CatDat

List of Implications

The following 93 implications and equivalences are available*.

• abelian is equivalent to   additive and  equalizers and  coequalizers and  mono-regular and  epi-regular

  by definition
• abelian implies   Malcev
• additive is equivalent to   preadditive and  finite products

  by definition
• additive implies   disjoint finite coproducts

  If $f : T \to A + B$ is a morphism that factors through $A$ and $B$, then $p_B f = 0$ and $p_A f = 0$, so $f = 0$.
• additive and  pullbacks and  subobject classifier implies   trivial

  see MSE/4086192
• binary products and  equalizers implies   pullbacks

  The pullback of $f : X \to S$ and $g : Y \to S$ is the equalizer of $p_1 \circ f, \, p_2 \circ g : X \times Y \rightrightarrows S$.
• binary products and  pullbacks implies   equalizers

  The equalizer of $f,g : X \rightrightarrows Y$ is the pullback of $(f,g) : X \to Y \times Y$ with the diagonal $Y \to Y \times Y$.
• binary products and  inhabited implies   connected

  For any two objects $A,B$ we have the zig-zag $A \to A \times B \to B$.
• cartesian closed and  finite coproducts implies   distributive

  Each functor $A \times -$ is left adjoint and hence preserves finite coproducts (in fact, all colimits).
• cartesian closed and  coproducts implies   infinitary distributive

  Each functor $A \times -$ is left adjoint and hence preserves coproducts (in fact, all colimits).
• cartesian closed implies   finite products

  by definition
• cartesian closed and  initial object implies   strict initial object

  See the nLab.
• complete implies   finitely complete and  filtered limits and  wide pullbacks and  connected limits

  trivial
• complete is equivalent to   products and  equalizers

  Mac Lane, V.2, Cor. 2
• connected implies   inhabited

  by definition
• connected limits is equivalent to   wide pullbacks and  equalizers

  The direction $\Rightarrow$ is trivial. The direction $\Leftarrow$ can be found at the nLab.
• countable products implies   finite products

  trivial
• discrete implies   essentially discrete and  locally small and  skeletal

  trivial
• disjoint coproducts is equivalent to   coproducts and  disjoint finite coproducts

  easy
• disjoint finite coproducts implies   finite coproducts

  by definition
• disjoint finite coproducts and  thin implies   trivial

  For every object $A$ the two inclusions $A \rightrightarrows A + A$ must be equal, so their equalizer is $A$, but also $0$ since the coproduct is disjoint. Hence $A = 0$.
• distributive implies   finite products and  finite coproducts

  by definition
• distributive implies   strict initial object

  See the nLab.
• distributive and  exact filtered colimits and  coproducts implies   infinitary distributive

  Each functor $A \times -$ preserves finite coproducts and filtered colimits, hence all coproducts.
• elementary topos is equivalent to   cartesian closed and  finitely complete and  subobject classifier

  by definition
• elementary topos implies   finitely cocomplete and  disjoint finite coproducts and  epi-regular

  Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.
• elementary topos and  locally essentially small implies   well-copowered

  This follows from Mac Lane & Moerdijk, Theorem IV.7.8 (and Prop. I.3.1).
• equalizers implies   Cauchy complete

  If $e : X \to X$ is an idempotent, then the equalizer of $e, \mathrm{id}_X : X \rightrightarrows X$ provides a splitting of $e$.
• equalizers and  countable products implies   sequential limits

  Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. Namely, the limit of $\cdots \to X_2 \to X_1 \to X_0$ is the equalizer of two suitable endomorphisms of $\prod_{n \geq 0} X_n$.
• essentially discrete is equivalent to   thin and  groupoid

  trivial
• essentially discrete implies   locally essentially small and  connected limits

  trivial
• essentially discrete and  connected implies   trivial

  trivial
• essentially finite and  finite products implies   thin

  Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.
• essentially finite implies   essentially small

  trivial
• essentially small implies   well-powered and  well-copowered and  locally essentially small

  trivial
• essentially small and  products implies   thin

  Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• essentially small and  thin and  complete implies   cocomplete

  The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• essentially small and  thin and  complete and  infinitary distributive implies   cartesian closed

  This is an application of the adjoint functor theorem. Specifically, if $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\int(t,-)$ is a left adjoint because it preserves all suprema.
• exact filtered colimits implies   filtered colimits and  finitely complete

  by definition
• filtered limits implies   sequential limits

  trivial
• finitary algebraic implies   locally finitely presentable

  Adamek-Rosicky, Cor. 3.7
• finite implies   small and  essentially finite

  trivial
• finite products is equivalent to   terminal object and  binary products

  The non-trivial direction follows since finite products can be constructed recursively via $X_1 \times \cdots \times X_{n+1} = (X_1 \times \cdots \times X_n) \times X_{n+1}$.
• finite products and  filtered limits implies   products

  The product $\prod_{i \in I} X_i$ is the filtered limit of the finite partial products $\prod_{i \in E} X_i$ where $E$ ranges over the finite subsets of $I$.
• finite products and  sequential limits implies   countable products

  If $X_1,X_2,\dotsc$ is an infinite sequence of objects, then their product is the limit of the sequence $\cdots \to X_2 \times X_1 \to X_1$.
• finitely complete is equivalent to   finite products and  equalizers

  Mac Lane, V.2, Cor. 1
• generator implies   inhabited

  trivial
• Grothendieck abelian is equivalent to   abelian and  coproducts and  generator and  exact filtered colimits

  by definition
• Grothendieck abelian implies   locally presentable

  See Deriving Auslander's formula, Cor. 5.2, or Sheafifiable homotopy model categories, Prop. 3.10.
• Grothendieck abelian implies   cogenerator

  Kashiwara-Schapira, Thm. 9.6.3
• Grothendieck abelian and  self-dual implies   trivial

  This follows since the dual of a non-trivial Grothendieck abelian category cannot be Grothendieck abelian. See Peter Freyd, Abelian categories, p. 116.
• Grothendieck topos is equivalent to   elementary topos and  coproducts and  generator and  locally essentially small

  Mac Lane & Moerdijk, Appendix, Prop. 4.4
• Grothendieck topos implies   locally presentable and  cogenerator

  For "locally presentable" see Prop. 3.4.16 in Handbook of Categorical Algebra Vol. 3. For "cogenerator" see the nLab.
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• groupoid and  equalizers implies   thin

  The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.
• groupoid and  binary products and  inhabited implies   trivial

  Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• groupoid and  initial object implies   trivial

  easy
• infinitary distributive implies   finite products and  coproducts

  by definition
• infinitary distributive implies   distributive

  trivial
• left cancellative and  coequalizers implies   thin

  If $f,g$ are two parallel morphisms, then their coequalizer is a regular epimorphism, but also a monomorphism by assumption, so it must be an isomorphism. But this means that $f = g$.
• left cancellative and  initial object implies   strict initial object

  It suffices to prove that in general any monomorphism $f : A \to 0$ into an initial object is an isomorphism. If $g : 0 \to A$ is the unique morphism, then $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $f$ is a split epimorphism and a monomorphism, hence an isomorphism.
• left cancellative implies   Cauchy complete

  Any idempotent monomorphism must be the identity and therefore splits.
• left cancellative and  right cancellative and  balanced implies   groupoid

  trivial
• locally finitely presentable implies   locally presentable

  Locally finitely presentable categories are by definition the locally $\aleph_0$-presentable categories.
• locally finitely presentable implies   exact filtered colimits

  Special case of Adamek-Rosicky, Prop. 1.59 with $\lambda = \aleph_0$.
• locally finitely presentable implies   locally ℵ₁-presentable

  trivial
• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• locally presentable and  self-dual implies   thin

  This follows from Adamek-Rosicky, Thm. 1.64.
• locally small implies   locally essentially small

  trivial
• locally ℵ₁-presentable implies   locally presentable

  trivial
• Malcev implies   finitely complete

  by definition
• mono-regular implies   balanced

  Any regular monomorphism that is an epimorphism must be an isomorphism.
• pointed is equivalent to   zero morphisms and  initial object

  easy
• pointed and  cartesian closed implies   trivial

  We have $X \cong X \times 1 \cong X \times 0 \cong 0$ for every object $X$.
• preadditive implies   locally essentially small and  zero morphisms

  trivial
• preadditive and  finite coproducts implies   finite products

  Mac Lane, VIII.2., Theorem 2
• products implies   finite products and  countable products

  trivial
• pullbacks and  terminal object implies   binary products

  If $1$ is a terminal object, then $X \times_1 Y = X \times Y$.
• right cancellative and  initial object implies   strict initial object

  Let $f : A \to 0$ be a morphism. Let $g : 0 \to A$ be the unique morphism. It is an epimorphism by assumption. Also, $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $g$ is a split monomorphism and an epimorphism, hence an isomorphism.
• small implies   locally small and  essentially small

  trivial
• split abelian implies   abelian

  by definition
• strict initial object implies   initial object

  by definition
• strict initial object and  pointed implies   trivial

  If $0$ is the zero object, then for every object $A$ the unique morphism $A \to 0$ is an isomorphism by assumption.
• subobject classifier and  locally essentially small implies   well-powered

  Mac Lane & Moerdijk, Prop. I.3.1
• subobject classifier implies   finitely complete and  mono-regular

  The first part holds by convention, and the second part: any monomorphism $U \to X$ is the equalizer of $$\chi_U,\chi_X : X \to \Omega$.
• terminal object implies   connected

  If $1$ denotes the terminal object, then for any two objects $A,B$ we have the zig-zag $A \to 1 \leftarrow B$.
• thin implies   equalizers and  left cancellative

  Any two parallel morphisms are equal, so their equalizer is the identity, and every morphism is a monomorphism as well.
• thin and  inhabited implies   generator

  Any object will be a generator for trivial reasons.
• thin and  finitely complete implies   Malcev

  In a thin category, every subobject of $X^2 = X$ containing $X$ is already $X$.
• trivial implies   finitary algebraic and  Grothendieck topos and  split abelian and  self-dual and  essentially discrete and  essentially finite

  trivial
• wide pullbacks and  terminal object implies   complete

  See the nLab.
• wide pullbacks is equivalent to   pullbacks and  filtered limits

  To prove $\Leftarrow$, a wide pullback can be constructed as a filtered limit of finite pullbacks, and finite pullbacks can be reduced to binary products (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in $\mathcal{C}$. For every object $A$ then the slice category $\mathcal{C} / A$ has wide pullbacks and a terminal object, hence is complete. Since a filtered limit can be finally reduced to such a slice, we are done.
• zero morphisms and  inhabited implies   connected

  trivial

*Deductions from these implications are automatically incorporated into each category whenever applicable. For instance, if a category is identified as complete, the property of having a terminal object is automatically inferred and added.

Moreover, implications are automatically dualized when the corresponding dual properties exist. For example, the statement that finitely complete categories with filtered limits are complete automatically implies that finitely cocomplete categories with filtered colimits are cocomplete. Similarly, if a category is self-dual and, for example, complete, it is automatically inferred to be cocomplete as well. Use this link to see all implications, including the dualizations.