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Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here"here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

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Reid Barton
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Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.

Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.

Source Link
Reid Barton
  • 25.2k
  • 1
  • 76
  • 133

Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.