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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 26th 2022
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   Author: nLab edit announcer
   Format: MarkdownItexthis factorization only holds for [[strict categories]], for [[univalent categories]] one needs to replace [[bijective-on-objects functors]] with [[essentially surjective functors]], and with general [[precategories]] one needs to replace them with [[equivalent-on-objects functors]], which generalize both bijective-on-objects functors and essentially surjective functors. Anonymous <a href="https://ncatlab.org/nlab/revision/diff/%28bo%2C+ff%29+factorization+system/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/%28bo%2C+ff%29+factorization+system/11">v11</a>, <a href="https://ncatlab.org/nlab/show/%28bo%2C+ff%29+factorization+system">current</a>

   this factorization only holds for strict categories, for univalent categories one needs to replace bijective-on-objects functors with essentially surjective functors, and with general precategories one needs to replace them with equivalent-on-objects functors, which generalize both bijective-on-objects functors and essentially surjective functors.

   Anonymous

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorvarkor
   • CommentTimeAug 26th 2022
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   Author: varkor
   Format: MarkdownItexI personally really don't like this trend of "correcting" usages of "category" to say "strict category" instead. Most of the readers of the nLab do not care about formalising category theory in Homotopy Type Theory: for them, a category has a class of objects and a class of morphisms. What you are calling a "strict category" is by definition a "category" for the vast majority of readers. I think this change serves little purpose other than obfuscation. If you really want to mention strictness issues, I think they would be better placed in their own new section.

   I personally really don’t like this trend of “correcting” usages of “category” to say “strict category” instead. Most of the readers of the nLab do not care about formalising category theory in Homotopy Type Theory: for them, a category has a class of objects and a class of morphisms. What you are calling a “strict category” is by definition a “category” for the vast majority of readers. I think this change serves little purpose other than obfuscation. If you really want to mention strictness issues, I think they would be better placed in their own new section.

3. • CommentRowNumber3.
   • CommentAuthormaxsnew
   • CommentTimeAug 26th 2022
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   Author: maxsnew
   Format: MarkdownItexI agree with @varkor, trying to make the nlab fully compatible with homotopy-theoretic foundations is not easy or desirable. Many pages on the nlab freely use the law of excluded middle and axiom of choice and have special comments on how to adapt them to constructive mathematics, so the idea of changing basic notions to homotopy-based ones doesn't make sense.

   I agree with @varkor, trying to make the nlab fully compatible with homotopy-theoretic foundations is not easy or desirable. Many pages on the nlab freely use the law of excluded middle and axiom of choice and have special comments on how to adapt them to constructive mathematics, so the idea of changing basic notions to homotopy-based ones doesn’t make sense.