Return to Answer

$\newcommand{\colim}{\mathrm{colim}}$ For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

$\DeclareMathOperator\colim{colim}\newcommand\uHom{\underline{\operatorname{Hom}}}$For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times \mathrm{id}}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \uHom(I, A\times_C B) \to A \times_C B $$ where $\uHom$ is the internal hom in $\mathrm{Cat}$. By the assumption one sees that this is identified with the functor on $$ \uHom(I, A) \times_{\uHom(I, C)} \uHom(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

$\newcommand{\colim}{\mathrm{colim}}$ For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), Y, f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.

For the second question, it sounds like you have the right idea already. For the first, one way to view the situation is as follows. In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), \colim(Y), f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible. In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$. (Co-base change is just another word for the arrow obtained by pushing out.) Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits. One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

$$\require{AMScd} \begin{CD} T\downarrow C @>>> C^{[1]} \\ @VVV @VV{(s,t)}V \\ C\times C @>>{T \times id}> C\times C \end{CD}$$ where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps. This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example. Hence it is sufficient to understand colimits in categories of the form $A\times_C B$. There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does). For example, consider the global colimit functor $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$ where $\underline{Hom}$ is the internal hom in Cat. By the assumption one sees that this is identified with the functor on $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$ induced by the global colimit functors in $A$, $B$ and $C$. The original claim then follows by a direct computation.