Conservative Functor: A Counterexample

Hey guys, let's dive into a super interesting topic in category theory today: conservative functors! We're going to explore a specific scenario where a functor is conservative but doesn't quite hit the mark when it comes to the Right Lifting Property (RLP). This might sound a bit niche, but trust me, understanding these nuances is crucial for anyone really getting their hands dirty with abstract mathematics, especially in areas like simplicial sets and higher category theory. We'll be working with the 'walking isomorphism', which is a foundational concept, and seeing how a particular functor behaves. So, grab your thinking caps, and let's unravel this! This exploration is key to understanding the subtle differences between being conservative and possessing the RLP, and why just one property doesn't guarantee the other. It's like knowing someone is a good listener (conservative) but not necessarily someone who always gives you the best advice (RLP). We'll be building up to a concrete example that illustrates this, so stick around.

The Walking Isomorphism and the Functor $J o S(|J|)$

Alright, let's start by setting the stage. We're talking about the walking isomorphism, often denoted by $J$. In category theory, this is a really neat way to represent the abstract notion of an isomorphism. Think of $J$ as a category with two objects, let's call them $0$ and $1$, and two non-identity morphisms: an arrow from $0$ to $1$ (let's call it $f$) and an arrow from $1$ to $0$ (let's call it $g$). Crucially, these are inverses of each other, meaning $f eq ext{id}_0$ and $g eq ext{id}_1$, and their compositions are the identities: g rown f = ext{id}_0 and f rown g = ext{id}_1. This structure perfectly captures what an isomorphism is: an invertible arrow. Now, we're going to consider a functor, let's call it $F$, that maps from this walking isomorphism category $J$ to the category of simplicial sets, denoted by $S$. So, we have $F: J o S$. What exactly does $F$ do? Well, it maps the objects of $J$ (0 and 1) to some simplicial sets, and the morphisms of $J$ ($f$ and $g$) to some morphisms between these simplicial sets. A very common and important example of such a functor is the one that maps $J$ to the nerve of $J$, specifically $|J|$, and then into the category of simplicial sets $S$. So, our functor in question is $F = |-|: J o S(|J|)$. This notation $|-|$ usually refers to the realization functor, which takes a simplicial set and produces a topological space, or sometimes, as in this context, the nerve construction which maps a category to a simplicial set. When we apply the $|-|$ functor to the walking isomorphism $J$, we get a simplicial set $|J|$. This $|J|$ is the nerve of the category $J$. The nerve of a category is a simplicial set that encodes the structure of the category. For the walking isomorphism, its nerve $|J|$ is a specific, simple simplicial set. It has a single 0-simplex (representing the objects) and two 1-simplices (representing the non-identity morphisms). The realization $|J|$ of this simplicial set is a topological space homeomorphic to the interval $[0,1]$. The functor we're interested in is $F: J o S(|J|)$, which is often understood as the Yoneda embedding of $J$ into the presheaf category $S^{J^{op}}$, and then taking the realization. However, the context here suggests a different interpretation, perhaps mapping $J$ to the category of simplicial sets, denoted as $ extSet}^{ ext{op}}$, where $ ext{Set}$ is the category of sets. When we talk about $S(|J|)$, it's more likely referring to the category of simplicial sets itself, and the functor maps from the walking isomorphism to this category. The key here is that the functor $F$ takes the structure of the walking isomorphism and represents it within the world of simplicial sets. We want to prove that this functor, $F$, is conservative. A functor $F C o D$ is conservative if for any morphism $f: X o Y$ in $C$, if $F(f): F(X) o F(Y)$ is an isomorphism in $D$, then $f$ itself must be an isomorphism in $C$. This means the functor preserves the property of being an isomorphism in a way that if the image is an isomorphism, the original must have been one too. So, we're examining if $F(|J|)$ preserves isomorphism. The statement says we're proving $J o S(|J|)$ is conservative. This notation is a bit dense, let's clarify. It likely means a functor $F: ext{Cat o extSet}^{ ext{op}}$, where $ ext{Cat}$ is the category of small categories and $ ext{Set}^{ ext{op}}$ is the category of simplicial sets (often denoted by $S$), and we are considering the image of the walking isomorphism $J$ under this functor, resulting in a simplicial set $|J|$. The functor we're analyzing maps from $J$ to $S(|J|)$, which is a bit meta. A more standard interpretation is a functor $F ext{Cat o S$ where $S$ is the category of simplicial sets. We are given the walking isomorphism $J$ as a category. The functor maps $J$ to $S$. Let's call this functor $F$. We want to show $F$ is conservative. The typical functor used here is the nerve functor, $N: ext{Cat} o S$. So, let's consider $F = N$. The nerve of the walking isomorphism $J$ is $|J|$, a specific simplicial set. The functor $N$ maps $J$ to $|J|$. The question is about a functor $J o S(|J|)$. This notation is still a bit tricky. Let's assume the question means a functor $F: ext{Cat} o S$ and we are looking at the behavior of $F$ on the walking isomorphism $J$. The statement might be implying a functor that takes a category $C$ and maps it to the category of simplicial sets over the nerve of $C$, i.e., $F(C): ext{Set}^{ ext{op}}_{/N(C)}$. However, given the context of