Filtered category

<h2 id="filtered-categories">Filtered categories</h2>
<p>A <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is filtered when
</p>
<ul><li>it is not empty,</li>
<li>for every two objects 
  
    
      
        j
      
    
    {\displaystyle j}
  
 and 
  
    
      
        
          j
          ′
        
      
    
    {\displaystyle j'}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
 there exists an object 
  
    
      
        k
      
    
    {\displaystyle k}
  
 and two arrows 
  
    
      
        f
        :
        j
        →
        k
      
    
    {\displaystyle f:j\to k}
  
 and 
  
    
      
        
          f
          ′
        
        :
        
          j
          ′
        
        →
        k
      
    
    {\displaystyle f':j'\to k}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
,</li>
<li>for every two parallel arrows 
  
    
      
        u
        ,
        v
        :
        i
        →
        j
      
    
    {\displaystyle u,v:i\to j}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
, there exists an object 
  
    
      
        k
      
    
    {\displaystyle k}
  
 and an arrow 
  
    
      
        w
        :
        j
        →
        k
      
    
    {\displaystyle w:j\to k}
  
 such that 
  
    
      
        w
        u
        =
        w
        v
      
    
    {\displaystyle wu=wv}
  
.</li></ul>
<p>A filtered colimit is a <a href="/facts/Colimit/JoLugLs3">colimit</a> of a <a href="/facts/Functor/l2u3rIBE">functor</a> 
  
    
      
        F
        :
        J
        →
        C
      
    
    {\displaystyle F:J\to C}
  
 where 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is a filtered category. 
</p>
<h2 id="cofiltered-categories">Cofiltered categories</h2>
<p>A category 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is cofiltered if the <a href="/facts/Opposite_category/swDy8NrJ">opposite category</a> 
  
    
      
        
          J
          
            
              o
              p
            
          
        
      
    
    {\displaystyle J^{\mathrm {op} }}
  
 is filtered.  In detail, a category is cofiltered when
</p>
<ul><li>it is not empty,</li>
<li>for every two objects 
  
    
      
        j
      
    
    {\displaystyle j}
  
 and 
  
    
      
        
          j
          ′
        
      
    
    {\displaystyle j'}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
 there exists an object 
  
    
      
        k
      
    
    {\displaystyle k}
  
 and two arrows 
  
    
      
        f
        :
        k
        →
        j
      
    
    {\displaystyle f:k\to j}
  
 and 
  
    
      
        
          f
          ′
        
        :
        k
        →
        
          j
          ′
        
      
    
    {\displaystyle f':k\to j'}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
,</li>
<li>for every two parallel arrows 
  
    
      
        u
        ,
        v
        :
        j
        →
        i
      
    
    {\displaystyle u,v:j\to i}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
, there exists an object 
  
    
      
        k
      
    
    {\displaystyle k}
  
 and an arrow 
  
    
      
        w
        :
        k
        →
        j
      
    
    {\displaystyle w:k\to j}
  
 such that 
  
    
      
        u
        w
        =
        v
        w
      
    
    {\displaystyle uw=vw}
  
.</li></ul>
<p>A cofiltered limit is a <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a> of a <a href="/facts/Functor/l2u3rIBE">functor</a> 
  
    
      
        F
        :
        J
        →
        C
      
    
    {\displaystyle F:J\to C}
  
 where 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is a cofiltered category.
</p>
<h2 id="ind-objects-and-pro-objects">Ind-objects and pro-objects</h2>
<p>Given a <a href="/facts/Small_category/7WzxUThf">small category</a> 
  
    
      
        C
      
    
    {\displaystyle C}
  
, a <a href="/facts/Presheaf_(category_theory)/gWTNAFo3">presheaf</a> of sets 
  
    
      
        
          C
          
            o
            p
          
        
        →
        S
        e
        t
      
    
    {\displaystyle C^{op}\to Set}
  
 that is a small filtered colimit of representable presheaves, is called an ind-object of the category 
  
    
      
        C
      
    
    {\displaystyle C}
  
. Ind-objects of a category 
  
    
      
        C
      
    
    {\displaystyle C}
  
 form a full subcategory 
  
    
      
        I
        n
        d
        (
        C
        )
      
    
    {\displaystyle Ind(C)}
  
 in the category of functors (presheaves) 
  
    
      
        
          C
          
            o
            p
          
        
        →
        S
        e
        t
      
    
    {\displaystyle C^{op}\to Set}
  
. The category 
  
    
      
        P
        r
        o
        (
        C
        )
        =
        I
        n
        d
        (
        
          C
          
            o
            p
          
        
        
          )
          
            o
            p
          
        
      
    
    {\displaystyle Pro(C)=Ind(C^{op})^{op}}
  
 of pro-objects in 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is the opposite of the category of ind-objects in the opposite category 
  
    
      
        
          C
          
            o
            p
          
        
      
    
    {\displaystyle C^{op}}
  
.
</p>
<h2 id="-filtered-categories">κ-filtered categories</h2>
<p>There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a <a href="/facts/Cone_(category_theory)/ST4dKlGE">cocone</a> over any diagram in 
  
    
      
        J
      
    
    {\displaystyle J}
  
 of the form 
  
    
      
        {
         
         
        }
        →
        J
      
    
    {\displaystyle \{\ \ \}\rightarrow J}
  
, 
  
    
      
        {
        j
         
         
         
        
          j
          ′
        
        }
        →
        J
      
    
    {\displaystyle \{j\ \ \ j'\}\rightarrow J}
  
, or 
  
    
      
        {
        i
        ⇉
        j
        }
        →
        J
      
    
    {\displaystyle \{i\rightrightarrows j\}\rightarrow J}
  
. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for <i>any</i> finite diagram; in other words, a category 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is filtered (according to the above definition) if and only if there is a cocone over any <i>finite</i> diagram 
  
    
      
        d
        :
        D
        →
        J
      
    
    {\displaystyle d:D\to J}
  
.
</p><p>Extending this, given a <a href="/facts/Regular_cardinal/wrGvPz2c">regular cardinal</a> κ, a category 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is defined to be κ-filtered if there is a cocone over every diagram 
  
    
      
        d
      
    
    {\displaystyle d}
  
 in 
  
    
      
        J
      
    
    {\displaystyle J}
  
 of cardinality smaller than κ. (A small <a href="/facts/Diagram_(category_theory)/kGTlM945">diagram</a> is of cardinality κ if the morphism set of its domain is of cardinality κ.)
</p><p>A κ-filtered colimit is a colimit of a <a href="/facts/Functor/l2u3rIBE">functor</a> 
  
    
      
        F
        :
        J
        →
        C
      
    
    {\displaystyle F:J\to C}
  
 where 
  
    
      
        J
      
    
    {\displaystyle J}
  
 is a κ-filtered category.
</p>

<ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, M.</a>, <a href="/facts/Grothendieck,_A./jLQGPY7M">Grothendieck, A.</a> and <a href="/facts/Jean-Louis_Verdier/nR3cH3Lj">Verdier, J.-L.</a> <i><a href="/facts/S%C3%A9minaire_de_G%C3%A9om%C3%A9trie_Alg%C3%A9brique_du_Bois_Marie/K961wdWq">Séminaire de Géométrie Algébrique du Bois Marie</a></i> (<i>SGA 4</i>). Lecture Notes in Mathematics 269, Springer Verlag, 1972.  Exposé I, 2.7.</li>
<li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (1998), <i><a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the Working Mathematician</a></i> (2nd ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98403-2, section IX.1.</li></ul>

Filtered category open-in-new

Filtered category