In a model category, is the full subcategory of fibrant objects a reflective subcategory? I apologize in advance if my question is utterly stupid, but I can't resist asking it. So...
Is it true that in a model category ( - for example $\mbox{Set}_\Delta$ with the Joyal model structure - ) the full subcategory of fibrant objects is a reflective subcategory? More concretely, is the fibrant replacement functor a left adjoint to the inclusion functor?
Thanks.
 A: Although most model structures appearing "in practice" do not have a reflective subcategory of fibrant objects (unless every object is fibrant), the following result of A. Salch provides many examples that do.
Theorem. Let $\mathcal{C}$ be a complete and cocomplete category, and let $\mathcal{A}$ be a full subcategory which is reflective and replete (i.e., closed under isomorphisms). Then there exists a model structure on $\mathcal{C}$ such that $\mathcal{A}$ consists of the fibrant objects.
See Corollary 6.3 here:
http://arxiv.org/abs/1501.00508v1
The model structure is obtained as a (left) Bousfield localization of the discrete model structure on $\mathcal{C}$. The weak equivalences are the maps that become isomorphisms upon applying the reflector $\mathcal{C} \to \mathcal{A}$.
A: I don't think this question is really meaningful at the level of 1-categories, but assuming that what your after is the model category theoretic version of reflecting into a reflexive $\infty$-subcategory, then what your looking for is the fibrant replacements arising from (left) Bousfield localizations of simplicial model categories. More precisely, suppose $C$ is a simplicial model category and let $C_{loc}$ denote a left Bousfield localization (if one exists, e.g. $C$ is combinatorial). Then by the basic properties of the localization, the identity functor 
$$id:C_{loc}\to C$$
preserves fibrations and acyclic fibrations, and the identity functor
$$id:C\to C_{loc}$$
preserves cofibrations and acyclic cofibrations. Passing to fibrant/cofibrant objects gives an inclusion
$$i:C_{loc}^{o}\to C^o.$$
In general, the fibrant replacement functor in the local model structure will not be a strict left adjoint. However, it will be a left $\infty$-adjoint. In fact, it's pretty straightforward to show (using the basic properites of left Bousfield localization) that for locally fibrant $Y$, we have a natural equivalence:
$$\mathrm{Map}(R(X),Y)\simeq \mathrm{Map}(X,Y)\;$$
at the level of mapping spaces. 
A: The 1-categorical question is well-defined, and its answer is no, in general. I will provide examples below.
Lemma 1. Let $\eta \colon X \to F$ be a map from $X$ to a fibrant object, initial among such maps. Then $\eta$ is a trivial cofibration. In particular, $F$ is a fibrant replacement of $X$.
Proof. $\require{AMScd}$ Factor $\eta$ as $\eta = pi$, where $i \colon X \to Z$ is a trivial cofibration and $p \colon Z \to F$ is a fibration. In particular, $Z$ is fibrant. By the universal property of $\eta$, there exists a (unique) map $\phi \colon F \to Z$ satisfying $i = \phi \eta$. The equality $p \phi \eta = p i = \eta$ then implies $p \phi = 1_F$, again by the universal property of $\eta$. The commutative diagram 
\begin{CD}
X @= X @= X\\@V{\eta}VV @V{i}VV @V{\eta}VV\\ F @>{\phi}>> Z @>{p}>> F
\end{CD}
exhibits $\eta$ as a retract of $i$, hence a trivial cofibration. $\blacksquare$
Next, I will describe a model category in which the full subcategory of cofibrant objects is not coreflective. Taking the opposite model category, this provides a negative answer to the original question.
Let $R$ be a ring and consider $\mathrm{Ch}_{\geq 0}(R)$ the category of non-negatively graded chain complexes of (left) $R$-modules, equipped with the projective model structure. More precisely, the weak equivalences are quasi-isomorphisms, the fibrations are epimorphisms in positive degrees, and the cofibrations are degreewise monomorphisms with projective cokernel. In particular, a complex is cofibrant if and only if it is degreewise projective.
Observation 2. Let $M$ be an $R$-module, viewed as a complex concentrated in degree $0$. Then a chain map $f \colon C \to M$ is zero if and only if it induces the zero map on homology.
Proposition 3. Let $\epsilon \colon P \to M$ be a map from a cofibrant object to $M$, terminal among such maps. Then the following holds.


*

*Let $Q$ be a cofibrant chain complex and $g \colon Q \to P$ a chain map satisfying $\epsilon g = 0 \colon Q \to M$. Then $g$ is the zero map.

*The chain complex $P$ has trivial differential. In particular, its homology $H_n P = P_n$ is projective for every $n \geq 0$.

*$M$ is projective.


Proof. (1) follows from the universal property of $\epsilon$ and the equality $\epsilon g = 0 = \epsilon 0$. For (2), consider the chain complex $P_n[n-1]$ concentrated in degree $n-1$. The differential $d_n \colon P_n \to P_{n-1}$ defines a chain map $d_n \colon P_n[n-1] \to P$ which is null-homotopic. By Observation 2, we have $\epsilon d_n = 0$, and thus $d_n=0$. Now (3) follows from the isomorphism $H_0 P \cong M$, using Lemma 1. $\blacksquare$
Corollary 4. The full subcategory of cofibrant objects in $\mathrm{Ch}_{\geq 0}(R)$ is coreflective if and only if every $R$-module is projective (in which case every object of $\mathrm{Ch}_{\geq 0}(R)$ is cofibrant).
A: In the case that all isomorphisms in the model category are fibration, cofibration, and weak equivalences, (trivial cofibrations, fibrations) and (fibrations, trivial cofibrations) form factorization systems in the sense of Borceux: 

A factorization system in a category $\mathbf B$ is defined as a pair
  $(E, F)$ of classes of arrows in $\mathbf B$ such that
  
  
*
  
*every isomorphism belongs to both E and M
  
*both E and M are closed under composition
  
*$\forall e \in E \forall m \in M. e \perp m$ (i.e. e has the LLP wrt m)
  
*every arrow f in B can be factored as f = me with e in E and m in M
  

Then the following theorem in Borceux's Handbook of Categorical Algebra, volume 1, page 211, gives an affirmative answer:

Suppose $\mathbf B$ is a category with a terminal object 1 and a factorization system (E, F). For each object $B \in \mathbf B$, factor the unique arrow $B \xrightarrow ! 1$ into $B \xrightarrow {e_B} r(B) \xrightarrow {f_B} 1$. Then $r$ is a reflection of $\mathbf B$ into the full subcategory spanned by all $r(B)$, and each $e_B$ is the unit of the reflection.

