Is there a concept of soft limit?












6












$begingroup$


Let $mathcal{C}$ and $mathcal{D}$ be categories, $mathcal{F}colonmathcal{C}tomathcal{D}$ be a functor. There is a notion of limit of $mathcal{F}$, namely a pair $(ell,varphi)$, where $ellintext{Obj}(mathcal{D})$ is an object and $varphicolonDelta_{ell}tomathcal{F}$ is a natural transformation from the constant functor to $mathcal{F}$, such that for any other pair $(d,alpha)$ of the same type (which is called a cone) there exists a unique morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$.



Obviously the concept of limit is a conjunction of the two properties:



1). for every cone $(d,alpha)$ there exists a morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$;



2). if such a morphism exists, then it is unique.



One may be interested what happens if we reject one of these requirements. For instance, if we reject the uniqueness condition, then we come to the concept of weak limit.




Question: What concept do we get if we reject the existence condition?




Of course, we may define it and call it a soft limit. We also may find some examples (empty set is a soft terminal object of $mathbf{Set}$). My question is whether it was studied anywhere or if this is a special case of another known category-theoreric concept.






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
    $endgroup$
    – Arnaud D.
    Dec 23 '18 at 17:51










  • $begingroup$
    @ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
    $endgroup$
    – Oskar
    Dec 23 '18 at 18:03
















6












$begingroup$


Let $mathcal{C}$ and $mathcal{D}$ be categories, $mathcal{F}colonmathcal{C}tomathcal{D}$ be a functor. There is a notion of limit of $mathcal{F}$, namely a pair $(ell,varphi)$, where $ellintext{Obj}(mathcal{D})$ is an object and $varphicolonDelta_{ell}tomathcal{F}$ is a natural transformation from the constant functor to $mathcal{F}$, such that for any other pair $(d,alpha)$ of the same type (which is called a cone) there exists a unique morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$.



Obviously the concept of limit is a conjunction of the two properties:



1). for every cone $(d,alpha)$ there exists a morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$;



2). if such a morphism exists, then it is unique.



One may be interested what happens if we reject one of these requirements. For instance, if we reject the uniqueness condition, then we come to the concept of weak limit.




Question: What concept do we get if we reject the existence condition?




Of course, we may define it and call it a soft limit. We also may find some examples (empty set is a soft terminal object of $mathbf{Set}$). My question is whether it was studied anywhere or if this is a special case of another known category-theoreric concept.






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
    $endgroup$
    – Arnaud D.
    Dec 23 '18 at 17:51










  • $begingroup$
    @ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
    $endgroup$
    – Oskar
    Dec 23 '18 at 18:03














6












6








6


0



$begingroup$


Let $mathcal{C}$ and $mathcal{D}$ be categories, $mathcal{F}colonmathcal{C}tomathcal{D}$ be a functor. There is a notion of limit of $mathcal{F}$, namely a pair $(ell,varphi)$, where $ellintext{Obj}(mathcal{D})$ is an object and $varphicolonDelta_{ell}tomathcal{F}$ is a natural transformation from the constant functor to $mathcal{F}$, such that for any other pair $(d,alpha)$ of the same type (which is called a cone) there exists a unique morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$.



Obviously the concept of limit is a conjunction of the two properties:



1). for every cone $(d,alpha)$ there exists a morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$;



2). if such a morphism exists, then it is unique.



One may be interested what happens if we reject one of these requirements. For instance, if we reject the uniqueness condition, then we come to the concept of weak limit.




Question: What concept do we get if we reject the existence condition?




Of course, we may define it and call it a soft limit. We also may find some examples (empty set is a soft terminal object of $mathbf{Set}$). My question is whether it was studied anywhere or if this is a special case of another known category-theoreric concept.






reference-request category-theory limits-colimits







share|cite|improve this question











$endgroup$




Let $mathcal{C}$ and $mathcal{D}$ be categories, $mathcal{F}colonmathcal{C}tomathcal{D}$ be a functor. There is a notion of limit of $mathcal{F}$, namely a pair $(ell,varphi)$, where $ellintext{Obj}(mathcal{D})$ is an object and $varphicolonDelta_{ell}tomathcal{F}$ is a natural transformation from the constant functor to $mathcal{F}$, such that for any other pair $(d,alpha)$ of the same type (which is called a cone) there exists a unique morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$.



Obviously the concept of limit is a conjunction of the two properties:



1). for every cone $(d,alpha)$ there exists a morphism $fcolon dtoell$, such that $varphicircDelta_f=alpha$;



2). if such a morphism exists, then it is unique.



One may be interested what happens if we reject one of these requirements. For instance, if we reject the uniqueness condition, then we come to the concept of weak limit.




Question: What concept do we get if we reject the existence condition?




Of course, we may define it and call it a soft limit. We also may find some examples (empty set is a soft terminal object of $mathbf{Set}$). My question is whether it was studied anywhere or if this is a special case of another known category-theoreric concept.






reference-request category-theory limits-colimits




reference-request category-theory limits-colimits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 23 '18 at 17:19







Oskar

















asked Dec 23 '18 at 16:51









OskarOskar

3,1731819




3,1731819








  • 2




    $begingroup$
    Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
    $endgroup$
    – Arnaud D.
    Dec 23 '18 at 17:51










  • $begingroup$
    @ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
    $endgroup$
    – Oskar
    Dec 23 '18 at 18:03














  • 2




    $begingroup$
    Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
    $endgroup$
    – Arnaud D.
    Dec 23 '18 at 17:51










  • $begingroup$
    @ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
    $endgroup$
    – Oskar
    Dec 23 '18 at 18:03








2




2




$begingroup$
Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
$endgroup$
– Arnaud D.
Dec 23 '18 at 17:51




$begingroup$
Your "soft terminal object" is generally called a subterminal object, because it is equivalent to a subobject of the terminal object (if there is one). So perhaps a better term would be "sublimit cone".
$endgroup$
– Arnaud D.
Dec 23 '18 at 17:51












$begingroup$
@ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
$endgroup$
– Oskar
Dec 23 '18 at 18:03




$begingroup$
@ArnaudD. Thanks! This is precisely what I was looking for, because limits may be reduced to terminal objects. If you post this comment as an answer, I accept it.
$endgroup$
– Oskar
Dec 23 '18 at 18:03










1 Answer
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$begingroup$

What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $Xtimes Xto X$ and the diagonal $Xto Xtimes X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.



More generally, a family of morphisms $alpha_i:Ato X_i$ for $iin I$ such that $alpha_icirc f=alpha_icirc g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $Ato prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.






share|cite|improve this answer









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    $begingroup$

    What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $Xtimes Xto X$ and the diagonal $Xto Xtimes X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.



    More generally, a family of morphisms $alpha_i:Ato X_i$ for $iin I$ such that $alpha_icirc f=alpha_icirc g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $Ato prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $Xtimes Xto X$ and the diagonal $Xto Xtimes X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.



      More generally, a family of morphisms $alpha_i:Ato X_i$ for $iin I$ such that $alpha_icirc f=alpha_icirc g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $Ato prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $Xtimes Xto X$ and the diagonal $Xto Xtimes X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.



        More generally, a family of morphisms $alpha_i:Ato X_i$ for $iin I$ such that $alpha_icirc f=alpha_icirc g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $Ato prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.






        share|cite|improve this answer









        $endgroup$



        What you call a "soft terminal object" is known as a subterminal object; it can be characterized as an object such that the two projections $Xtimes Xto X$ and the diagonal $Xto Xtimes X$ are isomorphisms, or equivalently as a subobject of the terminal object, if there is one.



        More generally, a family of morphisms $alpha_i:Ato X_i$ for $iin I$ such that $alpha_icirc f=alpha_icirc g$ for all $i$ implies $f=g$ is called a jointly monic family (or mono-source), and this condition is equivalent to the canonical map $Ato prod_i X_i$ being a monomorphism (if the product exists). So I guess you could call your "soft limits" sublimits, or jointly monic cones.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 23 '18 at 18:57









        Arnaud D.Arnaud D.

        16.2k52444




        16.2k52444






























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