Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: algebraic topology

Topic: Lifting a model structure to T-graphs

James Deikun (Jan 29 2023 at 19:10):

T-graphs for a [[Cartesian monad]] $T :: \mathcal{E} \to \mathcal{E}$ look like $TX_0 \xleftarrow{d_X} X_1 \xrightarrow{c_X} X_0$ and their morphisms look like:

$$\begin{CD} TX_0 @<{d_X}<< X_1 @>{c_X}>> X_0 \\ @V{Tf_0}VV @V{f_1}VV @V{f_0}VV \\ TY_0 @<{d_Y}<< Y_1 @>{c_Y}>> Y_0 \end{CD}$$

I want to lift a model structure consisting of [[algebraic weak factorization system]] s $(\mathcal{C},\mathcal{F}')$ and $(\mathcal{C'},\mathcal{F})$ from the underlying category $\mathcal{E}$ to the category of T-graphs. For various reasons I believe the correct notion has as acyclic cofibrations, morphisms where $f_0$ is an acyclic cofibration and $f_1$ is simply a cofibration, while for a cofibration they are both just cofibrations. I'm having trouble though lifting the algebraic factorizations.

$$\begin{CD} TX_0 @<{d_X}<< X_1 @>{c_X}>> X_0 \\ @V{T\mathcal{C}'f_0}VV @V{\mathcal{C}f_1}VV @V{\mathcal{C}'f_0}VV \\ TZ_0 @. Z_1 @. Z_0 \\ @V{T\mathcal{F}f_0}VV @V{\mathcal{F}'f_1}VV @V{\mathcal{F}f_0}VV \\ TY_0 @<{d_Y}<< Y_1 @>{c_Y}>> Y_0 \end{CD}$$

I can't figure out how to obtain the missing $d_Z$ and $c_Z$ to make the intermediate object $Z$ a T-graph. Am I missing something really obvious because I'm not used to working with weak factorization systems? Or is it actually difficult?

James Deikun (Jan 29 2023 at 19:46):

(Actually I'm not that sure my acyclic cofibs are right, maybe that's the problem. In fact, they're definitely wrong now that I look at it. They should be generated, but possibly not freely, by morphisms like this:

$$\begin{CD} T(X+Y) @<{\eta_T \circ \iota_1}<< X @>{c \circ \iota_2}>> X+Y \\ @V{T(ac+a)}VV @V{ac}VV @V{ac+a}VV \\ T(Z+Z) @<{\eta_T \circ \iota_1}<< Z @>{\iota_2}>> Z+Z \end{CD}$$

where $c$ is a cofibration and $a$ an acyclic cofibration.)

James Deikun (Jan 30 2023 at 13:59):

I guess a first stab at analyzing this is that the generating morphisms factor as

$$\begin{CD} T(X+Y) @<{\eta_T\circ\iota_1}<< X @>{c\circ\iota_2}>> X+Y \\ @V{T(a'+a)}VV @V{a'}VV @V{a'+a}VV \\ T(W+Z) @<{\eta_T\circ\iota_1}<< W @>{c'\circ\iota_2}>> W+Z \\ @V{T(c'+\mathrm{id})}VV @V{c'}VV @V{c'+\mathrm{id}}VV \\ T(Z+Z) @<{\eta_T\circ\iota_1}<< Z @>{\iota_2}>> Z+Z \end{CD}$$