Operad algebra

<h2 id="definitions">Definitions</h2>
<p>Given an operad <i>O</i> (say, a <a href="/facts/Symmetric_sequence/Ay7hEK9H">symmetric sequence</a> in a <a href="/facts/Symmetric_monoidal_%E2%88%9E-category/AxhumNny">symmetric monoidal ∞-category</a> <i>C</i>), an algebra over an operad, or <i>O</i>-algebra for short, is, roughly, a left module over <i>O</i> with multiplications parametrized by <i>O</i>.
</p><p>If <i>O</i> is a <a href="/facts/Topological_operad/RfUyH38R">topological operad</a>, then one can say an algebra over an operad is an <i>O</i>-monoid object in <i>C</i>. If <i>C</i> is symmetric monoidal, this recovers the usual definition.
</p><p>Let <i>C</i> be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If 
  
    
      
        f
        :
        O
        →
        
          O
          ′
        
      
    
    {\displaystyle f:O\to O'}
  
 is a map of operads and, moreover, if <i>f</i> is a homotopy equivalence, then the ∞-category of algebras over <i>O</i> in <i>C</i> is equivalent to the ∞-category of algebras over <i>O'</i> in <i>C</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/En-ring/C1BeC4BR">En-ring</a></li>
<li><a href="/facts/Homotopy_Lie_algebra/80FCTgoe">Homotopy Lie algebra</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Francis, John. <a href="http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf">"Derived Algebraic Geometry Over 
  
    
      
        
          
            
              E
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {E}}_{n}}
  
-Rings"</a> (PDF).</li>
<li>Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/q-alg/9702015">q-alg/9702015</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://ncatlab.org/nlab/show/operad">"operad"</a>, <i>ncatlab.org</i></li>
<li><a href="http://ncatlab.org/nlab/show/algebra+over+an+operad">http://ncatlab.org/nlab/show/algebra+over+an+operad</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Francis, Proposition 2.9. - Francis, John. "Derived Algebraic Geometry Over 
  
    
      
        
          
            
              E
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {E}}_{n}}
  
-Rings" (PDF). <a href="http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf" target="_blank">http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Operad algebra open-in-new

Operad algebra