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comment 
Contraction of simplicial presheaves
Well, if you assume $X$ is cofibrant and $Y$ is fibrant then you do get such a weak equivalence. 
Apr 23 
answered  Strict comma objects implies comma objects 
Apr 22 
comment 
Is any model category simplicially enriched?
If you had some extra structure on $F$, then you could compose. If you had even more structure on $F$, then the composition would be associative. But that is a lot of structure on $F$! 
Apr 14 
answered  Difference between coherent nerve of simplical model category and simplicial category 
Apr 11 
awarded  Nice Question 
Apr 10 
comment 
Constructing unnatural transformations
That is true, but I could easily modify the example by replacing $C_{/ c}$ with $C$ itself. Now there is no room to even insert a mediating 2cell. 
Apr 10 
comment 
Constructing unnatural transformations
Here's a natural example of an "unnatural" morphism: for each object $c$ in $C$, there is a functor $1 \to C_{/ c}$ picking out the object $(c, \mathrm{id}_c)$. $C_{/ c}$ is functorial in $c$ in an obvious way, and $1$ is just constant – so naturality amounts to saying that this defines a cone over the diagram $C_{/ \bullet}$, but it does not. 
Apr 10 
accepted  Is an open map with open relative diagonal necessarily a local homeomorphism? 
Apr 10 
comment 
Is an open map with open relative diagonal necessarily a local homeomorphism?
Apparently the definition does not appear explicitly in Stone spaces, so I have added it. 
Apr 10 
revised 
Is an open map with open relative diagonal necessarily a local homeomorphism?
added 1388 characters in body 
Apr 9 
comment 
Is an open map with open relative diagonal necessarily a local homeomorphism?
There is a definition, which can be found in e.g. Stone spaces. 
Apr 9 
comment 
Is the infinitygroupoid of a finite CW complex finitelypresented?
See Mike Shulman's answer. 
Apr 9 
comment 
Is the infinitygroupoid of a finite CW complex finitelypresented?
It appears to me that the OP is thinking in terms of homotopy type theory, so here "finitely presented ∞groupoid" should be "higher inductive type with finitely many constructors". 
Apr 4 
comment 
What is the applications of the dgenhancements of derived categories of sheaves
I don't understand how you go from simplicial localisation to having a dgcategory at the end there. 
Apr 2 
comment 
Finitely presented categories and limits
It seems to me that you are trying to reinvent sketches and normal sketches. 
Mar 31 
comment 
When is the category of small (pre)sheaves a(n elementary) topos?
But not every sieve is small as a presheaf? 
Mar 30 
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When is the category of small (pre)sheaves a(n elementary) topos?
It is possible for an infinitary pretopos to be a topos without being a Grothendieck topos – take presheaves on a large group, for example. 
Mar 28 
revised 
Is an open map with open relative diagonal necessarily a local homeomorphism?
edited tags 
Mar 27 
comment 
The bifunctoriality of co/limits
The short answer is: derivators. 
Mar 27 
comment 
Applications of set theory in physics
Isn't that the controversial Chaos, Solitons & Fractals? 