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Results tagged with enriched-category-theory
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user 294858
2
votes
Accepted
Interpreting a diagram in Borceux-Quinteiro's paper on enriched sheaves
Ordinary limits in $\mathcal K_0$ and conical limits in $\mathcal K$ exist and coincide whenever $\mathcal K$ has copowers (=tensors); see Kelly's book, page 50.
In your case you want $[\mathcal{C}^\t …
- 499
6
votes
Enriched vs ordinary filtered colimits
We deal with Question 1 in this paper, joint with Steve Lack.
In general it is not true that every $\kappa$-flat presheaf lies in the closure of the resperesntables under $\kappa$-filtered colimits an …
- 499
4
votes
Classification of absolute 2-limits?
A 2-category is Cauchy complete (in the sense you describe) if and only if idempotents splits, which is, if and only if its underlying ordinary category is Cauchy complete.
This holds more generally i …
- 499
7
votes
Relation between Ind-completion and "additive"-ind-completion
The equivalence you mention holds more generally whenever your base of enrichment has a finitely presentable unit. This certainly includes $Ab$ but also many other examples: $Cat$, $sSet$, $GAb$, $DGA …
- 499
6
votes
Accepted
Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical c...
Consider the terminal object $1\in\mathbf{Pos}$.
Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the co …
- 499