query: What is the meaning of $C^{op}$ if $C$ is a category without inverses?
positive_passages.docid: 155854
positive_passages.text: Do opposite categories always exist?
negative_passages.docid: 25159
negative_passages.text: Why does the inverse of the Hilbert matrix have integer entries?
negative_passages.docid: 57519
negative_passages.text: What is the opposite category of $\operatorname{Top}$?
negative_passages.docid: 93681
negative_passages.text: Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module?
negative_passages.docid: 42941
negative_passages.text: Pair of adjoint functors and showing that two certain maps are inverses
negative_passages.docid: 124003
negative_passages.text: Composition of adjoint functors
negative_passages.docid: 143648
negative_passages.text: Reconciling different definitions of left-exact functor
negative_passages.docid: 37041
negative_passages.text: Why is $\text{Aut}(F)$ of the forgetful functor $F$ on $G$-sets isomorphic to $G$?
negative_passages.docid: 70797
negative_passages.text: Why doesn't the definition of a category require an explicit notion of morphisms equality?
negative_passages.docid: 80260
negative_passages.text: Can it happen that the image of a functor is not a category?
negative_passages.docid: 52791
negative_passages.text: Invertibility of an operator $T$ if $\|I-T\|<1$
negative_passages.docid: 127000
negative_passages.text: When does $V = \ker(f) \oplus \operatorname{im}(f)$?
negative_passages.docid: 40793
negative_passages.text: What is the difference betwen equivalence and isomorphism of functors in categories.
negative_passages.docid: 152678
negative_passages.text: I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work?
negative_passages.docid: 109159
negative_passages.text: What's the intuition behind $(\exists y)(\mathscr{C}(y)\supset\mathscr{D})\supset((\forall x)\mathscr{C}(x)\supset\mathscr{D})$
negative_passages.docid: 64454
negative_passages.text: $L^2([0,1])$ is a set of first category in $L^1([0,1])$?
negative_passages.docid: 102899
negative_passages.text: Category theory with objects as logical expressions over $\{\vee,\wedge,\neg\}$ and morphisms as?