr
26
revised Theory of promonads
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Apr
25
comment Are complex submanifolds necessarily closed?
It is conceivable that ‘submanifold’ means closed submanifold. I attended a lecture course on complex manifolds using that definition.
Apr
25
answered Theory of promonads
Apr
25
comment Theory of promonads
Hmmm, I'm not convinced your characterisation is correct. A promonad on a small category $\mathbb{D}$ should correspond to a monad on $[\mathbb{D}^\textrm{op}, \textbf{Set}]$ whose underlying endofunctor preserves all colimits.
Apr
25
comment Is there a category whose objects are sets, in which both products and coproducts are Cartesian products in the classical sense?
Easy: let $\mathcal{C}$ be the category whose objects "are" sets, and whose morphisms $X \to Y$ are the group homomorphisms from the free abelian group generated by $X$ to the free abelian group generated by $Y$. This has the required property.
Apr
24
answered If monic, then *property*. Does the converse hold?
Apr
24
comment pullback square of regular epimorphisms is a pushout
It's just the statement that if $A = B \cup C$, then $A$ is the pushout of $B \leftarrow B \cap C \rightarrow C$ as well.
Apr
24
comment What is the homotopy colimit of the Cech nerve as a bi-simplical set?
The diagonal of a bisimplicial set is weakly equivalent to its homotopy colimit: see here.
Apr
23
comment Can we think of an adjunction as a homotopy equivalence of categories?
Homotopy, defined in the classical sense for topological spaces using the standard interval, is an equivalence relation, and it is clear that is what the OP is thinking about. (Also note that in the model structure on $\textbf{Cat}$, two functors are homotopic precisely if they are isomorphic, not merely linked by a natural transformation.)
Apr
23
comment Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?
@HaraldHanche-Olsen It's not entirely useless. It can be used to prove a whole range of negative results, like the non-existence of large cardinals!
Apr
22
comment Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?
I do not know of an example off the top of my head. CW complexes are much nicer than general topological spaces, so it is more plausible that the claim is true in that case.
Apr
22
answered Are homotopy equivalent path-connected spaces homotopy equivalent as pointed spaces?
Apr
22
comment Homotopical equivalence of complexes
You have defined the notion of quasi-isomorphism. Every chain homotopy equivalence is a quasi-isomorphism but not vice versa.
Apr
22
comment How does one charaterize functionhood (etc.) in the category of relations?
$\textbf{Poset}$ is a subcategory of $\textbf{Cat}$ that is closed under finite products, so a category enriched over $\textbf{Poset}$ is also enriched over $\textbf{Cat}$.
Apr
22
answered How does one charaterize functionhood (etc.) in the category of relations?
Apr
22
answered pullback square of regular epimorphisms is a pushout
Apr
22
answered How Co-Comma Categories are constructed?
Apr
22
revised How Co-Comma Categories are constructed?
added 2 characters in body
Apr
21
comment pullback square of regular epimorphisms is a pushout
I've not seen any lemma like that, but there is one involving a pullback–pushout square of monomorphisms.