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Proposition 10. The image of a projective quadric under a projective isomorphism is a projective quadric of the same rank. | PROOF. Let \( \pi \) be a projective isomorphism of \( \mathcal{P}\left( V\right) \) onto \( \mathcal{P}\left( {V}^{\prime }\right) \) . The special cases where pdim \( V \) equals 0 or 1 are obvious and so we may assume by Theorem 6 of Chapter III (p. 57) that \( \pi = \mathcal{P}\left( g\right) \) where \( g \) is a ... | Yes |
Given an affine quadric in an n-dimensional affine geometry \( A \), then a coordinate system for \( A \) can be chosen with respect to which the quadric has an equation of one of the following two types:\n\n\[ {d}_{1}{X}_{1}{}^{2} + \cdots + {d}_{n}{X}_{n}{}^{2} + {d}_{0} = 0, \]\n\n(1)\n\n\[ {X}_{1} + {d}_{2}{X}_{2}{... | PROOF. Let \( A \) be the affine geometry determined by the hyperplane \( H \) in \( \mathcal{P}\left( V\right) \) and let \( Q\left( \sigma \right) \) be a quadric in \( \mathcal{P}\left( V\right) \) that determines the given affine quadric \( {Q}^{\prime } \) in \( \mathrm{A} \).\n\n(i) If \( {H}^{ \bot } \subset H \... | Yes |
Proposition 13. Let \( \mathrm{A} \) be the affine geometry determined by the hyperplane \( H \) in \( \mathcal{P}\left( V\right) \) . If \( {Q}^{\prime } \) is an affine quadric in \( \mathrm{A} \) which is not contained in any hyperplane of \( \mathrm{A} \), then there is one and only one quadric \( Q \) in \( \mathc... | PROOF. We have seen above that there is always at least one quadric \( Q \) completing \( {Q}^{\prime } \) . What has to be done now is to show that the points of \( Q \) in \( H \) are determined by \( {Q}^{\prime } \) . We do this by proving that a point \( P \) of \( H \) does not lie on \( Q \) if, and only if, the... | Yes |
Proposition 14. An orthosymmetric sesquilinear form of rank \( \geq 2 \) is either alternating or is a scalar multiple of a hermitian form. | The lemma shows that if we wish to extend the structure theorems of \( §{5.4} \) to sesquilinear forms we need only consider forms that are not alternating. Theorem 3 is true for (non-alternating) hermitian forms and Theorem 4 is true for arbitrary (but non-alternating) sesquilinear forms. The proofs of both theorems a... | No |
Proposition 1. An isomorphism of euclidean geometries is an affinity whose underlying linear isomorphism is an isomorphism of euclidean spaces. | Proof. Suppose \( \dim S \geq 2 \) and that \( M,{M}^{\prime } \) are the subspaces belonging to the cosets \( S,{S}^{\prime } \), respectively. By Theorem 5 of Chapter III (p. 56), \( \alpha = \mathcal{A}\left( {{t}_{-a}g{t}_{a\alpha }}\right) \), where \( g \) is a semi-linear isomorphism of \( M \) onto \( {M}^{\pri... | Yes |
Proposition 2. The real polarity \( \bot \left( \sigma \right) \) is definite if, and only if, the bilinear form \( \sigma \) is either positive definite or negative definite. | PROOF. The \ | No |
Proposition 3. If \( f \) is a linear mapping of the (non-zero) complex vector space \( W \) into itself, then \( f \) has eigenvectors in \( W \) . | PROOF. Since \( W \) is finite dimensional there is a least positive integer \( r \) such that \( {f}^{r} \) is linearly dependent on \( {f}^{r - 1},\ldots, f,{1}_{W} \), say\n\n\[ \n{f}^{r} = {c}_{1}{f}^{r - 1} + \cdots + {c}_{r}{1}_{W} \n\]\n\nIf we put\n\n\[ \nm\left( X\right) = {X}^{r} - {c}_{1}{X}^{r - 1} - \cdots... | Yes |
Proposition 4. If \( \\left( {V,\\sigma }\\right) \) is a euclidean space and \( f \) is a \( \\sigma \) -symmetric linear mapping of \( V \) into \( V \) then there is a cartesian basis of \( \\left( {V,\\sigma }\\right) \) consisting entirely of eigenvectors of \( f \) . | PROOF. By Lemma 2 we can find an eigenvector \( {a}_{1} \) of \( f \) . Since \( \\sigma \) is positive definite we may choose \( {a}_{1} \) so that \( \\sigma \\left( {{a}_{1},{a}_{1}}\\right) = 1 \) . The subspace \( \\left\\lbrack {a}_{1}\\right\\rbrack \) is non-degenerate with respect to \( \\sigma \) and therefor... | Yes |
Lemma 6. If \( {Mf} = M \), then \( \left( {M}^{ \bot }\right) f = {M}^{ \bot } \) . | PROOF. For any \( a \) in \( M \) and \( b \) in \( {M}^{ \bot } \) we have \( a{f}^{-1} \in M \) and so\n\n\[ \sigma \left( {a,{bf}}\right) = \sigma \left( {a{f}^{-1}, b}\right) = 0. \]\n\nThus \( \left( {M}^{ \bot }\right) f \subset {M}^{ \bot } \) and equality follows because \( f \) is one-one. | Yes |
Proposition 5. Let \( \left( {A, \bot }\right) \) be the similarity euclidean geometry in \( \mathcal{P}\left( V\right) \) determined by \( H \) and the definite polarity \( \bot \) of \( \mathcal{P}\left( H\right) \) . Denote by \( {\left( \Pr \mathcal{P}\left( V\right) \right) }_{H, \bot } \) the subgroup of the proj... | PROOF. Let \( \mathcal{P}\left( f\right) \) be an element of \( {\left( \Pr \mathcal{P}\left( V\right) \right) }_{H} \) and let \( g \) be the restriction of \( f \) to \( H \) . By Lemma \( 7,\mathcal{P}\left( g\right) \) preserves \( \bot \) if, and only if, \( g \) preserves \( D \) . By our definition of the distan... | Yes |
Proposition 1. If \( R \) is a commutative ring with identity then \( R \) is a field if, and only if, \( \{ 0\} \) and \( R \) are the only ideals of \( R \) . | PROOF. Suppose that \( R \) is a field and that \( x \) is a non-zero element in an ideal \( \mathfrak{a} \) of \( R \) . For any \( r \) in \( R \) the element \( r{x}^{-1}x = r \) is in \( \mathfrak{a} \) and so \( \mathfrak{a} = R \ | No |
Proposition 2. Every ideal of \( \mathbf{Z} \) or \( F\left\lbrack X\right\rbrack \) is principal. | PROOF. Let \( \mathfrak{a} \) be a non-zero ideal of \( F\left\lbrack X\right\rbrack \) and \( p \) any non-zero element in a of smallest degree. Any \( q \) in a can be written in the form\n\n\[ q = {up} + v, \]\n\nwhere degree \( v < \) degree \( p \) . Since \( {up} \in \mathfrak{a}, v \in \mathfrak{a} \) and so \( ... | Yes |
Proposition 3. If \( M = \left\lbrack a\right\rbrack \) is a cyclic \( F\left\lbrack X\right\rbrack \) -module whose annihilator \( \mathfrak{a}\left( M\right) \) is generated by the non-zero polynomial \( p\left( X\right) \) of degree \( s \), then \( M \) is a vector space over \( F \) of dimension \( s \) . | PROOF. The \ | No |
Proposition 4. If \( M \) is a cyclic \( R \) -module, then \( M \) is isomorphic to \( R/\mathfrak{a}\left( M\right) \) . | PROOF. Let \( M = \left\lbrack a\right\rbrack \) and consider the mapping\n\n\[ f : x \rightarrow {xa} \]\n\nof \( R \) onto \( M \) . Clearly \( f \) is an \( R \) -homomorphism and the kernel is \( \mathfrak{a}\left( a\right) \) . Now \( \mathfrak{a}\left( a\right) = \mathfrak{a}\left( M\right) \) and hence the resul... | Yes |
Proposition 5. If \( f, g \in \mathcal{L}\left( {V, V}\right) \), then \( {V}_{f},{V}_{g} \) are isomorphic if, and only if, \( f, g \) are similar. | PROOF. (i) If there is an \( F\left\lbrack X\right\rbrack \) -isomorphism \( e \) of \( {V}_{f} \) onto \( {V}_{g} \), then \( e \) is a linear isomorphism of \( V \) onto \( V \) and also\n\n\[ \left( {p\left( X\right) v}\right) e = p\left( X\right) \left( {ve}\right) \]\n\nfor all \( v \) in \( V \) and all \( p\left... | Yes |
Proposition 6. Let \( M \) be a cyclic \( F\left\lbrack X\right\rbrack \) -module whose annihilator \( \mathfrak{a}\left( M\right) \) is generated by the non-zero polynomial \( p \) . If \( {p}_{1},\ldots ,{p}_{n} \) are the distinct monic irreducible factors of \( p \), then\n\n\[ M = \left\lbrack {a}_{1}\right\rbrack... | Let \( M = \left\lbrack a\right\rbrack \) and suppose that \( p = {st} \), where \( s, t \) are relatively prime. If \( b = {ta} \), then \( \mathfrak{a}\left( b\right) = \left\lbrack s\right\rbrack \) : for \( {qb} = 0 \) implies \( {qt} \in \mathfrak{a}\left( a\right) \) and so \( p \) divides \( {qt} \), i.e., \( s ... | No |
Proposition 7. Two finitely generated modules over \( F\left\lbrack X\right\rbrack \) are isomorphic if, and only if, they have the same invariant factors. | This result is an immediate consequence of Theorems 1 and 2 together with Proposition 4. | Yes |
Proposition 8. Let \( \left( {v}_{i}\right) ,\left( {{v}_{j}{}^{\prime }}\right) \) be ordered bases of the torsion-free \( F\left\lbrack X\right\rbrack \) - modules \( E,{E}^{\prime } \) of ranks \( m, n \), respectively. If \( g \) is a homomorphism of \( E \) into \( {E}^{\prime } \), then there exist ordered bases ... | \[ \left( {g;\left( {e}_{i}\right) ,\left( {{e}_{j}{}^{\prime }}\right) }\right) = \left( \begin{array}{ll} D & 0 \\ 0 & 0 \end{array}\right) \] where \( D \) is the diagonal matrix \[ \left( \begin{matrix} {d}_{1} & & 0 \\ & \ddots & \\ 0 & & {d}_{t} \end{matrix}\right) \] and (i) \( {d}_{i} \) divides \( {d}_{i + 1},... | Yes |
Proposition 10. \( {V}_{f} \) is isomorphic to \( {\left( {V}^{ * }\right) }_{{f}^{t}} \) . | PROOF. Let \( {V}_{f} = {\left\lbrack {a}_{1}\right\rbrack }_{f} \oplus \cdots \oplus {\left\lbrack {a}_{k}\right\rbrack }_{f} \) be the decomposition according to Theorem 1 and write \( {M}_{i} = {\left\lbrack {a}_{i}\right\rbrack }_{f} \) . If\n\n\[ \n{N}_{i} = \cap \left( {{M}_{j}^{ \circ } : j \neq i}\right) \n\] \... | Yes |
Theorem 1. Let \( P, Q \in A \) .\n\n(1) If \( P \subset Q \), then \( \operatorname{pdim}P \leq \operatorname{pdim}Q \) and \( \operatorname{pdim}P = \operatorname{pdim}Q \) implies \( P = Q \) . | Proof: (1) and (2) are immediate by Theorem 2 of Chapter I. | No |
Proposition 1. Let \( B, C \), and \( D \) be categories. For all objects \( c \in C \) and \( b \in B \), let\n\n\[ \n{L}_{c} : B \rightarrow D,\;{M}_{b} : C \rightarrow D \]\n\nbe functors such that \( {M}_{b}\left( c\right) = {L}_{c}\left( b\right) \) for all \( b \) and \( c \) . Then there exists a bifunctor \( S ... | Proof. If we write \( b \) and \( c \) for the corresponding identity arrows, the definition (1) of the composite in \( B \times C \) shows that\n\n\[ \n\left\langle {{b}^{\prime }, g}\right\rangle \circ \langle f, c\rangle = \left\langle {{b}^{\prime }f,{gc}}\right\rangle = \langle f, g\rangle = \left\langle {{fb},{c}... | Yes |
Theorem 1. Let \( G = \{ A \rightrightarrows O\} \) be a small graph. There is a small category \( C = {C}_{G} \) with \( O \) as set of objects and a morphism \( P : G \rightarrow {UC} \) of graphs from \( G \) to the underlying graph \( {UC} \) of \( C \) with the following property. Given any category \( B \) and an... | Proof. Take the objects of \( C \) to be those of \( G \) and the arrows of \( C \) to be the finite strings (or \ | Yes |
Corollary 2. To any set \( X \) there is a monoid \( M \) and a function \( p : X \rightarrow {UM} \), where \( {UM} \) is the underlying set of \( M \), with the following universal property: For any monoid \( L \) and any function \( h : X \rightarrow {UL} \) there is a unique morphism \( {h}^{\prime } : M \rightarro... | The elements of \( M \) are the identity and strings \( \left\langle {{x}_{1},\ldots ,{x}_{n - 1}}\right\rangle \), for \( {x}_{i} \in X \) . | No |
For a given category \( C \), let \( R \) be a function which assigns to each pair of objects \( a, b \) of \( C \) a binary relation \( {R}_{a, b} \) on the hom-set \( C\left( {a, b}\right) \) . Then there exist a category \( C/R \) and a functor \( Q = {Q}_{R} : C \rightarrow C/R \) such that (i) If \( f{R}_{a, b}{f}... | Sketch of proof. Call \( R \) a congruence on \( C \) if (i) for each pair \( a, b \) of objects, \( {R}_{a, b} \) is a reflexive, symmetric, and transitive relation on \( C\left( {a, b}\right) \) ; (ii) if \( f,{f}^{\prime } : a \rightarrow b \) have \( f{R}_{a, b}{f}^{\prime } \), then for all \( g : {a}^{\prime } \r... | No |
For a functor \( S : D \rightarrow C \) a pair \( \langle r, u : c \rightarrow {Sr}\rangle \) is universal from \( c \) to \( S \) if and only if the function sending each \( {f}^{\prime } : r \rightarrow d \) into \( S{f}^{\prime } \circ u : c \rightarrow {Sd} \) is a bijection of hom-sets\n\n\[ D\left( {r, d}\right) ... | The statement that \( \langle r, u\rangle \) is universal is exactly the statement that \( {f}^{\prime } \mapsto S{f}^{\prime } \circ u = f \) is a bijection. This bijection is natural in \( d \), for if \( {g}^{\prime } : d \rightarrow {d}^{\prime } \), then \( S\left( {{g}^{\prime }{f}^{\prime }}\right) \circ u = S{g... | Yes |
Proposition 2. Let \( * \) denote any one-point set and let \( D \) have small hom-sets. If \( \langle r, u : * \rightarrow {Kr}\rangle \) is a universal arrow from \( * \) to \( K : D \rightarrow \) Set, then the function \( \psi \) which for each object \( d \) of \( D \) sends the arrow \( {f}^{\prime } : r \rightar... | Proof. For any set \( X \), a function \( f : * \rightarrow X \) from the one-point set \( * \) to \( X \) is determined by the element \( f\left( *\right) \in X \) . This correspondence \( f \mapsto f\left( *\right) \) is a bijection \( \operatorname{Set}\left( {*, X}\right) \rightarrow X \), natural in \( X \in \) Se... | Yes |
If a category \( C \) has a terminal object \( t \) and a product diagram \( a \leftarrow a \times b \rightarrow b \) for any two of its objects, then \( C \) has all finite products. The product objects provide, by \( \langle a, b\rangle \mapsto a \times b \), a bifunctor \( C \times C \rightarrow C \) . For any three... | Proof. A product of one object \( c \) is just the diagram \( c \rightarrow c \) formed with the identity map of \( c \), so is present in any category. Now suppose that any two objects \( {a}_{1},{a}_{2} \) of \( C \) have a product. If we choose one such product diagram \( {a}_{1} \leftarrow {a}_{1} \times {a}_{2} \r... | Yes |
Proposition 1. If \( C \) is a category with finite products, then an object \( c \) is a group (or, a monoid) in \( C \) if and only if the hom functor \( C\left( {-, c}\right) \) is a group (respectively, a monoid) in the functor category \( {\operatorname{Set}}^{{c}^{\mathrm{{op}}}} \) . | Proof. Each multiplication \( \mu \) for \( c \) determines a corresponding multiplication \( \bar{\mu } \) for the hom-set \( C\left( {-, c}\right) : {C}^{\mathrm{{op}}} \rightarrow \) Set, as the composite\n\n\[ \nC\left( {-, c}\right) \times C\left( {-, c}\right) \overset{\theta }{ \rightarrow }C\left( {-, c \times ... | Yes |
Theorem 1. Any functor \( K : D \rightarrow \) Sets from a small category \( D \) to the category of sets can be represented (in a canonical way) as a colimit of a diagram of representable functors \( \hom \left( {d, - }\right) \) for objects \( d \) in \( D \) . | Proof. First, given \( K \), we construct the needed diagram category (for the colimit) \( J \) as the so-called \ | No |
Theorem 2. Each adjunction \( \langle F, G,\varphi \rangle : X \rightharpoonup A \) is completely determined by the items in any one of the following lists:\n\n(i) Functors \( F, G \), and a natural transformation \( \eta : {1}_{x} \rightarrow {GF} \) such that each \( {\eta }_{x} : x \rightarrow {GFx} \) is universal ... | Proof. Ad (i): The statement that \( {\eta }_{x} \) is universal means that to each \( f : x \rightarrow {Ga} \) there is exactly one \( g \) as in the commutative diagram\n\n\n\nThis states precisely that \( \theta \l... | Yes |
Corollary 1. Any two left-adjoints \( F \) and \( {F}^{\prime } \) of a functor \( G : A \rightarrow X \) are naturally isomorphic. | The proof is just an application of the fact that a universal arrow, like an initial object, is unique up to isomorphism. Explicitly, adjunctions \( \langle F, G,\varphi \rangle \) and \( \left\langle {{F}^{\prime }, G,{\varphi }^{\prime }}\right\rangle \) give to each \( x \) two universal arrows \( x \rightarrow {GFx... | Yes |
Corollary 2. A functor \( G : A \rightarrow X \) has a left adjoint if and only if, for each \( x \in X \), the functor \( X\left( {x,{Ga}}\right) \) is representable as a functor of \( a \in A \) . If \( \varphi : A\left( {{F}_{0}x, a}\right) \cong X\left( {x,{Ga}}\right) \) is a representation of this functor, then \... | This is just a restatement of part (ii) of the theorem. | No |
Theorem 3. If the additive functor \( G : A \rightarrow M \) between \( {Ab} \) -categories \( A \) and \( M \) has a left adjoint \( F : M \rightarrow A \), then \( F \) is additive and the adjunction bijections\n\n\[ \n\varphi : A\left( {{Fm}, a}\right) \cong M\left( {m,{Ga}}\right)\n\]\n\nare isomorphisms of abelian... | Proof. If \( \eta : I \rightarrow {GF} \) is the unit of the adjunction, then \( \varphi \) may be written as \( {\varphi f} = {Gf} \circ {\eta }_{m} \) for any \( f : {Fm} \rightarrow a \) . If also \( {f}^{\prime } : {Fm} \rightarrow a \), the additivity of \( G \) gives\n\n\[ \n\varphi \left( {f + {f}^{\prime }}\rig... | Yes |
Theorem 1. For an adjunction \( \langle F, G,\eta ,\varepsilon \rangle : X \rightharpoonup A \) : (i) \( G \) is faithful if and only if every component \( {\varepsilon }_{a} \) of the counit \( \varepsilon \) is epi,(ii) \( G \) is full if and only if every \( {\varepsilon }_{a} \) is a split monic. Hence \( G \) is f... | The proof depends on a lemma.\n\nLemma. Let \( {f}^{ * } : A\left( {a, - }\right) \rightarrow A\left( {b, - }\right) \) be the natural transformation induced by an arrow \( f : b \rightarrow a \) of \( A \) . Then \( {f}^{ * } \) is monic if and only if \( f \) is epi, while \( {f}^{ * } \) is epi if and only if \( f \... | Yes |
Theorem 1. The following properties of a functor \( S : A \rightarrow C \) are logically equivalent:\n\n(i) \( S \) is an equivalence of categories,\n\n(ii) \( S \) is part of an adjoint equivalence \( \langle T, S;\eta ,\varepsilon \rangle : C \rightharpoonup A \) ,\n\n(iii) \( S \) is full and faithful, and each obje... | Proof. Trivially, (ii) implies (i). To prove that (i) implies (iii), note that \( {ST} \cong I \) shows that each \( c \in C \) has the form \( c \cong S\left( {Tc}\right) \) for an \( a = {Tc} \in A \) . The natural isomorphism \( \theta : {TS} \cong I \) gives for each \( f : a \rightarrow {a}^{\prime } \) the commut... | Yes |
Theorem 1 (Galois connections are adjoint pairs). Let \( P, Q \) be two preorders and \( L : P \rightarrow {Q}^{\mathrm{{op}}}, R : {Q}^{\mathrm{{op}}} \rightarrow P \) two order-preserving functions. Then \( L \) (regarded as a functor) is a left adjoint to \( R \) if and only if, for all \( p \in P \) and \( q \in Q ... | Proof. Recall that \( P \) becomes a category in which there is (exactly) one arrow \( p \rightarrow {p}^{\prime } \) whenever \( p \leqq {p}^{\prime } \) . Thus the condition (1) states precisely that there is a bijection \( {\hom }_{{Q}^{\mathrm{{op}}}}\left( {{Lp}, q}\right) \cong {\hom }_{P}\left( {p,{Rq}}\right) \... | Yes |
Proposition 1. Given adjunctions (1) and functors \( K \) and \( L \) satisfying (2), the condition (3) on hom-sets is equivalent to \( {L\eta } = {\eta }^{\prime }L \) and also to \( {\varepsilon }^{\prime }K = {K\varepsilon } \) . | Proof. Given (3) commutative, set \( a = {Fx} \) and chase the identity arrow \( 1 : {Fx} \rightarrow {Fx} \) around (3) to get the units \( \eta ,{\eta }^{\prime } \) and the equality\n\n\[ \langle {L\eta } : L \rightarrow {LGF}\rangle = \left\langle {{\eta }^{\prime }L : L \rightarrow {G}^{\prime }{F}^{\prime }L}\rig... | Yes |
Theorem 2. Given the two adjunctions (4), the natural transformations \( \sigma \) and \( \tau \) are conjugate if and only if any one of the four following diagrams (of natural transformations) commutes | Proof. First,(5) implies (6) and (5) implies (7). For, put \( x = {G}^{\prime }a \) in (5), start with the identity arrow \( 1 : {G}^{\prime }a \rightarrow {G}^{\prime }a \) in the upper right and use the description of \( \varphi \) and \( {\varphi }^{\prime } \) by unit and counit to chase this element 1 around the d... | Yes |
Given a bifunctor \( F : X \times P \rightarrow A \), assume for each object \( p \in P \) that \( F\left( {-, p}\right) : X \rightarrow A \) has a right adjoint \( G\left( {p, - }\right) : A \rightarrow X \), via an adjunction\n\n\[ \hom \left( {F\left( {x, p}\right), a}\right) \cong \hom \left( {x, G\left( {p, a}\rig... | The condition that the adjunction (10) be natural in \( p \in P \) is the commutativity of the square\n\n\[ \hom \left( {F\left( {x, p}\right), a}\right) \cong \hom \left( {x, G\left( {p, a}\right) }\right) \]\n\n\[ F{\left( x, h\right) }^{ * }\; \uparrow G{\left( h, a\right) }_{ * } \]\n\n\[ \hom \left( {F\left( {x,{p... | Yes |
Given two adjunctions\n\n\[ \langle F, G,\eta ,\varepsilon \rangle : X \rightharpoonup A,\;\langle \bar{F},\bar{G},\bar{\eta },\bar{\varepsilon }\rangle : A \rightharpoonup D \]\n\nthe composite functors yield an adjunction\n\n\[ \langle \bar{F}F, G\bar{G}, G\bar{\eta }F \cdot \eta ,\bar{\varepsilon } \cdot \bar{F}\var... | Proof. With hom-sets, the two given adjunctions yield a composite isomorphism, natural in \( x \in X \) and \( d \in D \) :\n\n\[ D\left( {\bar{F}{Fx}, d}\right) \cong A\left( {{Fx},\bar{G}d}\right) \cong X\left( {x, G\bar{G}d}\right) . \]\n\nThis makes the composite \( \bar{F}F \) left adjoint to \( G\bar{G} \) . Sett... | Yes |
Theorem 2. Given two conjugate pairs\n\n\[ \langle \sigma ,\tau \rangle : \langle F, G,\eta ,\varepsilon \rangle \rightarrow \left\langle {{F}^{\prime },{G}^{\prime },{\eta }^{\prime },{\varepsilon }^{\prime }}\right\rangle : X \rightharpoonup A, \]\n\n\[ \langle \bar{\sigma },\bar{\tau }\rangle : \langle \bar{F},\bar{... | The proof may be visualized by the diagram of hom-sets\n\n\[ D\left( {{\bar{F}}^{\prime }{F}^{\prime }x, d}\right) \cong A\left( {{F}^{\prime }x,{\bar{G}}^{\prime }d}\right) \cong X\left( {x,{G}^{\prime }{\bar{G}}^{\prime }d}\right) \]\n\n\[ {\left( \bar{\sigma }\sigma x\right) }^{ * }\;{\left( \sigma x\right) }^{ * }{... | Yes |
Theorem 1 (Completeness of Set). If the category \( J \) is small, any functor \( F : J \rightarrow \) Set has a limit which is the set \( \operatorname{Cone}\left( {*, F}\right) \) of all cones \( \sigma : * \rightarrow F \) from the one point set \( * \) to \( F \), while the limiting cone \( v \), with\n\n\[ \n{v}_{... | Proof. Since \( J \) is small, Cone \( \left( {*, F}\right) \) is a small set, hence an object of Set. If \( u : j \rightarrow k \) is any arrow of \( J \), then \( {F}_{u}{\sigma }_{j} = {\sigma }_{k} \) because \( \sigma \) is a cone; hence \( v \) as defined in (2) is a cone to the base \( F \). To prove it universa... | Yes |
Theorem 2. Let \( U : \operatorname{Grp} \rightarrow \) Set be the forgetful functor. If \( H : J \rightarrow \operatorname{Grp} \) is such that the composite \( {UH} \) has a limit \( L \) and a limiting cone \( v : L \rightarrow {UH} \) in \( \mathbf{{Set}} \), then there is exactly one group structure on the set \( ... | Proof. By Theorem 1, take \( L = \) Cone \( \left( {*,{UH}}\right) \) ; define the product of two such cones \( \sigma ,\tau \in \operatorname{Cone}\left( {*,{UH}}\right) \) by \( {\left( \sigma \tau \right) }_{j} = {\sigma }_{j}{\tau }_{j} \) (the product in the group \( \left. {H}_{j}\right) \) and the inverse by \( ... | Yes |
Theorem 1. For categories \( C \) and \( J \), if \( C \) has equalizers of all pairs of arrows and all products indexed by the sets \( \operatorname{obj}\left( J\right) \) and \( \operatorname{arr}\left( J\right) \), then \( C \) has a limit for every functor \( F : J \rightarrow C \) . | The proof constructs the following diagram in stages, with \( i \) denoting an object and \( u : j \rightarrow k \) an arrow of the index category \( J \) . By assumption, the products \( {\Pi }_{i}{F}_{i} \) and \( {\Pi }_{u}{F}_{k} \) and their projections exist, where the second product is taken over all arrows \( u... | Yes |
Proposition 3 (Freyd). A small category \( C \) which is small-complete is simply a preorder which has a greatest lower bound for every small set of its elements. | Proof. Suppose \( C \) is not a preorder. Then there are objects \( a, b \in C \) with arrows \( f \neq g : a \rightarrow b \) . For any small set \( J \) form the product \( {\Pi }_{j}b \) of factors \( {b}_{j} \) all equal to \( b \) . Then an arrow \( h : a \rightarrow {\Pi }_{j}b \) is determined by its components,... | No |
Theorem 1. If \( S : J \rightarrow {X}^{P} \) is such that for each object \( p \in P \) the composite \( {E}_{p}S : J \rightarrow X \) has a limit \( {L}_{p} \) with a limiting cone \( {\tau }_{p} : {L}_{p} \rightarrow {E}_{p}S \), then there is a unique functor \( L : P \rightarrow X \) with object function \( p \map... | Proof. Let \( h : p \rightarrow q \) be any arrow of \( P \) . Then, writing \( {E}_{p}S \) as \( {S}_{p} \) , the given cones \( {\tau }_{p} \) and \( {\tau }_{q} \) for a typical arrow \( u : j \rightarrow k \) of \( J \) have the form\n\n with small hom-sets, each hom-functor \( C\left( {c, - }\right) : C \rightarrow \) Set preserves all limits; in particular, all small limits. | Proof. Let \( J \) be any category and \( F : J \rightarrow C \) a functor with a limiting cone \( v : \operatorname{Lim}F \rightarrow F \) in \( C \) . Apply the hom-functor \( C\left( {c, - }\right) \) ; there results a cone \( {v}_{ * } = C\left( {c, v}\right) \), as in the diagram ![1624acde-b97a-46fa-981e-53482323... | Yes |
Theorem 2. If \( V : A \rightarrow X \) creates limits for \( F : J \rightarrow A \) and the composite \( {VF} : J \rightarrow X \) has a limit, then \( V \) preserves the limit of \( F \). | Proof. Let \( \tau : a \rightarrow F \) and \( \sigma : x \rightarrow {VF} \) be limiting cones in \( A \) and \( X \) , respectively. Since \( V \) creates limits, there is a unique cone \( \varrho : b \rightarrow F \) in \( A \) with \( V\varrho : {Vb} \rightarrow {VF} \) equal to \( \sigma : x \rightarrow {VF} \) ; ... | Yes |
Theorem 1. If the functor \( G : A \rightarrow X \) has a left adjoint, while the functor \( T : J \rightarrow A \) has a limiting cone \( \tau : a \rightarrow T \) in \( A \), then \( {GT} \) has the limiting cone \( {G\tau } : {Ga} \rightarrow {GT} \) in \( X \) . | Proof. By composition, \( {G\tau } \) is indeed a cone from the vertex \( {Ga} \) in \( X \) . If \( F \) is a left adjoint to \( G \), and if we apply the adjunction isomorphism to every arrow of a cone \( \sigma : x \rightarrow {GT} \), we get arrows \( {\left( {\sigma }_{i}\right) }^{b} : {Fx} \rightarrow {Ti} \) fo... | Yes |
Theorem 1 (Existence of an initial object). Let \( D \) be a small-complete category with small hom-sets. Then \( D \) has an initial object if and only if it satisfies the following\n\nSolution Set Condition. There exists a small set \( I \) and an I-indexed family \( {k}_{i} \) of objects of \( D \) such that for eve... | Proof. This solution set condition is necessary: If \( D \) has an initial object \( k \), then \( k \) indexed by the one-point set realizes the condition, since there is always a (unique) arrow \( k \rightarrow d \) .\n\nConversely, assume the solution set condition. Since \( D \) is small-complete, it contains a pro... | Yes |
Theorem 2 (The Freyd Adjoint Functor Theorem). Given a small-complete category \( A \) with small hom-sets, a functor \( G : A \rightarrow X \) has a left adjoint if and only if it preserves all small limits and satisfies the following\n\nSolution Set Condition. For each object \( x \in X \) there is a small set \( I \... | Proof. If \( G \) has a left adjoint \( F \), then it must preserve all the limits which exist in its domain \( A \) ; in particular, all the small ones. Moreover, the universal arrow \( {\eta }_{x} : x \rightarrow {GFx} \) which is the unit of the adjunction satisfies the solution set condition for \( x \), with \( I ... | No |
Theorem 3 (The Representability Theorem). Let the category \( D \) be small complete with small hom-sets. A functor \( K : D \rightarrow \) Set is representable if and only if \( K \) preserves all small limits and satisfies the following\n\nSolution Set Condition. There exists a small set \( S \) of objects of \( D \)... | Proof. This is another reformulation of the existence Theorem 1 for initial objects. Indeed, a representation of \( K \) is a universal arrow from the one-point set \( * \) to \( K \) (Proposition III.2.2), hence an initial object in the comma category \( \left( {* \downarrow K}\right) \), which is small-complete becau... | Yes |
Theorem 1 (Special Initial-Object Theorem). If the category \( D \) is small-complete, has small hom-sets, and a small cogenerating set \( Q \), then \( D \) has an initial object provided every set of subobjects of each \( d \in D \) has an intersection. | Proof. Form the product \( {q}_{0} = {\Pi }_{q \in Q}q \) of all the objects in the small cogenerating set \( Q \) and take the intersection \( r \) of all subobjects of \( {q}_{0} \) . For any object \( d \in D \), there is at most one arrow \( r \rightarrow d \), for if there were two different arrows, their equalize... | Yes |
Proposition 1. If \( G : C \rightarrow D \) is a faithful functor, if \( D \) has equalizers, and if, for each \( x \in C,\left( {G \downarrow x}\right) : \left( {C \downarrow x}\right) \rightarrow \left( {D \downarrow {Gx}}\right) \) has a right-adjoint-right-inverse \( L \), then \( C \) has equalizers. | Proof. To get the equalizer of a parallel pair \( f,{f}^{\prime } : x \rightarrow y \), apply \( G \) , take the equalizer \( t : s \rightarrow {Gx} \) of \( {Gf}, G{f}^{\prime } \) in \( D \) and apply \( L \) ; the universal property of the adjunction shows \( {Lt} : {Ls} \rightarrow x \) an equalizer in \( C \) . | Yes |
Proposition 2. Haus, the full subcategory of all Hausdorff spaces in Top, is complete and cocomplete. The inclusion functor Haus \( \rightarrow \) Top has a left adjoint \( H \), as does the forgetful functor Haus \( \rightarrow \) Set. | Proof. The left adjoint \( H \) will be obtained by the adjoint functor theorem. First, any product of Hausdorff spaces or subspace of a Hausdorff space is also Hausdorff, hence Haus is complete and the inclusion functor is continuous (i.e., it preserves small limits). It remains only to verify the solution set conditi... | No |
Theorem 1 (Every monad is defined by its T-algebras). If \( \langle T,\eta ,\mu \rangle \) is a monad in \( X \), then the set of all T-algebras and their morphisms form a category \( {X}^{T} \). There is an adjunction\n\n\[ \left\langle {{F}^{T},{G}^{T};{\eta }^{T},{\varepsilon }^{T}}\right\rangle : X \rightharpoonup ... | The proof is straightforward verification. If \( f : \langle x, h\rangle \rightarrow \left\langle {{x}^{\prime },{h}^{\prime }}\right\rangle \) and \( g : \left\langle {{x}^{\prime },{h}^{\prime }}\right\rangle \rightarrow \left\langle {{x}^{\prime \prime },{h}^{\prime \prime }}\right\rangle \) are morphisms of \( T \)... | Yes |
Proposition 1. The monad on Set determined by the adjunction Set \( \rightarrow \) Smgrp is\n\n\[ \nW = \left\langle {W : \text{ Set } \rightarrow \text{ Set,}\eta : I \rightarrow W,\mu : {W}^{2} \rightarrow W}\right\rangle \]\n\nwhere \( {WX} = \mathop{\coprod }\limits_{{n = 1}}^{\infty }{X}^{n},{\eta }_{X}x = \langle... | Proof. By definition, \( {\eta x} = \langle x\rangle \), while \( \mu = {G\varepsilon F} : {W}^{2} \rightarrow W \) is determined by the formula above for \( {\varepsilon }_{S} \), where we have written each element of \( {W}^{2}X \) as a word (of length \( k \) ) in \( k \) words of the respective lengths \( {n}_{1},\... | Yes |
Proposition 2. For the above word-monad \( W \) in Set, the \( W \) -algebras have the form \( \left\langle {S,{v}_{1},{v}_{2},\ldots }\right\rangle : A \) set \( S \) equipped with one \( n \) -ary operation \( {v}_{n} : {S}^{n} \rightarrow S \) for each positive integer \( n \), such that \( {v}_{1} = 1 \) while for ... | Proof. Consider a \( W \) -algebra \( \langle S, h : {WS} \rightarrow S\rangle \) . Since \( {WS} = \coprod {S}^{n} \), the structure map \( h \) is a list of \( n \) -ary operations \( {v}_{n} : {S}^{n} \rightarrow S \), one for each \( n \) . The unit law for the algebra requires that \( h{\eta }_{X} = 1 \), hence th... | Yes |
Theorem 1 (The Kleisli category of a monad, [1965].) Given a monad \( \langle T,\eta ,\mu \rangle \) in a category \( X \), consider to each object \( x \in X \) a new object \( {x}_{T} \) and to each arrow \( f : x \rightarrow {Ty} \) in \( X \) a new arrow \( {f}^{b} : {x}_{T} \rightarrow {y}_{T} \) . These new objec... | Sketch of proof. The definition of the arrows \( {f}^{b} \) amounts to a bijection \( {X}_{T}\left( {{x}_{T},{y}_{T}}\right) \cong X\left( {x,{Ty}}\right) \) on hom-sets, while the definition of the composite in \( {X}_{T} \) refers to the composite\n\n\[ \nx\overset{f}{ \rightarrow }{Ty}\overset{Tg}{ \rightarrow }{T}^... | Yes |
Theorem 1. Let \( \Omega \) be a set of operators, \( E \) a set of identities (on the operators derived from \( \Omega \) ), \( G \) the forgetful functor from the category \( \langle \Omega, E\rangle \) -Alg of all small \( \langle \Omega, E\rangle \) -algebras to Set, and \( T \) the resulting monad in Set. Then the... | The proof will use Beck's theorem. Consider any parallel pair \( f, g : A \rightrightarrows B \) of morphisms of \( \langle \Omega, E\rangle \) -algebras for which the underlying functions have an absolute coequalizer \( e \) :\n\n\[ \n{GA}\xrightarrow[{Gg}]{Gf}{GB}\xrightarrow[]{e}X. \n\]\n\n(1)\n\nTo \ | No |
Theorem 1. For any monoidal category \( B \) and any object \( b \in B \), there is a unique morphism \( W \rightarrow B \) of monoidal categories with \( \left( -\right) \mapsto b \) . | Proof. We write the desired morphism as \( w \mapsto {w}_{b} \), to suggest that it means \ | No |
Proposition 1 (General Associative Law). For \( \langle c,\mu ,\eta \rangle \) a monoid in B, the iterated products \( {\mu }_{v} \) and \( {\mu }_{w} \) for any two words \( v \) and \( w \) of the same length \( n \) satisfy\n\n\[ \n{\mu }_{w} \circ {\operatorname{can}}_{c}\left( {v, w}\right) = {\mu }_{v} : {v}_{c} ... | Proof. The axioms (1) and (2) for a monoid are exactly those cases of (4) where the canonical arrow in question is \( \alpha ,\lambda \), or \( \varrho \) . From these cases, (4) may be verified by induction, since all canonical arrows are composites of \( \alpha \) ’s, \( \lambda \) ’s, and \( \varrho \) ’s. | No |
Theorem 2 (Construction of free monoids). If the monoidal category \( B \) has denumerable coproducts, and if for each \( a \in B \) the functors \( a▱ - \) and \( - ▱a : B \rightarrow B \) preserve these coproducts, then the forgetful functor \( U : {\operatorname{Mon}}_{B} \rightarrow B \) has a left adjoint. | Proof. The distributive law \( \theta : { \coprod }_{n}\left( {a▱{b}_{n}}\right) \cong a▱{ \coprod }_{n}{b}_{n} \) holds for each denumerable coproduct \( { \coprod }_{n}{b}_{n} \) of objects \( {b}_{n} \in B \) because \( a▱ - \) preserves coproducts. Indeed, the definition of the coproduct injections \( {i}_{n} : {b}... | Yes |
Given a monoid \( \left\langle {c,{\mu }^{\prime },{\eta }^{\prime }}\right\rangle \) in a strict monoidal category \( \langle B,▱, e\rangle \), there is a unique morphism \( F : \langle \Delta , + ,0\rangle \rightarrow \langle B,▱, e\rangle \) such that \( {F1} = c,{F\mu } = {\mu }^{\prime } \) and \( {F\eta } = {\eta... | The proof depends on showing that the arrows of \( \Delta \) are exactly the iterated formal products (for the binary product \( \mu \) ). In detail, write \( {\mu }^{\left( k\right) } \) for the unique arrow \( {\mu }^{\left( k\right) } : k \rightarrow 1 \) . Thus \( {\mu }^{\left( 0\right) } = \eta ,{\mu }^{\left( 1\... | No |
Proposition 2. The category \( \Delta \), with objects all finite ordinals, is generated by the arrows \( {\delta }_{i}^{n} : n \rightarrow n + 1 \) and \( {\sigma }_{j}^{n} : n + 1 \rightarrow n \) subject to the relations (11), (12), and (13). | Proof. These relations suffice to put any composite of \( \delta \) ’s and \( \sigma \) ’s into the unique form (10) of the Lemma. | No |
Proposition 1. CGHaus is a full coreflective subcategory of Haus. | It is a full subcategory by definition. To each Hausdorff space \( Y \) we construct a compactly generated space \( {KY} \) with the same points as \( Y \) (the \ | No |
Proposition 2. CGHaus is (small) complete and cocomplete. | The category Haus is complete (Proposition V.9.2) and a right adjoint such as \( K \) preserves limits. Hence CGHaus is complete. In particular, the product (written \( ▱ \) ) of two spaces \( X \) and \( Y \) in CGHaus is obtained from their \ | No |
Lemma 1. If also \( f = {m}^{\prime }{q}^{\prime } \), where \( {m}^{\prime } \) is a kernel, then in the commutative square\n\n\n\n(3)\n\nthere is a (unique) diagonal arrow \( t \) with \( m = {m}^{\prime }t \) and ... | Proof. By assumption, \( {m}^{\prime } = \ker {p}^{\prime } \) where \( {p}^{\prime } = \operatorname{coker}{m}^{\prime } \) ; take also\n\n\n\n\( p = \operatorname{coker}m = \operatorname{coker}f \) . Then \( {p}^{\... | Yes |
Proposition 1. The following properties of an object \( z \) in an Ab-category A are equivalent: (i) \( z \) is initial; (ii) \( z \) is terminal; (iii) \( {1}_{z} = 0 : z \rightarrow z \) ; (iv) the abelian group \( A\left( {z, z}\right) \) is the zero group. In particular, any initial (or any terminal) object in \( A... | Proof. If \( z \) is initial, there is a unique map \( z \rightarrow z \), hence \( {1}_{z} = 0 \) and \( A\left( {z, z}\right) = 0 \) . If \( {1}_{z} = 0 \), then any \( f : b \rightarrow z \) has \( f = {1}_{z}f = {0f} = 0 : b \rightarrow z \), so there is a unique arrow, namely 0, from \( b \) to \( z \), and \( z \... | Yes |
Theorem 2. Two objects \( a \) and \( b \) in an \( {Ab} \) -category \( A \) have a product in \( A \) if and only if they have a biproduct in \( A \) . Specifically, given a biproduct diagram (1), the object \( c \) with the projections \( {p}_{1} \) and \( {p}_{2} \) is a product of a and \( b \), while, dually, \( ... | Proof. First assume we have the biproduct diagram (1) with the condition (2). Then\n\n\[ \n{p}_{1}{i}_{2} = {p}_{1}\left( {{i}_{1}{p}_{1} + {i}_{2}{p}_{2}}\right) {i}_{2} = 1 \circ {p}_{1}{i}_{2} + {p}_{1}{i}_{2} \circ 1 = {p}_{1}{i}_{2} + {p}_{1}{i}_{2}; \n\]\n\nsubtracting, \( {p}_{1}{i}_{2} = 0 \) ; symmetrically \(... | Yes |
Proposition 3. For parallel arrows \( f,{f}^{\prime } : a \rightarrow b \) in an additive category \( A \) , \n\n\[ \nf + {f}^{\prime } = {\delta }^{b}\left( {f \oplus {f}^{\prime }}\right) {\delta }_{a} : a \rightarrow b, \n\] \n\nwhere \( {\delta }_{a} : a \rightarrow a \times a \) is the diagonal map, \( {\delta }^{... | Here the diagonal is defined by \( {p}_{1}{\delta }_{a} = {1}_{a} = {p}_{2}{\delta }_{a} \) and the codiagonal by \( {\delta }^{b}{i}_{1} = {1}_{b} = {\delta }^{b}{i}_{2} \) . The proof is a direct calculation: \n\n\[ \n{\breve{\delta }}^{b}\left( {f \oplus {f}^{\prime }}\right) {\delta }_{a} = {\breve{\delta }}^{b}\le... | Yes |
Proposition 4. If \( A \) and \( B \) are \( {Ab} \) -categories, while \( A \) has all binary biproducts, then a functor \( T : A \rightarrow B \) is additive if and only if \( T \) carries each binary biproduct diagram in \( A \) to a biproduct diagram in \( B \) . | Proof. Each of the equations \( {p}_{1}{i}_{1} = 1,{p}_{2}{i}_{2} = 1 \), and \( {i}_{1}{p}_{1} + {i}_{2}{p}_{2} = 1 \) describing a biproduct in terms of its insertions \( {i}_{j} \) and projections \( {p}_{j} \) is preserved by an additive functor; therefore each additive functor preserves biproducts.\n\nConversely, ... | Yes |
In an abelian category \( A \), every arrow \( f \) has a factorization \( f = {me} \), with \( m \) monic and e epi; moreover,\n\n\[ m = \ker \left( {\operatorname{coker}f}\right) ,\;e = \operatorname{coker}\left( {\ker \mathrm{f}}\right) . \] | To construct such a factorization of \( f \), take \( m = \ker \left( {\operatorname{coker}f}\right) \) . Since \( \left( {\operatorname{coker}f}\right) \circ f = 0, f \) factors as \( f = {me} \) for a unique \( e \), and by Lemma 1 of \( §1, e \) is epi. Now \( m \) is monic, so for any composable \( t,{ft} = 0 \) if... | Yes |
Lemma 1 (The short five lemma). In any commutative diagram (3) with short exact rows, f and \( h \) monic imply \( g \) monic, and f and \( h \) epi imply \( g \) epi. | In Ab, take any element \( x \) in ker \( g \) ; then \( g\left( x\right) = 0 \) :\n\n\n\nso \( {he}\left( x\right) = 0, e\left( x\right) = 0 \) . By exactness of the first row, there must be an element \( {x}^{\prim... | No |
Proposition 2. Given a pullback square (on the right below) in an abelian category, fepi implies \( {f}^{\prime } \) epi. Also, the kernel \( k \) offfactors as \( k = {g}^{\prime }{k}^{\prime } \) for a \( {k}^{\prime } \) which is the kernel of \( {f}^{\prime } \) . | Proof. The pullback \( s \) (like any pullback) is constructed from products and equalizers thus: Take \( b \oplus d \) with projections \( {p}_{1} \) and \( {p}_{2} \) , form the left exact sequence \[ 0 \rightarrow s\xrightarrow[]{m}b \oplus d\xrightarrow[]{f{p}_{1} - g{p}_{2}}c \] (i.e., \( m \) is a kernel), and se... | Yes |
Theorem 3 (Elementary rules for chasing diagrams). For the members in any abelian category\n\n(i) \( f : a \rightarrow b \) is monic if and only if, for all \( x{\epsilon }_{m}a,{fx} \equiv 0 \) implies \( x \equiv 0 \) ; (ii) \( f : a \rightarrow b \) is monic if and only if, for all \( x,{x}^{\prime }{ \in }_{m}a,{fx... | Proof. Rules (i) and (ii) are just the definition of a monic. In (iii), if \( g \) is epi, then one can construct \( y{\epsilon }_{m}b \) with \( {gy} \equiv z \) by pullback (using Proposition 2); conversely, if \( g \) is not epi, the member \( {1}_{c}{ \in }_{m}c \) is not of the form \( {gy} \equiv {1}_{c} \) for a... | Yes |
Lemma 4 (The Five Lemma). In a commutative diagram\n\n\n\n(4)\n\nwith exact rows, \( {f}_{1},{f}_{2},{f}_{4},{f}_{5} \) isomorphisms imply \( {f}_{3} \) an isomorphism. | Proof. By duality, it suffices to prove \( {f}_{3} \) monic. In Ab one would \ | No |
Lemma 5 (Ker-coker sequence \( = \) Snake lemma). Given a morphism \( \langle f, g, h\rangle \) of short exact sequences, as in (3), there is an arrow \( \delta : \) Keh \( \rightarrow \) Cof such that the following sequence is exact:\n\n\[ 0 \rightarrow \operatorname{Kef}\overset{{m}_{0}}{ \rightarrow }\operatorname{K... | Proof. From the map of short exact sequences we first build a different diagram; on the left in \n\n(7)\n\n\( {c}_{0} = \operatorname{Keh}, d \) is the pullback of \( e \) and \( k = \ker h \), so that \( u \) is epi... | Yes |
Theorem 1. A category \( C \) with finite coproducts and colimits over all (small) directed preorders has all (small) coproducts. | Proof. We wish to construct a colimit for a functor \( F : J \rightarrow C \), where \( J \) is a set ( \( = \) a discrete category). Let \( {J}^{ + } \) be the preorder with objects all finite subsets \( S \subset J \), ordered by inclusion; clearly, \( {J}^{ + } \) is filtered. Let \( {F}^{ + } \) assign to each fini... | Yes |
Corollary 3. Grp has all (small) colimits. | Proof. First, the one-element group is an initial object in Grp. Next, any two groups \( G \) and \( H \) have a coproduct \( G * H \) . Indeed, any pair of homomorphisms \( G \rightarrow L, H \rightarrow L \) to a third group \( L \) factors through the subgroup of \( L \) generated by the images of \( G \) and \( H \... | Yes |
Theorem 1. If the category \( P \) is finite while \( J \) is small and filtered, then for any bifunctor \( F : P \times J \rightarrow \) Set the canonical arrow\n\n\[ \kappa : {\underline{\mathrm{{Colim}}}}_{j}\overset{{\mathrm{{Lim}}}_{p}}{ \leftarrow }F\left( {p, j}\right) \rightarrow {\underline{\mathrm{{Lim}}}}_{p... | Proof. By the construction of colimits in terms of coproducts and coequalizers (dual of Theorem V.2.2 with \( J \) filtered),\n\n\[ {\operatorname{Colim}}_{j}F\left( {p, j}\right) = { \coprod }_{j}F\left( {p, j}\right) /E, \]\n\n (5)\n\nwhere \( {\mathrm{{LI}}}_{i} \) is the disjoint union and \( E \) is the equivalenc... | No |
Theorem 1. If \( L : {J}^{\prime } \rightarrow J \) is final and \( F : J \rightarrow X \) is a functor such that \( x = \underline{\operatorname{Colim}}{FL} \) exists, then \( \underline{\operatorname{Colim}}F \) exists and the canonical map (1) is an isomorphism. | Proof. Given a colimiting cone \( \mu : {FL} \rightarrow \) Colim \( {FL} = x \), we construct arrows \( {\tau }_{k} : {Fk} \rightarrow x \) for each \( k \in J \) by choosing an arrow \( u : k \rightarrow L{j}^{\prime } \) and taking \( {\tau }_{k} \) to be the composite\n\n\[ \n{Fk}\xrightarrow[]{Fu}{FL}{j}^{\prime }... | Yes |
For each functor \( T : C \rightarrow X \) let \( S \) be the composite functor \n\nwhere \( Q \) is the second projection \( \left\langle {c,{c}^{\prime }}\right\rangle \mapsto {c}^{\prime } \) of the product. Then ... | Proof. The components \( {\tau }_{c} \) of a cone \( e \rightarrow T \) make the triangle \( {\tau }_{c} = {Tf} \circ {\tau }_{b} \) commute (naturality condition!) for each \( f : b \rightarrow c \) in \( C \) . This amounts to saying that every square\n\n. Given a natural transformation \( \gamma : S \rightarrow {S}^{\prime } \) between functors \( S,{S}^{\prime } : {C}^{\mathrm{{op}}} \times C \rightarrow X \) which both have ends \( \langle e,\omega \rangle \) and \( \left\langle {{e}^{\prime },{\omega }^{\prime... | Proof. The composites \( {\gamma }_{c, c} \circ {\omega }_{c} \) define a wedge, so \( g \) exists and is unique by the universality of the wedge \( {\omega }^{\prime } \) . | Yes |
Theorem 2 (Parameter Theorem for ends and limits). Let \( T : P \times {C}^{\mathrm{{op}}} \times C \rightarrow X \) be a functor such that \( T\left( {p,-, - }\right) \) for each object \( p \in P \) has an end\n\n\[{\omega }_{p} : {\int }_{c}T\left( {p, c, c}\right) \rightarrow T\left( {p,-, - }\right)\]\n\nin \( X \... | Proof. Each arrow \( s : p \rightarrow q \) of \( P \) defines a natural transformation \( \gamma = T\left( {s,-, - }\right) : T\left( {p,-, - }\right) \rightarrow T\left( {q,-, - }\right) \) . Hence the arrow function of the desired functor \( U \) must have \( {Us} = {\int }_{c}T\left( {s, c, c}\right) \), defined as... | Yes |
Theorem 3 (Parameter Theorem, continued). Under the same hypotheses on \( T \), the functor \( {T}^{\sharp } \) has the end\n\n\[ \n{\omega }^{\sharp } : {\int }_{c}T\left( {-, c, c}\right) \rightarrow {T}^{\sharp } \n\] \n\nwhere \( {\left( {\omega }_{c}^{\sharp }\right) }_{p} = {\left( {\omega }_{p}\right) }_{c} \) f... | Proof. The end \( \int T\left( {-, c, c}\right) \) is an object of \( {X}^{P} \), while \( {T}^{\sharp } \) is a functor with codomain \( {X}^{P} \) . By the previous theorem, the arrows \( {\left( {\omega }_{p}\right) }_{c} \) of \( X \) provide for each \( c \) an arrow of \( {X}^{P} \) (a natural transformation) \n\... | Yes |
Lemma 1. If \( \lambda : d \rightarrow {\operatorname{Id}}_{C} \) is a cone over the identity functor and \( F : J \rightarrow C \) is a functor such that \( {\lambda F} : d \rightarrow F \) is a limiting cone for \( F \), then \( d \) is initial in \( C \) . | Proof. Since \( \lambda \) is a cone, the triangles\n\n\n\ncommute for each \( i \in J \) and each arrow \( f \) in \( C \) . But \( {\lambda F} \) is a limiting cone, so the first two triangles prove \( {\lambda }_{... | Yes |
Theorem 2 (Formal criterion for the existence of an adjoint). \( A \) functor \( G : A \rightarrow X \) has a left adjoint if and only if both\n\n(i) \( G \) preserves all limits which exist in \( A \) ;\n\n(ii) For each \( x \in X,\operatorname{Lim}\left( {Q : \left( {x \downarrow G}\right) \rightarrow A}\right) \) ex... | Proof. Since right adjoints preserve all limits, (i) is necessary. Since a left adjoint \( F \) to \( G \) has each \( \left\langle {{\eta }_{x} : x \rightarrow {GFx},{Fx}}\right\rangle \) an initial object in \( \left( {x \downarrow G}\right) \), any functor on this comma category has a limit (namely, its value on tha... | Yes |
Theorem 1. If \( D \) is a small complete category with small hom-sets, then \( D \) has an initial object if and only if it has a small set \( S \) of objects which is weakly initial: For every \( d \in D \) there exists \( s \in S \) and an arrow \( s \rightarrow d \) . | Proof. Let \( S \) also denote the full subcategory of \( D \) with the objects \( s \) ; since \( D \) has small hom-sets, \( S \) is still small, so by completeness the inclusion functor \( F : S \rightarrow D \) has a limiting cone \( \mu : v \rightarrow F \) . We shall prove \( v = \operatorname{Lim}F \) initial in... | Yes |
Theorem 1 (Right Kan extension as a point-wise limit). Given \( K : M \rightarrow C \), let \( T : M \rightarrow A \) be a functor such that the composite \( \left( {c \downarrow K}\right) \rightarrow M \rightarrow A \) has for each \( c \in C \) a limit in \( A \), with limiting cone \( \lambda \) , written \( {Rc} = ... | Proof. First, \( {Rg} \) is defined in (4) by the fact that the limit is a functor of \( \left( {c \downarrow K}\right) \) and hence of \( c \) . Specifically, given \( g : c \rightarrow {c}^{\prime } \) and the projection \( {Q}^{\prime } : \left( {{c}^{\prime } \downarrow K}\right) \rightarrow A \), each \( {f}^{\pri... | Yes |
Corollary 3. If the functor \( K \) in the theorem is full and faithful, then the universal arrow \( \varepsilon : {RK} \rightarrow T \) for the Kan extension \( R \) of \( T \) along \( K \) is a natural isomorphism \( \varepsilon : {RK} \cong T \) . | Proof. For \( n \in M,{RKn} \) is obtained from a limit over the comma category \( \left( {c \downarrow K}\right) \) with \( c = {Kn} \) . Because \( K \) is full and faithful, every object \( f : {Kn} \rightarrow {Km} \) in this comma category can be written as \( f = {Kh} \) for a unique \( h : n \rightarrow m \) . T... | Yes |
Corollary 4. If \( M \) is a full subcategory of a category \( C \) and \( T : M \rightarrow A \) is a functor such that each composite \( \left( {c \downarrow K}\right) \rightarrow M \rightarrow A \) has a limit in \( A \) , then there is a functor \( R : C \rightarrow A \) with \( {RK} = T \) (i.e., \( R \) extends \... | Proof. Apply Corollary 3 to the insertion \( M \rightarrow C \) . | No |
Theorem 1. Given functors \( K : M \rightarrow C \) and \( T : M \rightarrow A \) such that for all \( m,{m}^{\prime } \in M \) and all \( c \in C \) the copowers \( C\left( {K{m}^{\prime }, c}\right) \cdot {Tm} \) exist in \( A \), then \( T \) has a left Kan extension \( L = {\operatorname{Lan}}_{K}T \) along \( K \)... | Proof. By the parameter theorem, we may regard this coend as a functor of \( c \) . Compare it with any other functor \( S : C \rightarrow A \) . Then\n\n\[ \nA\left( {{Tm},{SKm}}\right) \cong \operatorname{Nat}\left( {C\left( {{Km}, - }\right), A\left( {{Tm}, S - }\right) }\right) \n\]\n\n\[ \n\cong {\int }_{c}\mathbf... | Yes |
Theorem 2 (Kan extensions as coends, continued.) For the Kan extension (1) above the universal arrow \( \eta : T \rightarrow {LK} \) is given for each \( n \in M \) as the composite of an injection \( {i}_{{1}_{Kn}} \) of the copower (for \( f = {1}_{Kn} : {Kn} \rightarrow {Kn} \) ) with a component of the ending wedge... |  | No |
Theorem 1. If \( G : A \rightarrow X \) has a left adjoint \( F \), it preserves all right Kan extensions which exist in \( A \) . | Proof. First a preliminary, for an adjunction\n\n\[ A\left( {{Fx}, a}\right) \cong X\left( {x,{Ga}}\right) ,\;x \in X, a \in A. \]\n\nIf in place of \( x \) we have a functor \( H : C \rightarrow X \) and in place of \( a \) a functor \( L : C \rightarrow A \), then applying this adjunction at every \( {Lc} \) and \( {... | Yes |
Corollary 2. If \( R,\varepsilon : {RK}\overset{ * }{ \rightarrow }T \) is a right Kan extension and \( A \) has small hom-sets and all small copowers, then for each \( a \in A, A\left( {a, R - }\right) : C \rightarrow \) Set, is the right Kan extension of \( A\left( {a, T - }\right) : M \rightarrow \) Set, with counit... | Proof. The functor \( A\left( {a, - }\right) : A \rightarrow \) Set has the left adjoint \( X \mapsto X \cdot a \) , the copower. | No |
Theorem 3. A functor \( T : M \rightarrow A \) has a pointwise right Kan extension along \( K : M \rightarrow C \) if and only if the limit of \( \left( {c \downarrow K}\right) \rightarrow M \rightarrow A \) exists for all \( c \) . When this is the case, \( {\operatorname{Ran}}_{K}T \) is given by the formulas of Theo... | Proof. Since \( A\left( {a, - }\right) \) preserves limits, any Kan extension given by the limit formula is pointwise.\n\nConversely, suppose for each \( a \in A \) that \( A\left( {a, T - }\right) : M \rightarrow \) Set has a right Kan extension \( {R}^{a} = A\left( {a, R - }\right) \), as in the figure\n\n![1624acde-... | Yes |
Proposition 2. The functor \( K : M \rightarrow C \) is codense if and only if the correspondence \( f \mapsto C\left( {f, K - }\right) \) above is for all a and \( c \in C \) a bijection | \[ C\left( {a, c}\right) \cong \operatorname{Nat}\left( {C\left( {c, K - }\right), C\left( {a, K - }\right) }\right) ; \] (4) that is, if and only if the functor \( {C}^{\mathrm{{op}}} \rightarrow {\mathbf{{Ens}}}^{M} \) defined by \[ c \mapsto C\left( {c, K - }\right) : M \rightarrow {\mathbf{{Ens}}}^{M} \] (5) is ful... | Yes |
Corollary 3. If the hom-sets of \( M \) lie in a full category Ens of sets, then Yoneda embedding \( Y : M \rightarrow {\left( {\mathbf{{Ens}}}^{M}\right) }^{\text{op }} \), given by \( {Ym} = M\left( {m, - }\right) \) is codense. | Proof. By the Yoneda Lemma itself, for each \( F : M \rightarrow \) Ens,\n\n\[{\left( {\mathbf{{Ens}}}^{M}\right) }^{\mathrm{{op}}}\left( {F,{Ym}}\right) = {\mathbf{{Ens}}}^{M}\left( {{Ym}, F}\right) \cong {Fm}.\n\]\n\nThus the right side of (4) above, with \( C = {\left( {\mathbf{{Ens}}}^{M}\right) }^{\mathrm{{op}}}, ... | Yes |
Theorem 1. A functor \( T : M \rightarrow A \) has a colimit if and only if it has a left Kan extension along the (unique) functor \( {K}_{1} : M \rightarrow \mathbf{1} \), and then Colim \( T \) is the value of \( {\operatorname{Lan}}_{{K}_{1}}T \) on the unique object of 1 . | Proof. A functor \( S : 1 \rightarrow A \) is just an object \( a \in A \), and a natural transformation \( \alpha : T \rightarrow S{K}_{1} \), for \( {K}_{1} : M \rightarrow \mathbf{1} \), is just a cone with base \( T \) and vertex \( a \) . Since the left Kan extension \( L = {\operatorname{Lan}}_{{K}_{1}}T \) is co... | Yes |
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