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clarified second bullet point.
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Morgan Rogers
Morgan Rogers
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Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies (where I should reiterate that $\Phi$ contains all finite diagrams). Is this more than a linear order? (edited)

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies. Is this more than a linear order?

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies (where I should reiterate that $\Phi$ contains all finite diagrams). Is this more than a linear order? (edited)

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

clarified second bullet point in response to comment
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Morgan Rogers
Morgan Rogers
  • 519
  • 4
  • 16

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-colimit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies. Is this more than a linear order?

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-colimit-commutation classes?

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies. Is this more than a linear order?

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-commutation classes strictly contained in filtered colimits? (edited)

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies. Is this more than a linear order?

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.

Source Link
Morgan Rogers
Morgan Rogers
  • 519
  • 4
  • 16

Are there any interesting classes of limits containing finite limits?

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any other examples? To make this more concrete, here are two specific questions I am interested in:

  • Considering that filtered colimits commute with finite limits, and $\kappa$-filtered colimits commute with $\kappa$-small limits, are there any other limit-colimit-commutation classes?

  • There is a heirarchy of 2-categories of categories having all $\Phi$-limits as $\Phi$ varies. Is this more than a linear order?

I'm just asking about ordinary/conical limits here, but more exotic answers/counterexamples could be of interest.