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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2010
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   Author: Urs
   Format: Markdowncreated [[monomorphism in an (infinity,1)-category]]

   created monomorphism in an (infinity,1)-category

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeFeb 24th 2010
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   Author: zskoda
   Format: MarkdownIt is a bit confusing: a direct summand of an infinity groupoid. How does this look like, e.g. what is in reality a direct sum of Kan complexes ? (I don't mean the uni property but explicit description).

   It is a bit confusing: a direct summand of an infinity groupoid. How does this look like, e.g. what is in reality a direct sum of Kan complexes ? (I don't mean the uni property but explicit description).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2010
   • (edited Feb 24th 2010)
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   Author: Urs
   Format: MarkdownIt just means: the two oo-groupoids each consist of a bunch of connected components <latex> C = \coprod_i C_i </latex> and <latex> D = \coprod_j D_j </latex> and a morphism <latex> C \to D</latex> satisfies this condition if the component maps <latex> C_i \to D_{j(i)} </latex> are equivalences and the function <latex> i \mapsto j(i) </latex> is injective.

   It just means: the two oo-groupoids each consist of a bunch of connected components and and a morphism satisfies this condition if the component maps are equivalences and the function is injective.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2010
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   Author: Urs
   Format: MarkdownI added that to the entry, now.

   I added that to the entry, now.

5. • CommentRowNumber5.
   • CommentAuthorMatanP
   • CommentTimeAug 1st 2012
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   Author: MatanP
   Format: MarkdownItexSo in particular, in the $(\infty,1)$-topos of topological spaces, an inclusion (even a nice one) need not be an $(\infty,1)$-monomorphism. Do anyone here knows of a different notion for a monomorphism in an $(\infty,1)$-topos that will include (nice) inclusions of topological spaces?

   So in particular, in the -topos of topological spaces, an inclusion (even a nice one) need not be an -monomorphism. Do anyone here knows of a different notion for a monomorphism in an -topos that will include (nice) inclusions of topological spaces?

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeAug 2nd 2012
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   Author: TobyBartels
   Format: MarkdownItexI don't have such a good feel for that $(\infty,1)$-category as I should, but here's one general fact that may be relevant: in an $(\infty,1)$-category (and already in a $(2,1)$-category, a [[regular monomorphism in an (infinity,1)-category|regular monomorphism]] need not be a monomorphism. So if nice inclusions aren't monomorphisms, they may yet be regular (or strong, extremal, etc) monomorphisms.

   I don’t have such a good feel for that -category as I should, but here’s one general fact that may be relevant: in an -category (and already in a -category, a regular monomorphism need not be a monomorphism. So if nice inclusions aren’t monomorphisms, they may yet be regular (or strong, extremal, etc) monomorphisms.

7. • CommentRowNumber7.
   • CommentAuthorMatanP
   • CommentTimeAug 3rd 2012
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   Author: MatanP
   Format: TextThanks. I want to carefully examine the other notions you mentioned (regular, strong external), but except the nLab stubs I couldn't find any references. Can you (or someone) refer me to a place with a bit more details, i.e. other than the definition itself?

   Thanks. I want to carefully examine the other notions you mentioned (regular, strong external), but except the nLab stubs I couldn't find any references. Can you (or someone) refer me to a place with a bit more details, i.e. other than the definition itself?
8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeAug 5th 2012
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   Author: TobyBartels
   Format: MarkdownItexIn $(\infty,1)$-categories, I don't know; Urs would. In $1$-categories, the nLab has somewhat better coverage of the dual epimorphisms; see [[epimorphism#Variations]] for discussion of how the different forms relate.

   In -categories, I don’t know; Urs would. In -categories, the nLab has somewhat better coverage of the dual epimorphisms; see epimorphism#Variations for discussion of how the different forms relate.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeAug 5th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI think it's misleading to talk about "the $(\infty,1)$-topos of topological spaces"; it's better to call it the $(\infty,1)$-topos of $\infty$-groupoids. Topological spaces are a particular way to present $\infty$-groupoids, but a *lot* of information is lost in passing from a topological space to its fundamental $\infty$-groupoid. In particular, *every* morphism of $\infty$-groupoids can be presented by a subspace inclusion of topological spaces, namely a relative cell complex (a cofibration in the Quillen model structure).

   I think it’s misleading to talk about “the -topos of topological spaces”; it’s better to call it the -topos of -groupoids. Topological spaces are a particular way to present -groupoids, but a lot of information is lost in passing from a topological space to its fundamental -groupoid.

   In particular, every morphism of -groupoids can be presented by a subspace inclusion of topological spaces, namely a relative cell complex (a cofibration in the Quillen model structure).