How to show the following two definitions of homotopy monomorphism are equivalent?

Question

Let $M$ be a model category. In Toen's The homotopy theory of dg-categories and derived Morita theory Page 11 it is written:

a morphism $x \to y$ in a model category $M$ is called a homotopy monomorphism if for any $z\in M$ the induced morphism $$ Map_M(z,x)\to Map_M(z,y) $$ induces an injection on $\pi_0$ and isomorphisms on all $\pi_i$ for $i > 0$ (for all base points).

He then gives another definition of homotopy monomorphism:

This is also equivalent to say that the natural morphism $$ x\to x\times_y^h x $$ is an isomorphism in $Ho(M)$.

This two definition seems to be quite different. How to prove that they are actually equivalent?

Answer 1

Let $sSet$ be the category of simplicial sets with the Quillen model structure. Define a homotopy monomorphism in $sSet$ to be a morphism whose homotopy fibres are empty or weakly contractible. In a model category $C$, define a homotopy monomorphism $f : x \to y$ to be a morphism such that $f_* : Map(z,x) \to Map(z,y)$ is a homotopy monomorphism in $sSet$ for each object $z$.

Then it is easy to show that $f : x \to y$ is a homotopy monomorphism if and only if the diagonal $x \to x \times^h_y x$ is a weak equivalence. Indeed one reduces to the case $C = sSet$ easily. Then one can assume that $f$ is a Kan fibration between Kan complexes, so that this homotopy fibred product can be replaced by the ordinary fibred product $x \times_y x$. Similarly the homotopy fibres of $f$ can be replaced by ordinary fibres. Then the claim is clear.

Finally note that the above definition of homotopy monomorphism in $sSet$ is equivalent to the definition you wrote using homotopy groups: this follows directly by looking at the long exact sequence on homotopy groups.

Answer 2

Here is some philosophizing which won't directly answer your question because I won't say the words "model category" anywhere.

We know that in ordinary category theory a monomorphism $f : x \to y$ is a map such that for all $z$ the induced map $\text{Hom}(z, x) \to \text{Hom}(z, y)$ is injective. Let me rephrase this as follows: it is a map such that, if you give me a map $g : z \to y$, then "lifts of $g$ to $x$ along $f$" is a truth value. This means that I either can't do it or I can do it uniquely.

The universal map admitting such a lift is the map $f : x \to y$ itself. Lifts of this map correspond to sections of one of the projections $x \times_y x \to x$. There is a canonical lift, namely the identity, which corresponds to the section given by the natural relative diagonal map $x \to x \times_y x$, and the statement that this is an isomorphism precisely encodes the uniqueness of lifts (when they exist).

Now suppose that $f : x \to y$ is a morphism in an $(\infty, 1)$-category, so the induced map $\text{Hom}(z, x) \to \text{Hom}(z, y)$ is now a map of spaces. What is the correct analogue of "injective" here? Well, if $g : z \to y$ is a map, then "lifts of $g$ to $x$ along $f$" is now a space (the homotopy fiber of the above map based at $g$), and "truth value" now means "either empty or contractible." That is, either I can't lift $g$ to $x$ or I can lift it uniquely up to a contractible space of choices. An inspection of the long exact sequence in homotopy shows that this is equivalent to the first definition.

Again there is a universal map admitting such a lift, namely $f : x \to y$ itself. Lifts (here I always mean homotopy lifts) now correspond to sections of one of the projections $x \times^h_y x \to x$, and up to replacing pullbacks with homotopy pullbacks the story looks exactly the same as above, and we get the second definition.